Homogeneous isotropic turbulence in dilute polymers: scale by scale budget

The turbulent energy cascade in dilute polymers solution is addressed here by considering a direct numerical simulation of homogeneous isotropic turbulence of a FENE-P fluid in a triply periodic box. On the basis of the DNS data, a scale by scale analysis is provided by using the proper extension to visco-elastic fluids of the Karman-Howarth equation for the velocity. For the microstructure, an equation, analogous to the Yaglom equation for scalars, is proposed for the free-energy density associated to the elastic behavior of the material. Two mechanisms of energy removal from the scale of the forcing are identified, namely the classical non-linear transfer term of the standard Navier-Stokes equations and the coupling between macroscopic velocity and microstructure. The latter, on average, drains kinetic energy to feed the dynamics of the microstructure. The cross-over scale between the two corresponding energy fluxes is identified, with the flux associated with the microstructure dominating at small separations to become sub-leading above the cross-over scale, which is the equivalent of the elastic limit scale defined by De Gennes-Tabor on the basis of phenomenological assumptions.