On the Energy Spectrum of Non-Newtonian Turbulence

Abstract

The goal of this paper is to propose a theoretical framework to study homogeneous and isotropic turbulence in a viscoelastic fluid, regarded as a perturbation of a Newtonian incompressible fluid, where the fluid relaxation time, or else the Weissenberg number, plays the role of small parameter. We use a Martin-Siggia-Rose framework to obtain a formal expression for the velocity correlation function of the non-Newtonian flow, and we expand this formal expression to linear order in the relaxation time. The coefficients in this expansion are correlation functions of the base Newtonian flow. We do not derive these correlations, instead we replace them by their values according to K41 theory, which could be regarded as an extreme form of renormalization. While substantial work will be necessary to validate the model against numerical and experimental data, preliminary results are encouraging.