In this chapter it has been presented a theoretical and numerical investigation of the effects of polymer additives on two-dimensional turbulence by means of Oldroyd-B viscoelastic model.
In the passive case, i.e. neglecting the polymer feedback,
I showed that for values of the 
number below the
coil-stretch transition the polymer elongations reaches a steady
probability distribution functions with power-law tail. Its slope
is related to the statistics of finite-time Lyapunov exponents of the flow,
in quantitative agreement with theoretical predictions.
Above the coil-stretch transition the statistics became unsteady.
Restoring the polymer feedback I showed that the kinetic energy of the fluid is drastically reduced in the viscoelastic case, as observed in soap film experiments. Oldroyd-B model provides a clear explanations of this phenomenon: part of the energy is converted into elastic energy of polymers, and consequently dissipated by their relaxation.
I showed that polymers cause a strong reduction of Lagrangian chaos, a phenomenon which is probably independent on the dimension of the space.
Finally I studied the effects of polymers on the inverse energy cascade, showing that for large enough elasticity the inverse cascade can be completely suppressed by the polymer feedback even in absence of friction.