Self-preservation relation to the Kolmogorov similarity hypotheses
Résumé
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 [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. and the mean turbulent kinetic energy dissipation rate (epsilon) over bar (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 [equivalent to(nu(3)/(epsilon) over bar) 1/4] and velocity scale nu(K) [equivalent to(nu(epsilon) over bar )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 ((delta u) over bar)(4) is derived from the analysis carried out in the inertial range. The expression, which involves the product of (delta u)(2) and partial derivative delta p/partial derivative x, does not require the use the volume-averaged dissipation (epsilon) over bar (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 (epsilon) over bar (r) is not, like (epsilon) over bar, a quantity issuing from the Navier-Stokes equations.