Scaling hypothesis leading to generalized extended self-similarity in turbulence

A scaling hypothesis leading to generalized extended self-similarity (GESS) for velocity structure functions, valid for intermediate scales in isotropic, homogeneous turbulence, is proposed. By introducing an effective scale r̂, monotonically depending on the physical scale r, with the use of the large deviation theory, the asymptotic forms of the probability densities for the velocity differences u_r and for the coarse-grained energy-dissipation rate fluctuations ε_r, compatible with this GESS, are proposed. The probability density for ε_r is shown to have the form P _r(ε)∼ε^-1(r̂/L) ^{sr̂(zr̂(ε))} with zr̂(ε)=ln(ε/ ε_L)/ln(L/r̂), where L and ε_L are the stirring scale and the coarse-grained energy-dissipation rate over the scale L. The concave function Sr̂(z), the spectrum, plays the central role of the present approach. Comparing the results with numerical and experimental data, we explicitly obtain the fluctuation spectra Sr̂(z).