Energy balance

The suppression of velocity fluctuations by polymer additives in two-dimensional turbulence can be easily explained in the context of the randomly driven viscoelastic model. Indeed, the average kinetic energy balance in the statistically stationary state reads

$$F=\epsilon +\frac{2\eta\nu}{\tau^2} (\langle \mathrm{tr} {\m... ...$}}) +\alpha \langle \vert{\mbox{\boldmath$u$}}\vert^2 \rangle$$ (4.17)

where $\epsilon=\nu\langle \vert{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$u$}}\vert^2 \rangle$ is the viscous dissipation and $F$ is the average energy input, which is flow-independent for a Gaussian, $\delta$-correlated random forcing ${\mbox{\boldmath$f$}}$. To obtain Eq. (4.17) we multiply Eq. (4.1) by ${\mbox{\boldmath$u$}}$, add to it the trace of Eq. (4.2) times $\eta \nu/\tau$, and average over space and time. Since in two dimensions kinetic energy flows towards large scales, it is mainly drained by friction, and viscous dissipation is vanishingly small in the limit of very large Reynolds numbers [71]. Neglecting $\epsilon$ and observing that in the Newtonian case ($\eta =0$) the balance (4.8) yields $F=\alpha \langle \vert{\mbox{\boldmath$u$}}\vert^2 \rangle_{N}$, we obtain

$$\langle \vert{\mbox{\boldmath$u$}}\vert^2 \rangle = \langle ... ...ldmath$\sigma$}}\rangle -\mathrm{tr} {\mbox{\boldmath$1$}})\;.$$ (4.18)

According to Eq (4.11), incompressibility of the flow ensures that $\mathrm{tr} {\mbox{\boldmath$\sigma$}} \ge \mathrm{tr} {\mbox{\boldmath$1$}}$, and we finally have $\langle \vert{\mbox{\boldmath$u$}}\vert^2 \rangle \le \langle \vert{\mbox{\boldmath$u$}}\vert^2 \rangle_{N}$, in agreement with numerical results. This simple energy balance argument can be generalized to nonlinear elastic models. As viscosity tends to zero, the average polymer elongation increases so as to compensate for the factor $\nu$ in eq. (4.18), resulting in a finite effect also in the infinite $Re$ limit. Since energy is essentially dissipated by linear friction, the depletion of $\langle \vert{\mbox{\boldmath$u$}}\vert^2 \rangle$ entails immediately the reduction of energy dissipation. The main difference between two-dimensional ``friction reduction'' and three-dimensional drag reduction resides in the lengthscales involved in the energy drain, i.e. large scales in 2D vs small scales in 3D.