Fluid Dynamics

, Volume 53, Issue 4, pp 485–499 | Cite as

On the Model of Generation of Vortex Structures in an Isotropic Turbulent Flow

  • K. P. Zybin
  • A. V. KopyevEmail author


It is known that turbulence is characterized by intermittence which is closely related to the development of unsteady nonisotropic intense small-scale vortex structures. In this study, small fluid particles from the inertial range of isotropic turbulence are considered. It is shown that the phenomenon of rotation intensification and stretching of the particles can be analyzed theoretically. In recent experimental and numerical studies, where this phenomenon was called “the pirouette effect”, its significance in the mechanism of the intense small-scale structures generation was discussed. In this study, a linear stochastic Lagrangian model for the effect is developed. In this model, the kinetic equation for the distribution function of the squared cosine of the angle between the vorticity and the eigenvector of the strain rate tensor of a fluid particle is derived and time history asymptotics of this quantity are analytically calculated at large and small times. The results are in good agreement with the recent experiments and numerical calculations. An analysis made in this study shows that the linear processes probably play the crucial role in certain processes in the isotropic turbulence, which is known to be a principally nonlinear phenomenon. The model developed makes it possible to analyze the statistics of the Lagrangian dynamics of small fluid particles in the inertial range which can be useful in some computational approaches to turbulence.


intermittence homogeneous and isotropic turbulence inertial range vorticity strain rate tensor Gaussian process Furutsu–Novikov formula 


