Energy transfer in turbulent polymer solutions

Abstract

The paper addresses a set of new equations concerning the scale-by-scale balance of turbulent fluctuations in dilute polymer solutions. The main difficulty is the energy associated with the polymers, which is not of a quadratic form in terms of the traditional descriptor of the micro-structure. A different choice is however possible, which, at least for mild stretching of the polymeric chains, directly leads to an L2 structure for the total free-energy density of the system thus allowing the extension of the classical method to polymeric fluids. On this basis, the energy budget in spectral space is discussed, providing the spectral decomposition of the energy of the system. New equations are also derived in physical space, to provide balance equations for the fluctuations in both the kinetic field and the micro-structure, thus extending, in a sense, the celebrated Kármán–Howarth and Kolmogorov equations of classical turbulence theory. The paper is limited to the context of homogeneous turbulence. However the necessary steps required to expand the treatment to wall-bounded flows of polymeric liquids are indicated in detail.