Stretching of polymers around the Kolmogorov scale in a turbulent shear flow

We present numerical studies of stretching of Hookean dumbbells in a turbulent Navier-Stokes flow with a linear mean profile, <u_x>=Sy. In addition to the turbulence features beyond the viscous Kolmogorov scale \eta, the dynamics at the equilibrium extension of the dumbbells significantly below eta is well resolved. The variation of the constant shear rate S causes a change of the turbulent velocity fluctuations on all scales and thus of the intensity of local stretching rate of the advecting flow. The latter is measured by the maximum Lyapunov exponent lambda_1 which is found to increase as \lambda_1 ~ S^{3/2}, in agreement with a dimensional argument. The ensemble of up to 2 times 10^6 passively advected dumbbells is advanced by Brownian dynamics simulations in combination with a pseudospectral integration for the turbulent shear flow. Anisotropy of stretching is quantified by the statistics of the azimuthal angle $\phi$ which measures the alignment with the mean flow axis in the x-y shear plane, and the polar angle theta which determines the orientation with respect to the shear plane. The asymmetry of the probability density function (PDF) of phi increases with growing shear rate S. Furthermore, the PDF becomes increasingly peaked around mean flow direction (phi= 0). In contrast, the PDF of the polar angle theta is symmetric and less sensitive to changes of S.