Geometric Intermittency in Turbulence

Abstract

Equal-time scaling exponents in fully developed turbulence typically exhibit non anomalous scaling in the inverse cascade of two-dimensional (2D) turbulence and anomalous scaling in three dimensions. We demonstrate that multiscaling is not confined to longitudinal, scalar velocity increments, but also emerges in increments associated with the magnitude and orientation of the velocity vector. This decomposition uncovers a multiscaling in the 2D inverse cascade, which remains obscured when using conventional structure functions. Our results highlight a decoupling between velocity amplitude and flow geometry, offering new insight into the statistical structure of turbulent cascades as well as showing how different classes of multiscaling emerge.

Figures (7)

• Figure 1: A comparison of the mixed correlator $\langle 2u_{\rm A}u_{\rm B}\cos\theta_{\rm A}\cos\theta_{\rm B} \rangle$ and $\langle 2u_{\rm A}u_{\rm B}\rangle \langle\cos\theta_{\rm A}\cos\theta_{\rm B} \rangle$ as well as (inset) $\langle \cos \theta_{\rm A} \cos \theta_{\rm B} \rangle$ and $\langle u_{\rm A} u_{\rm B} \rangle$, normalised to 1 at $r = 0$, in three-dimensional turbulence.
• Figure 2: Log-log plots of the second-order structure functions vs $r/\eta$. Upper inset: Local slopes of $S^{\rm u}_p$ and $S^{\delta \cos \theta}_p$ for $p = 2$ (lower set of curves) and $p=6$ (upper set of curves); the pair of vertical lines denote the inertial range. Lower inset: Analogous plots as those in the upper inset but for local slopes extracted via ESS.
• Figure 3: Plots of the scaling exponent ratios $\tilde{\xi}_p$, $\tilde{\chi}_p$, $\tilde{\beta}_p$, $\tilde{\gamma}_p$ and $\tilde{\eta}_p$ (obtained via ESS) and (inset) the bare exponents $\xi_p$, $\chi_p$, $\beta_p$, $\gamma_p$ and $\eta_p$ for three-dimensional turbulence.
• Figure 4: Plots of the scaling exponent ratios $\tilde{\chi}_p$, $\tilde{\xi}_p$, $\tilde{\beta}_p$, $\tilde{\gamma}_p$ and $\tilde{\eta}_p$ (obtained via ESS) and (inset) the bare exponents $\chi_p$, $\xi_p$, $\beta_p$, $\gamma_p$ and $\eta_p$ for two-dimensional turbulence.
• Figure 5: Loglog plots of the different second-order structure functions for increments of (a) the longitudinal velocity (b) velocity amplitudes and (c) the cosine-angles. Each panel is accompanied by the scaling form (valid in the inertial range) from which factors in the anomalous part of the scaling; for (c) we do not have a dimensional form for the scaling since the cosine-angle structure functions are dimensionaless.