Inverse energy cascade

The strong influence of polymers on the energy balance pose the intriguing question of the possible effects on the inverse energy cascade which occurs in the Newtonian case. Indeed, in absence of friction, the growth rate of kinetic energy $E(t) = 1/2 \langle \vert{\mbox{\boldmath$u$}(t)}\vert^2 \rangle$ can be obtained repeating the derivation of the energy balance (4.17) but averaging only over space and over the statistics of the random forcing ${\mbox{\boldmath$f$}}$:

$${dE \over dt} = F - \epsilon -\frac{2\eta\nu}{\tau^2} (\lan... ...dmath$\sigma$}}(t) \rangle - \mathrm{tr}{\mbox{\boldmath$1$}})$$ (4.20)

As already shown, the polymer contribution has a definite sign, acting as a dissipative term. Neglecting the viscous dissipation $\epsilon$, which in the limit of infinite Reynolds number is vanishingly small, it is clear from Eq. (4.20) that the energy growth rate in the viscoelastic case is reduced with respect to the Newtonian case where it is essentially given by the input of the random forcing.

In order to measure the energy growth rate, I performed numerical simulations of the viscoelastic model using a slightly different configuration: I turned off the friction term and put the forcing on a smaller scale, allowing the energy to give origin to an inverse cascade. After an initial growth, the polymer elongation reaches indeed a statistically steady state, and consequently the energy growth rate is reduced of a constant fraction depending on the concentration and the Weissenberg number of the polymer solution, in quantitative agreement with Eq. (4.20) (see Fig. (4.11)).

Figure 4.11: Linear growth of kinetic energy growth in absence of friction. The energy growth rate is reduced by the polymer feedback at increasing $Wi$ number. If the Weissenberg number is large enough the energy growth can be completely stopped.

We remark the striking fact that at sufficiently high $Wi$ numbers, the energy growth rate in the viscoelastic case can be reduced to zero when the polymer dissipation balances exactly the forcing input. This means that the inverse cascade can be completely suppressed by the polymer feedback even in absence of friction.

The feedback of polymers reacts on the fluid in order to reduce its velocity gradients. In the case of two-dimensional turbulence the power spectrum of velocity gradients is peaked at the forcing length-scale, thus is reasonable to assume that polymer feedback is essentially localized at the scale of forcing and does not entail the inertial range. Thus we expect to observe also for the viscoelastic two-dimensional solution the development of the inverse cascade, with a constant flux $\Pi(k) \simeq \epsilon$ reduced of a fraction depending on the elongation of the polymers:

$$\epsilon_{Viscoelastic} = \epsilon_{Newtonian} -\frac{2\eta... ...math$\sigma$}}(t) \rangle - \mathrm{tr}{\mbox{\boldmath$1$}})$$ (4.21)

Figure 4.12: Inverse energy cascade in viscoelastic simulations. Increasing the $Wi$ number the energy flux in the cascade is reduced and consequently the friction term stops the cascade at smaller scale.

A direct measurement of the energy flux requires a large scaling range in the inverse energy cascade, but at the same time it is necessary to resolve also the direct enstrophy cascade whose smooth flow is responsible of the polymer stretching. Unfortunately this task is unaffordable with actual computational resource. Nevertheless it is possible to have an indirect check of our prediction. The hypothesis of an inverse cascade with constant energy flux, leads to a Kolmogorov-like scaling law for the velocity fluctuations $u_{\ell} = \epsilon^{1/3} \ell^{1/3}$ with the reduced flux given by Eq. (4.21). The friction length scale $\ell_f$ where the friction term balance the nonlinear term responsible for the energy transfer, can be estimated by dimensional arguments (see Chapter 1) as $\ell_f \sim \epsilon^{1/2} \alpha^{-3/2}$. The reduction of the energy flux in the viscoelastic case should then reflect in a reduction of the friction length-scale. Restoring the friction term in our simulations, we checked that at increasing values of $Wi$ number the inverse energy cascade is indeed stopped by friction at smaller scale as shown in Figure (4.12).

While the mean square polymer elongation $\langle \mathrm{tr} {\mbox{\boldmath$\sigma$}}(t) \rangle$ quickly reaches statistically constant values depending on the value of $Wi$, the kinetic energy grows up with different rates until it reaches the steady state fixed by the energy balance (4.17) (see Figure (4.13))

Figure 4.13: Average kinetic energy in viscoelastic simulations of the inverse energy cascade. In the initial stage the energy grows linearly with smaller rates at larger $Wi$ numbers then it is stopped by the friction term at values determined by the energy balance (4.17)