Decaying turbulence

The organization into coherent vortices, starting from from a disordered background is a characteristic feature of decaying two-dimensional turbulence (see Chapter 1). This complex and interesting phenomenology is suppressed by the presence of a strong friction which halts the flux of energy toward large scales. Since in this case the energy is mainly dissipated by the linear friction, the decay of total energy trivially display an exponential behavior $E(t) \simeq E(0) e^{2 \alpha t}$ where $\alpha$ is the friction coefficient. (see Fig. 4.10).

Figure: Decay of mean velocity fluctuation $\langle u^2(t) \rangle$. While in the Newtonian case (dashed line ) the decay is exponential with a rate fixed by the friction coefficient $\langle u^2(t) \rangle = \langle u^2(0) \rangle e^{-2 \alpha t}$ the viscoelastic solution shows an oscillatory behavior with an exponential trend fixed by the polymer relaxation time. The oscillations are anti-correlated with those of the mean elastic energy (see inset). At late stage there is a sharp decoupling of the polymer and velocity fields which recovers the Newtonian behavior.

The coupling with polymer dynamics changes in a different way the decay of two-dimensional turbulence. Starting from different configuration randomly chosen from the statistically steady state above the coil-stretch transition $Wi \simeq 3.2$ we turned off the forcing on the velocity field and let the coupled system decay. In the first stage the balance is dominated by polymer contribution. An oscillatory decay of the kinetic energy is observable (see Fig. 4.10), with an exponential trend fixed by the polymer relaxation time, while the friction term in the energy balance seems to be sub-dominant. Thus the mean square elongation of polymers decay exponentially as $tr ({\mbox{\boldmath$\sigma$}}) \sim e^{2t / \tau}$, with over-imposed strong oscillations which are anti-correlated to those of the kinetic energy. In this stage there is a continuous exchange of energy between the velocity field and polymers and the decay of the two fields is strongly coupled. Since the trend of decay imposed by the polymers is steeper than the exponential decay predicted by the friction, at a certain moment the feedback term which slaves the kinetic energy decay becomes smaller than friction one. From this point there is a sharp decoupling of the dynamics of the two fields: the oscillations disappear and each field decays exponentially with his own characteristic time: $\tau$ for polymers and $1 \over \alpha$ for velocity, which in this late stage recovers the Newtonian behavior.