Renormalization of shell model of turbulence

Abstract

Renormalization enables a systematic scale-by-scale analysis of multiscale systems. In this paper, we employ \textit{renormalization group} (RG) to the shell model of turbulence and show that the RG equation is satisfied by $|u_n{|}^2=K_{\mathrm{Ko}}{\epsilon }^{2/3}{k}_n^{-2/3}$ and ${\nu }_n={\nu }_{\ast }\sqrt{K_{\mathrm{Ko}}}{\epsilon }^{1/3}{k}_n^{-4/3}$, where $k_n,u_n$ are the wavenumber and velocity of shell $n$; ${\nu }_{\ast },K_{\mathrm{Ko}}$ are RG and Kolmogorov's constants; and $\epsilon$ is the energy dissipation rate. We find that ${\nu }_{\ast }\approx 0.5$ and $K_{\mathrm{Ko}}\approx 1.7$, consistent with earlier RG works on Navier-Stokes equation. We verify the theoretical predictions using numerical simulations.

arXiv commentTo appear in Phys. Rev. E

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