Extreme velocity gradients in turbulent flows

Abstract

Fully turbulent flows are characterized by intermittent formation of very localized and intense velocity gradients. These gradients can be orders of magnitude larger than their typical value and lead to many unique properties of turbulence. Using direct numerical simulations of the Navier–Stokes equations with unprecedented small-scale resolution, we characterize such extreme events over a significant range of turbulence intensities, parameterized by the Taylor-scale Reynolds number ( ${R}_{\lambda }$ ). Remarkably, we find the strongest velocity gradients to empirically scale as ${\tau }_{{\rm{K}}}^{-1}{R}_{\lambda }^{\beta }$ , with $\beta \approx 0.775\pm 0.025$ , where ${\tau }_{{\rm{K}}}$ is the Kolmogorov time scale (with its inverse, ${\tau }_{{\rm{K}}}^{-1}$ , being the rms of velocity gradient fluctuations). Additionally, we observe velocity increments across very small distances $r\leqslant \eta $ , where η is the Kolmogorov length scale, to be as large as the rms of the velocity fluctuations. Both observations suggest that the smallest length scale in the flow behaves as $\eta {R}_{\lambda }^{-\alpha }$ , with $\alpha =\beta -\frac{1}{2}$ , which is at odds with predictions from existing phenomenological theories. We find that extreme gradients are arranged in vortex tubes, such that strain conditioned on vorticity grows on average slower than vorticity, approximately as a power law with an exponent $\gamma \lt 1$ , which weakly increases with ${R}_{\lambda }$ . Using scaling arguments, we get $\beta ={(2-\gamma )}^{-1}$ , which suggests that β would also slowly increase with ${R}_{\lambda }$ . We conjecture that approaching the mathematical limit of infinite ${R}_{\lambda }$ , strain and vorticity would scale similarly resulting in $\gamma =1$ and hence extreme events occurring at a scale $\eta {R}_{\lambda }^{-1/2}$ corresponding to $\beta =1$ .

Keywords

• turbulence
• extreme events
• intermittency
• direct numerical simulations