Elastic turbulence

The phenomenon of elastic turbulence in viscoelastic solutions has been discovered very recently [65,66]. While the drag reduction is a high Reynolds number phenomenon the elastic turbulence occurs at low $Re$ numbers. The presence of polymer changes the stability of the laminar flow, and polymers with large elasticity (i.e. large relaxation time) can be stretched even by a weak primary shear flow, producing elastic instabilities which causes irregular secondary flow. This flow stretch further the polymer molecules, and because of their back reaction becomes increasingly turbulent, until a kind of saturated dynamic state is reached.

Transition to elastic turbulence has been observed at extremely small $Re$ numbers (e.g. $Re \simeq 10^{-3}$, see [65]). Although the $Re$ numbers can be arbitrarily small, the resulting flow displays all the main features of developed turbulence, as the enhancement of mixing and the power law spectrum of velocity fluctuations.

In some sense this phenomenon acts in the opposite direction of the drag reduction: at high $Re$ numbers the polymers tend to suppress the small velocity fluctuations, reducing the turbulent drag, while at low $Re$ number they can destroy the laminar flow pumping energy to small scale motions through elastic instabilities.

Indeed, a linear stability analysis of Oldroyd-B model shows the presence of elastic instabilities in shear flows at small $Re$ numbers and large Deborah numbers $De = U \tau /L$. For concentrations of polymer larger than the critical value $\eta* = 3/7$ it can be shown that the critical $Re$ number for elastic instabilities vanishes at large enough $De$ numbers, allowing for a possible transition to elastic turbulence at arbitrarily small $Re$ number.

It seems thus that the simple Oldroyd-B model is able to reproduce, at least qualitatively, both the drag reduction and the elastic turbulence phenomena, and it constitutes an optimal tool for numerical and theoretical investigations.