Turbulent energy transfer in a nonlinear isotropic fluid

In this paper we consider a hypothetical non-Newtonian fluid, in which the usual Newtonian constitutive relationship is perturbed by the presence of a small quadratic term. It is argued that this model is worth studying because (a) such non-linearities are believed to lead to anomalous turbulent effects, (b) it is a necessary preliminary to a full treatment of a more realistic model, which takes account of fluid anisotropy. The equations of motion are derived in the Fourier wavenumber-time domain and specialised to the case of isotropic turbulence. Recent developments in the theory of turbulence are used to obtain a closed equation for the energy spectrum and this is solved for the dissipation-range energy spectrum. It is concluded that the inertial transfer of turbulent energy is reduced in such a way that the spectrum contains more energy, but falls off more rapidly, than in the Newtonian case.