Viscous spectral energy coupling across scales in generalised Newtonian fluids

Abstract

We investigate the spectral energy dynamics of turbulent flows with variable viscosity using direct numerical simulation of homogeneous isotropic turbulence of generalised Newtonian fluids described by the Carreau constitutive model, covering both shear-thinning and shear-thickening regimes. The spectral evolution equations for the variable viscosity Navier-Stokes system show that the viscous term becomes nonlinear and gives rise to a convolution product in spectral space, formally analogous to that of the convective term. Unlike the constant viscosity case, where it acts as a purely local dissipation mechanism, the variable viscosity term carries both conservative (transfer) and non-conservative (dissipation) contributions entangled in the convolution product. We present novel computations of the viscous \mtm coupling $\hat{V}(\k, \kP)$, which does not satisfy a detailed conservation property analogous to that of the convective term. The viscous coupling maps reveal two distinct spectral regions: a sign-definite non-conservative region near $\kP \approx \bm{0}$, and a transfer-like dipole near $\kP \approx \k$ in shear-thickening fluids. The dipole satisfies the approximate antisymmetry $\hat{V}(\k, \kP) \approx -\hat{V}(\kP, \k)$, which is the defining signature of a conservative energy transfer. This demonstrates that energy transfer across scales, a role traditionally attributed exclusively to the convective nonlinearity, can arise from any nonlinear term in the momentum equation. The viscous energy transfer participates in the forward cascade alongside the convective transfer, eventually taking over the latter in the dissipation range. Its presence is connected to the emergence of power-law spectral decay replacing the classical exponential cutoff in shear-thickening fluids.