Intermittency in the not-so-smooth elastic turbulence
Rahul K. Singh, Prasad Perlekar, Dhrubaditya Mitra, Marco Edoardo Rosti
Abstract
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 ( $${{{{{{{\rm{Re}}}}}}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Re</mml:mi> </mml:math> ) numbers. Our direct numerical simulations show that elastic turbulence, though a low $${{{{{{{\rm{Re}}}}}}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Re</mml:mi> </mml:math> phenomenon, has more in common with classical, Newtonian turbulence than previously thought. In particular, we find power-law spectra for kinetic energy E ( k ) ~ k −4 and polymeric energy E p ( k ) ~ k −3/2 , independent of the Deborah (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 , δ u ~ 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 , ε r , from which we compute the multifractal spectrum.