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 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
numbers (e.g.
, see [65]).
Although the
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 numbers the polymers
tend to suppress the small velocity fluctuations,
reducing the turbulent drag, while at low
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 numbers
and large Deborah numbers
.
For concentrations of polymer larger than the critical value
it can be shown that the critical
number
for elastic instabilities vanishes at large enough
numbers,
allowing for a possible transition to elastic turbulence at
arbitrarily small
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.