Non-Gaussian statistical model of turbulence

We study probability density function (PDF) P(x|r) in turbulence, x ≡ |Δu _r|/ <Δu _r ² > ^1/2, where Δu _r is the longitudinal velocity increment across a distance r, ⟨⟩ means a statistical average. DNS and experimental data of P(x|r) published recently support the non-Gaussian PDF model proposed in earlier papers (Qian 1998, 2000). The non-Gaussian PDF model implies the quasi-closure of turbulence statistics: higher-order statistical moments (for example, structure functions) can be derived from few lower-order moments. It is shown that experimental data of structure functions confirm the quasi-closure. By Kolmogorov's 4/5 law, the (approximate) scaling range η ≪ r ≪ L at the experimental Reynolds number is not the same as Kolmogorov's inertial range, here η is Kolmogorov's scale, and L is the large scale. Hence it is expected that the scaling exponents ξ _p of pth order structure function observed in experiments, may deviate from the real inertial-range scaling exponents ζ _p. By applying the non-Gaussian PDF model, we confirm this expectation, and obtain ξ ₂ > ζ ₂ and ξ _p < ζ _p if p > 3.

Acknowledgments

This author is grateful to P. Tabeling and T. Gotoh for making their experimental and DNS data available to us. The work was supported by the National Natural Science Foundation of China.