Polymer stress tensor in turbulent shear flows

The interaction of polymers with turbulent shear flows is examined. We focus on the structure of the elastic stress tensor, which is proportional to the polymer conformation tensor. We examine this object in turbulent flows of increasing complexity. First is isotropic turbulence, then anisotropic (but homogenous) shear turbulence, and finally wall bounded turbulence. The main result of this paper is that for all these flows the polymer stress tensor attains a universal structure in the limit of large Deborah number $\mathrm{De}⪢1$. We present analytic results for the suppression of the coil-stretch transition at large Deborah numbers. Above the transition the turbulent velocity fluctuations are strongly correlated with the polymer’s elongation: there appear high-quality “hydroelastic” waves in which turbulent kinetic energy turns into polymer potential energy and vice versa. These waves determine the trace of the elastic stress tensor but practically do not modify its universal structure. We demonstrate that the influence of the polymers on the balance of energy and momentum can be accurately described by an effective polymer viscosity that is proportional to the cross-stream component of the elastic stress tensor. This component is smaller than the streamwise component by a factor proportional to ${\mathrm{De}}^2$. Finally we tie our results to wall bounded turbulence and clarify some puzzling facts observed in the problem of drag reduction by polymers.

Figure 1

Comparison of the DNS data Ref. 20 for the mean profiles of $R_{xx}$ and $10\times R_{yy}$, the components of the dimensionless conformation tensor, with analytical predictions. In our notation ${\Pi }_0^{ij}\equiv {\nu }_{0p}{R}_{\max }^2{R}_{ij}∕\left({\tau }_p{R}_{\mathrm{eq}}^2\right)$. Black squares: DNS data for the streamwise diagonal component $R_{xx}$, that according to our theory, Eqs. (5.10), has to decrease as $1∕y$ with the distance to the wall. Black solid line: the function $1∕y^{+}$. Red empty circles: DNS data for the wall-normal component $10R_{yy}$, for which we predicted a linear increase with $y^{+}$ in the log-law turbulent region. Red dashed line: linear dependence, $\propto y^{+}$.