Insights into the Energy Transfers in Hydrodynamic Turbulence Using Field-theoretic Tools

Turbulent flows exhibit intriguing energy transfers. In this paper, we compute the renormalized viscosities, mode-to-mode energy transfers, energy fluxes, and shell-to-shell energy transfers for the two-dimensional (2D) and three-dimensional (3D) hydrodynamic turbulence (HDT) using field-theoretic methods. We employ Craya-Herring basis that provides separate renormalized viscosities and energy transfers for its two components. In addition, Craya-Herring basis eliminates complex tensor algebra and simplifies the calculations considerably. In the $k^{-5/3}$ spectral regime of 2D HDT, the energy transfers between neighbouring (local) wavenumbers are forward, but they are backwards for distant (nonlocal) wavenumbers. The individual transfers between the distant wavenumber shells are small, but their cumulative sum is significant and it overcomes the forward local transfer to yield a constant inverse energy cascade. For 3D HDT, the mode-to-mode and shell-to-shell energy transfers reveal forward energy transfers for both local and nonlocal wavenumbers. More importantly, using scale-by-scale energy transfers we show that the cumulative nonlocal and local energy transfers are of the same order, which is contrary to the assumption of local energy transfers in turbulence. For 3D HDT, the renormalized viscosity, $\nu_2(k)$, computed by us and other authors are in general agreement, and it varies as $k^{-4/3}$. For 2D HDT, our the renormalized viscosity, $\nu_1(k)>0$, but $\nu_1(k)$ reported in literature shows significant variations including a negative $\nu_1(k)$. In this paper, we argue that the inconsistencies in $\nu_1(k)$ indicate inadequacy of renormalization group analysis that takes into account only local interactions and excludes nonlocal ones. In 2D HDT, the opposing nature of local and nonlocal energy transfers amplifies the error in $\nu_1(k)$.