Application of extended self-similarity in turbulence

From Navier-Stokes turbulence numerical simulations we show that for the extended self-similarity (ESS) method it is essential to take the third order structure function taken with the modulus and called $D_3^{*}\mathrm{}\left(r\right)\mathrm{}$, rather than the standard third order structure function $D_3\mathrm{}\left(r\right)\mathrm{}$ itself. If this is done, we find ESS towards scales larger than order $\sim 10\eta$, where $\eta$ is the Kolmogorov scale. If $D_3\mathrm{}\left(r\right)\mathrm{}$ is used, there is no ESS. We also analyze ESS within the Batchelor parametrization of the second and third order longitudinal structure function and focus on the scaling of the transversal structure function. The Re-asymptotic inertial range scaling develops only beyond a Taylor-Reynolds number ${\mathrm{Re}}_{\lambda }\gtrsim 500$.