Intermittency in turbulence

Definition

Let $\mathbf{u}(x, t)$ be a statistically homogeneous and isotropic turbulent velocity field in $\R^3$, and let $\mathbf{e}$ be a unit vector. For $r > 0$, the longitudinal velocity increment is $\delta u(r) = (\mathbf{u}(x + r\mathbf{e}, t) - \mathbf{u}(x, t)) \cdot \mathbf{e}$. Its structure function of order $p$ is $S_p(r) = \langle \delta u(r)^p \rangle$, where $\langle \cdot \rangle$ is the ensemble average. Its flatness is $F(r) = S_4(r) / S_2(r)^2$, the kurtosis of $\delta u(r)$. The flow is intermittent at small scales if the law of the normalised increment $\delta u(r) / S_2(r)^{1/2}$ changes with $r$ in the inertial range. Its probability density function then has a flatness $F(r)$ that grows as $r$ decreases. When $S_p(r)$ behaves like $r^{\zeta_p}$ in the inertial range, this means that $p \mapsto \zeta_p$ is not a linear function.