Use of the 4 ∕ 5 Kolmogorov equation for describing some characteristics of fully developed turbulence

• Published in: Physics of fluids (1994) Vol. 17; no. 3; pp. 035110.1 - 035110.12
• Main Author: Tatarskii, V. I.
• Format: Journal Article
• Language: English
• Published: Melville, NY American Institute of Physics 01.03.2005
• Subjects: Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Physics; Turbulence simulation and modeling; Turbulent flows, convection, and heat transfer; Probability density function; Structure functions; Modelling; Kolmogorov equation; Turbulence structure; Selfsimilarity; Intermittency
• ISSN: 1070-6631; 1089-7666
• DOI: 10.1063/1.1858531

The Kolmogorov equation, which relates the second-order structure function D l l ( r ) and the third-ordrer structure function D l l l ( r ) , may be presented as a closed linear integrodifferential equation for the probability distribution function W ( U , r ) of longitudinal velocity difference U . In general, without any restrictions, the function of two variables W ( U , r ) may be presented as [ D l l ( r ) ] − 1 ∕ 2 F ( U ∕ D l l ( r ) , r ) . As a first approximation, we neglect dependence of F on the second argument r . In this approximation (self-similarity of the probability density function), the integrodifferential equation for W reduces to the ordinary nonlinear differential equation for D l l ( r ) . It follows from this equation that for r → ∞ the function D l l ( r ) ∼ r 2 ∕ 3 . This consideration does not use the 1941 Kolmogorov hypothesis that is based on dimension analysis. Any deviations from the 2 ∕ 3 law, including intermittency effects, must be related to the violation of the above-mentioned self-similarity of W , i.e., with the additional dependence of F on the second argument r . On the basis of experimental data, we suggest a simple model of W , which allows us to describe deviations from the 2 ∕ 3 law, caused by intermittency, and describe the local exponents κ in the structure functions ⟨ ∣ U ( r ) ∣ ρ ⟩ ∼ r κ ( ρ ) for moderate ρ . The so called “bottleneck effect” also can be described by 4 ∕ 5 Kolmogorov equation.