Intermittency in the not-so-smooth elastic turbulence

Abstract

Elastic turbulence is the chaotic fluid motion resulting from elastic instabilities due to the addition of polymers in small concentrations at very small Reynolds ($\mbox{Re}$) numbers. Our direct numerical simulations show that elastic turbulence, though a low $\mbox{Re}$ phenomenon, has more in common with classical, Newtonian turbulence than previously thought. In particular, we find power-law spectra for kinetic energy $E(k)\sim k^{-4}$ and polymeric energy $E_{\mathrm{p}}(k)\sim k^{-3/2}$, independent of the Deborah ($\mbox{De}$) number. This is further supported by calculation of scale-by-scale energy budget which shows a balance between the viscous term and the polymeric term in the momentum equation. In real space, as expected, the velocity field is smooth, i.e., the velocity difference across a length scale $r$, $\hat{I}´u\sim r$ but, crucially, with a non-trivial sub-leading contribution $r^{3/2}$ which we extract by using the second difference of velocity. The structure functions of second difference of velocity up to order $6$ show clear evidence of intermittency/multifractality. We provide additional evidence in support of this intermittent nature by calculating moments of rate of dissipation of kinetic energy averaged over a ball of radius $r$, ${\epsilon }_r$, from which we compute the multifractal spectrum.