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  1. 1.
    G. I. Taylor, “Statistical Theory of Turbulence,” Proc. Roy. Soc. London. A 151, 421 (1935).ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    A. N. Kolmogorov, “Local Structure of Turbulence in an Incompressible Fluid at very Large Reynolds Numbers,” Dokl. Akad. Nauk SSSR 30 (4), 299 (1941).ADSGoogle Scholar
  3. 3.
    A. N. Kolmogorov, “Energy Scattering in Locally Isotropic Turbulence,” Dokl. Akad. Nauk SSSR 32 (1), 19 (1941).ADSGoogle Scholar
  4. 4.
    U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov (Cambridge Univ. Press, Cambridge, 1995).CrossRefzbMATHGoogle Scholar
  5. 5.
    V. R. Kuznetsov and V. A. Sabel’nikov, Turbulence and Combustion [in Russian] (Nauka, Moscow, 1986).Google Scholar
  6. 6.
    J. M. Wallace, “Twenty Years of Experimental and Direct Numerical Simulation Access to the Velocity Gradient Tensor: What HaveWe Learned about Turbulence?” Phys. Fluids 21, 021301 (2009).ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    M. Kholmyansky and A. Tsinober, “On an Alternative Explanation of Anomalous Scaling and How Well-Defined is the Concept of Inertial Range,” Phys. Lett. A 373, 2364 (2009).ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    K. P. Zybin, V. A. Sirota, A. S. Il’iyn, and A. V. Gurevich, “Small-Scale Structure generation in Developed Turbulence,” Zh. Eksp. Teor. Fiz. 132 (2(8)), 510 (2007).Google Scholar
  9. 9.
    K. P. Zybin and V. A. Sirota, “Model of Sretching Vortex Filaments and Foundations of the Statistical Theory of Turbulence,” Phys. Usp. 58, 556 (2015).ADSCrossRefGoogle Scholar
  10. 10.
    V. F. Kopiev and S. A. Chernyshev, “Refraction Effect in Correlation Model of Quadrupole Noise Sources in Turbulent Jets,” AIAA Paper No. 3130 (2015).CrossRefGoogle Scholar
  11. 11.
    N. P. Mikhailova, E. U. Repik, and Yu. P. Sosedko, “Reynolds Number Effect on the Grid Turbulence Degeneration Law,” Fluid Dynamics 40 (5), 714 (2005).ADSCrossRefGoogle Scholar
  12. 12.
    R. Gomes-Fernandes, B. Ganapathisubramani, and J. C. Vassilicos, “Evolution of the Velocity-Gradient Tensor in a Spatially Developing Turbulent Flow,” J. Fluid Mech. 756, 252 (2014).ADSCrossRefGoogle Scholar
  13. 13.
    J.-P. Minier, S. Chibarro, and S. B. Pope, “Guidelines for the Formation of Lagrangian StochasticModels for Particle Simulations of Single-Phase and Dispersed Two-Phase Turbulent Flows,” Phys. Fluids 26, 113303 (2014).ADSCrossRefGoogle Scholar
  14. 14.
    N. A. Lebedeva and A. N. Osiptsov, “A Combined Lagrangian Method for Simulation of Axisymmetric Gas-Particle Vortex Flows,” Fluid Dynamics 51 (5), 647 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. B. Lebedev, A. N. Sekundov, and K. Ya. Yakubovskii, “Possible Mechanism of Self-Oscillations in a CombustorWorking on a PremixedMethane/Air Mixture,” Fluid Dynamics 52 (3), 388 (2017).MathSciNetCrossRefGoogle Scholar
  16. 16.
    G. A. Voth, K. Satyanarayan, and E. Bodenschatz, “Lagrangian Acceleration Measurements at Large Reynolds Numbers,” Phys. Fluids 10, 2268 (1998).ADSCrossRefGoogle Scholar
  17. 17.
    H. Xu, A. Pumir, and E. Bodenschatz, “The Pirouette Effect in Turbulent Flows,” Nat. Phys. 7, 709 (2011).CrossRefGoogle Scholar
  18. 18.
    A. Pumir, E. Bodenschatz, and H. Xu, “Tetrahedron Deformation and Alignment of Perceived Vorticity and Strain in a Turbulent Flow,” Phys. Fluids 25, 035101 (2013).ADSCrossRefGoogle Scholar
  19. 19.
    M. Chertkov, A. Pumir, and B. I. Shraiman, “Lagrangian Tetrad Dynamics and the Phenomenology of Turbulence,” Phys. Fluids 11, 2394 (1999).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    L. G. Loitsyanskii, Mechanics of Liquids and Gases (Pergamon Press, Oxford, 1966).zbMATHGoogle Scholar
  21. 21.
    L. Chevillard and C. Meneveau, “Lagrangian Time Correlations of Vorticity Alignments in Isotropic Turbulence Observations and Model Predictions,” Phys. Fluids 23, 101704 (2011).ADSCrossRefGoogle Scholar
  22. 22.
    V. I. Klyatskin, Dynamics of Stochastic Systems (Elsevier, Amsterdam, 2005).zbMATHGoogle Scholar
  23. 23.
    P. E. Hamlington, J. Schumacher, and W. J. A. Dahm, “Local and Nonlocal Strain Rate Fields and Vorticity Alignment in Turbulent Flows,” Phys. Rev. E 77, 026303 (2008).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    P.E. Hamlington, J. Schumacher, and W. J.A. Dahm, “Direct Assessment of Vorticity Alignment with Local and Nonlocal Strain Rate Fields in Turbulent Flows,” Phys. Fluids 20, 111703 (2008).ADSCrossRefzbMATHGoogle Scholar
  25. 25.
    A. N. Shiryaev, Probability-1 (Springer-Verlag New York, 2016).CrossRefzbMATHGoogle Scholar
  26. 26.
    V. S. Vladimirov, Equations of Mathematical Physics [in Russian] (Nauka, Moscow, 1981).Google Scholar
  27. 27.
    I. M. Gel’fand and G. E. Shilov, Generalized Functions. Volume I: Properties and Operations (AMS Chelsea Publishing, 1964)).zbMATHGoogle Scholar
  28. 28.
    B. J. Cantwell, “Exact Solution of a Restricted Euler Equation for the VelocityGradient Tensor,” Phys. Fluids A 4, 782 (1992).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    A. V. Kopyev, “Degeneracy of Velocity Strain-Rate Tensor Statistics in Random Isotropic Incompressible Flows,” Phys. Rev. Fluids 3, 024603 (2018).ADSCrossRefGoogle Scholar
  30. 30.
    L. Shtilman, M. Spector, and A. Tsinober, “On Some Kinematic Versus Dynamic Properties of Homogeneous Turbulence,” J. Fluid Mech. 247, 65 (1993).ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    M. L. Mehta, Random Matrices (Acad. Press, New York, 2004).zbMATHGoogle Scholar
  32. 32.
    G. A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Researchers (McGraw-Hill, New York, 1961).zbMATHGoogle Scholar

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© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  1. 1.P.N. Lebedev Physical Institute of the Russian Academy of SciencesMoscowRussia
  2. 2.National Research University Higher School of EconomicsMoscowRussia

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