Skip to main content
SpringerLink
Log in

The correlation dimension of an attractor determined on the base of the theory of equivalence of measures and stochastic equations for continuum

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

The physical law of the equivalence of measures between the random process and the regular process and the stochastic equations of continuum have opened the new way in stochastic theory of turbulence. An experimental method for determining the dimension of an attractor for hydrodynamic flows suggests re-conducting an enormous complex of experiments for flows for which data on the measurement of statistical moments have already been obtained. This article proposes the dependence for the calculation of the dimensions of the attractor based on statistical moments. In addition, applying this formula and the results obtained in the stochastic theory of turbulence based on the theory of the equivalence of measures, the new dependence for the dimension of the attractor as a function of initial perturbations in a hydrodynamic flow is presented. Calculated portraits of the correlation dimension of the attractor in the cross section of a circular pipe and in the cross section of the boundary layer on a flat plate are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Landau, L.D.: Toward the problem of turbulence. Dokl. Akad. Nauk SSSR 44, 339–342 (1944)

    Google Scholar 

  2. Kolmogorov, A.N.: A new metric invariant of transitive dynamic sets and automorphisms of the Lebesgue spaces. Dokl. Akad. Nauk SSSR 119(5), 861–864 (1958)

    MathSciNet  MATH  Google Scholar 

  3. Kolmogorov, A.N.: About the entropy per time unit as a metric invariant of automorphisms. Dokl. Akad. Nauk SSSR 124(4), 754–755 (1959)

    MathSciNet  MATH  Google Scholar 

  4. Kolmogorov, A.N.: Mathematical models of turbulentmotion of an incompressible viscous fluid. Usp. Mat. Nauk 59(1(355)), 5–10 (2004)

    Google Scholar 

  5. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963). https://doi.org/10.1175/1520-0469

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)

    ADS  MathSciNet  MATH  Google Scholar 

  7. Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 23, 343–344 (1971). https://doi.org/10.1007/bf01646553

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Feigenbaum, M.: The transition to aperiodic behavior in turbulent sets. Commun. Math. Phys. 77(1), 65–86 (1980)

    ADS  MATH  Google Scholar 

  9. Rabinovich, M.I.: Stochastic self-oscillations and turbulence. Usp. Fiz. Nauk 125, 123–168 (1978)

    ADS  MathSciNet  Google Scholar 

  10. Monin, A.S.: On the nature of turbulence. Usp. Fiz. Nauk 125, 97–122 (1978)

    ADS  MathSciNet  Google Scholar 

  11. Rabinovich, M.I., Sushchik, M.M.: Coherent structures in turbulent flows. In: Gaponov, A.V., Rabinovich, M.I. (eds.) Nonlinear Waves. Self-Organization, pp. 58–84. Nauka, Moscow (1983). (in Russian)

    Google Scholar 

  12. Zaslavskii, G.M.: Stochasticity of Dynamic Sets. Nauka, Moscow (1984). (in Russian)

    MATH  Google Scholar 

  13. Struminskii, V.V.: Origination of turbulence. Dokl. Akad. Nauk SSSR 307(3), 564–567 (1989)

    ADS  MathSciNet  Google Scholar 

  14. Samarskii, A.A., Mazhukin, V.I., Matus, P.P., Mikhailik, I.A.: Z/2 conservative schemes for the Korteweg–de Vries equations. Dokl. Akad. Nauk 357(4), 458–461 (1997)

    MathSciNet  Google Scholar 

  15. Klimontovich, Y.L.: Problems of the statistical theory of open sets: criteria of the relative degree of the ordering of states in the self-organization processes. Usp. Fiz. Nauk 158(1), 59–91 (1989). https://doi.org/10.1070/pu1999v042n01abeh000445

    Article  MathSciNet  Google Scholar 

  16. Sreenivasan, K.R.: Fractals and multifractals in fluid turbulence. Annu. Rev. Fluid Mech. 23, 539–600 (1991)

    ADS  MathSciNet  Google Scholar 

  17. Orzag, S.A., Kells, L.C.: Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 96(1), 159–205 (1980). https://doi.org/10.1017/s0022112080002066/

    Article  ADS  MATH  Google Scholar 

  18. Fursikov, A.V.: Moment theory for Navier–Stokes equations with a random right-hand side. Izv. Ross. Akad. Nauk 56(6), 1273–1315 (1992)

    Google Scholar 

  19. Gilmor, R.: Catastrophe Theory for Scientists and Engineers. Dover, New York (1993)

    Google Scholar 

  20. Haller, G.: Chaos Near Resonance. Springer, Berlin (1999). https://doi.org/10.1007/978-1-4612-1508-0

    Book  MATH  Google Scholar 

  21. Priymak, V.G.: Splitting dynamics of coherent structures in a transitional round-tube flow. Dokl. Phys. 58(10), 457–465 (2013)

