An analytical representation of a turbulence spectrum in the inertial interval is given based on stochastic equations for the continual laws of continuous medium and the laws of the equivalence of measures between random and deterministic motions in the theory of stochastic hydrodynamics. The analytical solution of these equations is presented in the form of spectral function E(k) ~ kn corresponding to the classical dependence E(k) ~ k–5/3 obtained earlier by A. N. Kolmogorov in the statistical theory on the basis of dimensional considerations. The presented solution confirms the possibility of determining partial solutions for the spectral function depending on the wave number on the base of single implications of the theory of stochastic hydrodynamics within the framework of which the solutions in the fi eld of wave numbers of turbulence generation were obtained.
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References
L. D. Landau, Toward the problem of turbulence, Dokl. Akad. Nauk SSSR, 44, No. 8, 339–342 (1944). 126
A. N. Kolmogorov, A new metric invariant of transitive dynamic systems and automorphisms of the Lebesgue spaces, Dokl. Akad. Nauk SSSR, 119, No. 5, 861–864 (1958).
A. N. Kolmogorov, About the entropy per time unit as a metric invariant of automorphisms, Dokl. Akad. Nauk SSSR, 124, No. 4, 754–755 (1959).
A. N. Kolmogorov, Mathematical models of turbulent motion of an incompressible viscous fluid, Usp. Mat. Nauk, 59, Issue 1 (355), 5–10 (2004).
E. N. Lorenz, Deterministic nonperiodic fl ow, J. Atmos. Sci., 20, 130–141 (1963); DOI:https://doi.org/10.1175/1520-0469.
D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys., 20, 167–192 (1971); http://dx.doi. org/10.1007/bf01646553.
M. Feigenbaum, The transition to aperiodic behavior in turbulent systems, Commun. Math. Phys., 77, No. 1, 65–86 (1980).
M. I. Rabinovich, Stochastic self-oscillations and turbulence, Usp. Fiz. Nauk, 125, No. 1, 123–168 (1978).
A. S. Monin, On the nature of turbulence, Usp. Fiz. Nauk, 125, No. 1, 97–122 (1978).
M. I. Rabinovich and M. M. Sushchik, Coherent structures in turbulent fl ows, in: A. V. Gaponov and M. I. Rabinovich (Eds.), Nonlinear Waves. Self-Organization [in Russian], Nauka, Moscow (1983), pp. 58–84.
G. M. Zaslavskii, Stochasticity of Dynamic Systems [in Russian], Nauka, Moscow (1984).
V. V. Struminskii, Origination of turbulence, Dokl. Akad. Nauk SSSR, 307, No. 3, 564–567 (1989).
A. A. Samarskii, V. I. Mazhukin, P. P. Matus, and I. A. Mikhailik, Z/2-conservative schemes for the Korteweg–de Vries equation, Dokl. Akad. Nauk, 357, No. 4, 458–461 (1997).
Yu. L. Klimontovich, Problems of statistical theory of open systems: criteria of the relative degree of the ordiness of states in the processes of self-organization, Usp. Fiz. Nauk, 158, Issue 1, 59–91 (1989).
K. R. Sreenivasan, Fractals and multifractals in fluid turbulence, Ann. Rev. Fluid Mech., 23, 539–600 (1991).
S. A. Orzag and L. C. Kells, Transition to turbulence in plane Poiseuille and plane Couette flow, J. Fluid Mech., 96, No. 1, 159–205 (1980).
V. G. Priymak, Splitting dynamics of coherent structures in a transitional round-pipe fl ow, Dokl. Phys., 58, No. 10, 457–465 (2013).
A. V. Fursikov, Moment theory for the Navier–Stokes equations with a random right part, Izv. Ross. Akad. Nauk, Ser. Mat., 56, No. 6, 1273–1315 (1992).
A. V. Dmitrenko, Equivalence of measures and stochastic equations for turbulent fl ows, Dokl. Akad. Nauk, 450, No. 6, 651–658 (2013).
A. V. Dmitrenko, Equivalent measures and stochastic equations for determination of the turbulent velocity fields and correlation moments of the second order, in: Proc. Int. Conf. "Turbulence and Wave Processes," November 26–28, 2013, Lomonosov Moscow State University, Moscow (2013), pp. 39–40.
A. V. Dmitrenko, Theory of Equivalent Measures and Sets with Repeated Denumerable Fractal Elements. Stochastic Thermodynamics and Turbulence. Determinacy–Randomness Correlator [in Russian], Galleya-Print, Moscow (2013).
A. V. Dmitrenko, Regular Coupling between (Laminar) and Random (Turbulent) Motions — Equivalence of Measures, Scientific Discovery Diploma No. 458, Reg. No. 583 of 02.12.2013.
