The relation between self-preservation (SP) and the Kolmogorov similarity hypotheses (Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk SSSR 30, 301 (1941) [Proc. R. Soc. London A 434, 9 (1991)]) is investigated through the transport equations for the second- and third-order moments of the longitudinal velocity increments $\mathrm{[}\delta u(r,t)=u(x,t)-u(x+r,t)$, where $x,t$, and $r$ are the spatial point and the time and longitudinal separation between two points, respectively]. It is shown that the fluid viscosity $\nu$ and the mean turbulent kinetic energy dissipation rate $\overline{\epsilon }$ (the overbar represents an ensemble average) emerge naturally from the equations of motion as controlling parameters for the velocity increment moments when SP is assumed. Consequently, the Kolmogorov length scale $\eta$ $[\equiv {({\nu }^3/\overline{\epsilon })}^{1/4}]$ and velocity scale $v_K$ $[\equiv {\left(\nu \overline{\epsilon }\right)}^{1/4}]$ also emerge as natural scaling parameters in conformity with SP, indicating that Kolmogorov's first hypothesis is subsumed under the more general hypothesis of SP. Further, the requirement for a very large Reynolds number is also relaxed, at least for the first similarity hypothesis. This requirement however is still necessary to derive the two-thirds law (or the four-fifths law) from the analysis. These analytical results are supported by experimental data in wake, jet, and grid turbulence. An expression for the fourth-order moment of the longitudinal velocity increments $\overline{{\left(\delta u\right)}^4}$ is derived from the analysis carried out in the inertial range. The expression, which involves the product of ${\left(\delta u\right)}^2$ and $\partial \delta p/\partial x$, does not require the use the volume-averaged dissipation ${\overline{\epsilon }}_r$, introduced by Oboukhov [Oboukhov, Some specific features of atmospheric turbulence, J. Fluid Mech. 13, 77 (1962)] on a phenomenological basis and used by Kolmogorov to derive his refined similarity hypotheses [Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech. 13, 82 (1962)], suggesting that ${\overline{\epsilon }}_r$ is not, like $\overline{\epsilon }$, a quantity issuing from the Navier-Stokes equations.

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• Turbulence
• Turbulent mixing
• Wakes & jets