    ADS  Google Scholar 

  22. Newton, P.K.: The fate of random initial vorticity distributions for two-dimensional Euler equations on a sphere. J. Fluid Mech. 786, 1–4 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  23. Ladyzhenskaya, O.A.: On a dynamical system generated by Navier–Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 27, 91–115 (1972)

    MathSciNet  Google Scholar 

  24. Ladyzhenskaya, O.A.: On a dynamical system generated by Navier–Stokes equations. J. Sov. Math. 3, 458–479 (1975)

    MATH  Google Scholar 

  25. Vishik, M.I., Zelik, S.V., Chepyzhov, V.V.: Regular attractors and nonautonomous perturbations of them. Mat. Sb. 204(1), 3–46 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Vishik, M.I., Zelik, S.V., Chepyzhov, V.V.: Regular attractors and nonautonomous perturbations of them. Sb. Math. 204(1), 1–42 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Vishik, M.I., Chepyzhov, V.V.: Trajectory attractors of equations of mathematical physics. Uspekhi Mat. Nauk 66:4(400), 3–102 (2011)

    MathSciNet  MATH  Google Scholar 

  28. Vishik, M.I., Chepyzhov, V.V.: Trajectory attractors of equations of mathematical physics. Russ. Math. Surv. 66:4, 637–731 (2011)

    MathSciNet  MATH  Google Scholar 

  29. Carvalho, A.N., Langa, J.A., Robinson, J.C., Su’arez, A.: Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system’. J. Differ. Equ. 236(2), 570–603 (2007)

    ADS  MathSciNet  MATH  Google Scholar 

  30. Carvalho, A.N., Langa, J.A.: An extension of the concept of gradient semigroups which is stable under perturbation. J. Differ. Equ. 246(7), 2646–2668 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  31. Dmitrenko, A.V.: Equivalence of measures and stochastic equations for turbulent flows. Dokl. Phys. 58(6), 228–235 (2013). https://doi.org/10.1134/s1028335813060098

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Dmitrenko, A.V.: Theory of Equivalent Measures and Sets with Repeating Denumerable Fractal Elements. Stochastic Thermodynamics and Turbulence. Determinacy–Randomness Correlator. Galleya-Print, Moscow (2013) (in Russian). http://search.rsl.ru/ru/catalog/record/6633402

  33. Dmitrenko, A.V.: Regular Coupling between Deterministic (Laminar) and Random (Turbulent) Motions—Equivalence of Measures. Scientific Discovery Diploma No. 458, Registration No. 583 of December 2 (2013)

  34. Dmitrenko, A.V.: Equivalent measures and stochastic equations for determination of the turbulent velocity fields and correlation moments of the second order. In: Abstracts International Conference “Turbulence and Wave Processes,” Lomonosov Moscow State University, November 26–28, 2013, pp. 39–40. Moscow (2013). http://www.dubrovinlab.msu.ru/turbulencemdm100ru

  35. Dmitrenko, A.V.: Some analytical results of the theory of equivalence measures and stochastic theory of turbulence for nonisothermal flows. Adv. Stud. Theor. Phys. 8(25), 1101–1111 (2014). https://doi.org/10.12988/astp.2014.49131

    Article  Google Scholar 

  36. Dmitrenko, A.V.: Analytical estimation of velocity and temperature fields in a circular tube on the basis of stochastic equations and equivalence of measures. J. Eng. Phys. Thermophys. 88(6), 1569–1576 (2015). https://doi.org/10.1007/s10891-015-1344-x

    Article  Google Scholar 

  37. Dmitrenko, A.V.: Determination of critical Reynolds numbers for nonisothermal flows using stochastic theories of turbulence and equivalent measures. Heat Transf. Res. 47(1), 41–48 (2016). https://doi.org/10.1615/HeatTransRes.2015014191

    Article  MathSciNet  Google Scholar 

  38. Dmitrenko, A.V.: The theory of equivalence measures and stochastic theory of turbulence for non-isothermal flow on the flat plate. Int. J. Fluid Mech. Res. 43(2), 182–187 (2016). https://doi.org/10.1615/InterJFluidMechRes.v43.i2

    Article  Google Scholar 

  39. Dmitrenko, A.V.: An estimation of turbulent vector fields, spectral and correlation functions depending on initial turbulence based on stochastic equations. The Landau fractal equation. Int. J. Fluid Mech. Res. 43(3), 82–91 (2016). https://doi.org/10.1615/InterJFluidMechRes.v43.i3

