Non-Gaussian statistical model of turbulence

Abstract

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.