A. V. Dmitrenko, Some analytical results of the theory of equivalence measures and stochastic theory of turbulence for non-isothermal flows, Adv. Stud. Theor. Phys., 8, No. 25, 1101–1111 (2014).
A. V. Dmitrenko, 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, No. 6, 1569–1576 (2015).
A. V. Dmitrenko, Determination of critical Reynolds numbers for nonisothermal flows using stochastic theories of turbulence and equivalent measures, Heat Transf. Res., 47, Issue 1, 41–48 (2016).
A. V. Dmitrenko, The theory of equivalence measures and stochastic theory of turbulence for non-isothermal flow on the flat plate, Int. J. Fluid Mech. Res., 43, Issue 2, 182–187 (2016).
A. V. Dmitrenko, Turbulent velocity field and the correlation moments of the second order determined by stochastic equations. Fractal equation of Landau, Int. J. Fluid Mech. Res., 43, Issue 3, 271–280 (2016).
A. V. Dmitrenko, 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, No. 1, 1–9 (2017).
A. V. Dmitrenko, 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, No. 6, 1197–1205 (2017). 127
A. V. Dmitrenko, 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, No. 6, 1288–1294 (2017).
A. V. Dmitrenko, Estimation of the critical Rayleigh number as a function of an initial turbulence in the boundary layer of the vertical heated plate, Heat Transf. Res., 48(13), No. 12, 1102–1112 (2017).
A. V. Dmitrenko, Results of investigations of nonisothermal turbulent fl ows based on stochastic equations of the continuum and equivalence of measures, J. Phys.: Conf. Ser.; https://doi.org/10.1088/1742-6596/1009/1/012017.
A. V. Dmitrenko, The stochastic theory of turbulence, IOP Conf. Ser.: Mater. Sci. Eng.; https://doi.org/10.1088/1757- 899X/468/1/01202.
A. V. Dmitrenko, Determination of the correlation dimension of an attractor in a pipe based on the theory of stochastic equations and equivalence of measures, J. Phys.: Conf. Series; DOI:https://doi.org/10.1088/1742-6596/1291/1/012001.
A. V. Dmitrenko, The correlation dimension of an attractor determined on the base of the theory of equivalence of measures and stochastic equations for continuum, Contin. Mech. Thermodyn.; doi.org/10.1007/s00161-019-00784-0.
A. V. Dmitrenko, Calculated portrait of thе correlation dimension of an attractor in the boundary layer of Earth′s atmosphere, J. Phys.: Conf. Ser.; DOI:https://doi.org/10.1088/1742-6596/1332/1/012004.
A. V. Dmitrenko, The uncertainty relation in the turbulent continuous medium, Contin. Mech. Thermodyn.; doi. org/10.1007/s00161-019-00792-0.
A. V. Dmitrenko, Stochastic Hydrodynamics and Heat Transfer Turbulence and Dimensionality of the Attractor. Theory of Equivalent Measure and Sets with Repeated, Countable Fractal Elements. Stochastic Thermodynamics and Turbulence. Correlator Determinacy Randomness [in Russian], Part 2, Galleya-Print, Moscow (2018).
A. V. Dmitrenko, Calculation of pulsations of pressure of heterogeneous turbulent fl ows, Dokl. Akad. Nauk, 415, No. 1, 44–47 (2007).
A. V. Dmitrenko, Calculation of the boundary layer of a two-phase medium, High Temp.,40, Issue 5, 706–715 (2002).
A. V. Dmitrenko, Principles of Heat and Mass Transfer and Hydrodynamics of Single Phase and Two-Phase Media. Criterial, Integral, Statistical, and Direct Numerical Methods of Simulation [in Russian], Galleya-Print, Moscow (2008).
P. A. Davidson, Turbulence, Oxford Univ. Press (2004).
H. Schlichting, Boundary-Layer Theory, 6th edn., McGraw-Hill, New York (1968).
A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics, M.I.T. Press (1971).
J. O. Hinze, Turbulence, McGraw-Hill, New York (1975).
S. B. Pope, Turbulent Flows, Cambridge Univ. Press, Cambridge (2000).
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Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 93, No. 1, pp. 128–133, January–February, 2020.
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Dmitrenko, A.V. Formation of a Turbulence Spectrum in the Inertial Interval on the Basis of the Theory of Stochastic Equations and Equivalence of Measures. J Eng Phys Thermophy 93, 122–127 (2020). https://doi.org/10.1007/s10891-020-02098-4
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DOI: https://doi.org/10.1007/s10891-020-02098-4