    Article  Google Scholar 

  40. Dmitrenko, Artur V.: Stochastic equations for continuum and determination of hydraulic drag coefficients for smooth flat plate and smooth round tube with taking into account intensity and scale of turbulent flow. Contin. Mech. Thermodyn. 29(1), 1–9 (2017). https://doi.org/10.1007/s00161-016-0514-1

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Dmitrenko, A.V.: Analytical determination of the heat transfer coefficient for gas, liquid and liquid metal flows in the tube based on stochastic equations and equivalence of measures for continuum. Contin. Mech. Thermodyn. 29(6), 1197–1205 (2017). https://doi.org/10.1007/s00161-017-0566-x

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Dmitrenko, A.V.: Determination of the coefficients of heat transfer and friction in supercritical-pressure nuclear reactors with account of the intensity and scale of flow turbulence on the basis of the theory of stochastic equations and equivalence of measures. J. Eng. Phys. Thermophys. 90(6), 1288–1294 (2017). https://doi.org/10.1007/s10891-017-1685-8

    Article  MathSciNet  Google Scholar 

  43. Dmitrenko, A.V.: Results of investigations of non-isothermal turbulent flows based on stochastic equations of the continuum and equivalence of measures. In: 11th International Conference “Aerophysics and Physical Mechanics of Classical and Quantum Systems”—APhM-2017. 21–24 November 2017, Moscow, Russian Federation. IOP Conf. Ser. J. Phys. Conf. Ser. 1009 (2018) 012017. https://doi.org/10.1088/1742-6596/1009/1/012017

    Google Scholar 

  44. Landau, L.D., Lifshits, E.F.: Fluid Mechanics. Perg. Press, Oxford (1959)

    Google Scholar 

  45. Constantin, P., Foais, C., Temam, R.: On dimensions of the attractors in two-dimensional turbulence. Physica D 30, 284–296 (1988)

    ADS  MathSciNet  Google Scholar 

  46. Babin, A.V., Vishik, M.I.: Attractors of evolution partial differential equations and estimates of their dimension. Russ. Math. Surv. 384, 151–213 (1983)

    MATH  Google Scholar 

  47. Babin, A.V., Vishik, M.I.: Existence of and an estimate for the dimension of attractors in quasilinear parabolic equations, and Navier-Stokes systems. Russ. Math. Surv. 37(3), 195–197 (1983)

    MATH  Google Scholar 

  48. Vishik, M.I., Komech, A.I.: Kolmogorov equations corresponding to a two-dimensional stochastic Navier–Stokes system. Tr. Mosk. Mat. Obs. 46, MSU, M. 3–43 (1983)

  49. Packard, N.H., Crutchfield, J.P., Farmer, J.D., Shaw, R.S.: Geometry from a time series. Phys. Rev. Lett. 45(9), 712–715 (1980)

    ADS  Google Scholar 

  50. Malraison, B., Berge, P., Dubois, M.: Dimension of strange attractors: an experimental determination for the chaotic regime of two convective systems. J. Phys. Lett. 44, L897–L902 (1983)

    Google Scholar 

  51. Procaccia, I., Grassberger, P.: Characterization of strange attractors. Phys. Rev. Lett. 50, 346–349 (1983)

    ADS  MathSciNet  MATH  Google Scholar 

  52. Procaccia, I., Grassberger, P.: Estimation of the Kolmogorov entropy from a chaotic signal. Phys. Rev. A 28(4), 2591–2593 (1983)

    ADS  Google Scholar 

  53. Procaccia, I., Grassberger, P., Hentschel, H.G.E.: Dynamical Systems and Chaos, pp. 212–222. Springer, Berlin (1983)

    Google Scholar 

  54. Grassberger, P., Procaccia, I.: Measuring the strangeness of strange attractors. Phys. D Nonlinear Phenom. 9(1), 189–208 (1983)

    ADS  MathSciNet  MATH  Google Scholar 

  55. Grassberger, P., Procaccia, I.: Dimensions and entropies of strange attractors from a fluctuating dynamics approach. Phys. D Nonlinear Phenom. 13(1–2), 34–54 (1984). https://doi.org/10.1016/0167-2789(84)90269-0

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Gromov, P.R., Zobin, F.B., Rabinovich, M.I., Reiman, A.M., Sushchik, M.M.: Finite dimensional attractors in shear flow with feedback. Dokl. Akad. Nauk 292(2), 284 (1987)

    ADS  MathSciNet  Google Scholar 

  57. Rabinovich, M.I., Reiman, A.M., Sushchik, M.M., et al.: Correlation dimension of the flow and spatial development of dynamic chaos in the boundary layer. JETP Lett. 13(16), 987 (1987)

