Self-preservation relation to the Kolmogorov similarity hypotheses

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 [δu(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 ν and the mean turbulent kinetic energy dissipation rate ⋶ (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 η [≡(ν³/⋶)^1/4] and velocity scale v_K [≡(ν⋶)^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 (δu)⁴ is derived from the analysis carried out in the inertial range. The expression, which involves the product of (δu)² and ∂δp/∂x, does not require the use the volume-averaged dissipation ⋶_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 ⋶_r is not, like ⋶, a quantity issuing from the Navier-Stokes equations.