# Universality of small scale statistics in various turbulent flows

The talk focuses on the universality of small scale statistics in different laboratory flows, which are either locally isotropic or anisotropic. Four issues will be addressed, summarized as follows.

1) Second-order moments transport equations, derived from the Navier-Stokes equations written at two points in space, reflect the way energy is transferred across the scales and account for the large-scale effects such as the decay, production, turbulent and pressure diffusion. These equations are also called scale-by-scale (s-b-s) energy budget equations, because the second-order moments represent the energy contained in eddies of scales smaller or equal to r. The exact mathematical form of these equations depend on the flow, and each region of the same flow.

Therefore, they allow finite Reynolds number effects to be appraised correctly, at least at the level of second-order moments.

The equations comply with the one-point energy budget equations for very large scales. More importantly, at very small scales, they reduce to the transport equations for the mean energy dissipation rate.

2) We further show that the collapse of the turbulent dissipative range on Kolmogorov scales does not require either of the two major assumptions in Kolmogorov's (1941) first similarity hypothesis, i.e. the Taylor microscale Reynolds number is very large and local isotropy is satisfied. In particular, the Kolmogorov velocity and length scales are shown to be the appropriate normalization scales.

3) Whereas the first two points concern both locally isotropic and anisotropic turbulence, for the sake of simplicity, we shall focus on the locally isotropic form of the transport equation for the mean energy dissipation rate εiso. In each flow, the equation can be expressed in the form S + 2G/Rλ =C/Rλ, where S is the velocity derivative skewness, G is the non-dimensional rate of destruction of εiso , C is a flow dependent constant and Rλ is the Taylor microscale Reynolds number.

It is shown that G/Rλ is independent of the Reynolds number, as a direct consequence of point 2). Experimental and numerical collected results indicate that G/Rλ is indeed very nearly constant for Rλ >70. Therefore, S should approach a universal constant when Rλ is sufficiently large, but the way this constant is approached is flow dependent. For all the flows considered, the approach is reasonably well supported by experimental and numerical data. The constancy of S at large Rλ violates the modified similarity hypothesis introduced in Kolmogorov (1962), but is consistent with the original similarity hypotheses of Kolmogorov (1941).