    Google Scholar 

  58. Brandstater, A., Swift, J., Swinney, H.L., Wolf, A., Farmer, D.J., Jen, E., Crutchfield, P.J.: Low-dimensional chaos in hydrodynamic system. Phys. Rev. Lett. 51(16), 1442–1446 (1983)

    ADS  MathSciNet  Google Scholar 

  59. Priymak, V.G.: Splitting dynamics of coherent structures in a transitional round-pipe flow. Dokl. Phys. 58(10), 457–465 (2013)

    ADS  Google Scholar 

  60. Hinze, J.O.: Turbulence, 2nd edn. McGraw-Hill, New York (1975)

    Google Scholar 

  61. Monin, A.S., Yaglom, A.M.: Statistical Fluid Mechanics, vols. 1 and 2. MIT Press, Cambridge (1971)

  62. Schlichting, H.: Boundary-Layer Theory, 6th edn. McGraw-Hill, New York (1968)

    MATH  Google Scholar 

  63. Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000). https://doi.org/10.1017/cbo9780511840531

    Book  MATH  Google Scholar 

  64. Davidson, P.A.: Turbulence. Oxford University Press, New York (2004)

    MATH  Google Scholar 

  65. Dmitrenko, A.V.: Fundamentals of Heat and Mass Transfer and Hydrodynamics of Single-Phase and Two-Phase Media. Integral and Statistical Methods, and Direct Numerical Simulation. Galleya-Print, Moscow, Criteria (2008) (in Russian)

  66. Dmitrenko, A.V.: Calculation of pressure pulsations for a turbulent heterogeneous medium. Dokl. Phys. 52(7), 384–387 (2007). https://doi.org/10.1134/s1028335807120166

    Article  ADS  MATH  Google Scholar 

  67. Dmitrenko, A.V.: Calculation of the boundary layer of atwo-phase medium. High Temp. 40(5), 706–715 (2002). https://doi.org/10.1023/A:1020436720213

    Article  Google Scholar 

  68. Dmitrenko, A.V.: Heat and mass transfer and friction in injection to a supersonic region of the Laval nozzle. Heat Transf. Res. 31(6–8), 338–399 (2000). https://doi.org/10.1615/HeatTrasRes.v31.i6-8.30

    Article  Google Scholar 

  69. Dmitrenko, A.V.: Film cooling in nozzles with large geometric expansion using method of integral relation and second moment closure model for turbulence. In: 33th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit AIAA Paper 97-2911 (1997). https://doi.org/10.2514/6.1997-2911

  70. Dmitrenko, A.V.: Heat and mass transfer in combustion chamber using a second-moment turbulence closure including an influence coefficient of the density fluctuation in film cooling conditions. In: 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit. AIAA Paper 98-3444 (1998). https://doi.org/10.2514/6.1998-3444

  71. Dmitrenko, A.V.: Nonselfsimilarity of a boundary-layer fl ow of a high-temperature gas in a Laval nozzle. Aviats. Tekh. 1, 39–42 (1993)

    Google Scholar 

  72. Dmitrenko, A.V.: Computational investigations of a turbulent thermal boundary layer in the presence of external flow pulsations. In: Proceedings of 11th Conference on Young Scientists, Moscow Physicotechnical Institute, Part 2, Moscow (1986), pp. 48–52. Deposited at VINITI 08.08.86, No. 5698-B86

  73. Rotta, J.C.: Turbulent boundary layers with heat transfer in compressible flow. AGARD Report 281 (1960)

  74. Lauffer, J.: The structure of turbulence in fully developed pipe flow. TN Report 2954 NACA (1953)

  75. Klebanoff, P.S.: Characteristics of turbulence in boundary layer with zero pressure gradient. TN Report 1247 NACA (1954)

Download references

Acknowledgements

This work was supported by the program of increasing the competitive ability of National Research Nuclear University MEPhI (agreement with the Ministry of Education and Science of the Russian Federation of August 27, 2013, Project No. 02.a03.21.0005).

Author information

Authors and Affiliations

  1. Department of Thermal Physics, National Research Nuclear University (MEPhI), Kashirskoye shosse 31, Moscow, Russian Federation, 115409

    Artur V. Dmitrenko

  2. Department of Power Engineering, Russian University of Transport (MIIT), Obraztsova Street 9, Moscow, Russian Federation, 127994

    Artur V. Dmitrenko

Authors
  1. Artur V. Dmitrenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Artur V. Dmitrenko.

Additional information

Communicated by Andreas Öchsne.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dmitrenko, A.V. The correlation dimension of an attractor determined on the base of the theory of equivalence of measures and stochastic equations for continuum. Continuum Mech. Thermodyn. 32, 63–74 (2020). https://doi.org/10.1007/s00161-019-00784-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-019-00784-0

Keywords

Search

Navigation