Turbulent cascade, bottleneck and thermalized spectrum in hyperviscous flows

TL;DR

This study investigates how hyper-viscosity affects decaying turbulence, revealing that higher hyper-viscosity orders lead to a thermalized energy spectrum resembling equilibrium states, yet energy dissipation persists contrary to theoretical predictions.

Contribution

The paper provides a detailed numerical analysis of hyper-viscous turbulence, showing the transition to thermalized spectra and clarifying the conditions under which hyper-viscous flows resemble truncated Euler systems.

Findings

Higher hyper-viscosity orders induce a thermalized spectrum with a positive power-law.

Energy dissipation remains finite even at high hyper-viscosity orders.

The spectrum's thermalization correlates with the flow's behavior akin to truncated Euler equations.

Abstract

In many simulations of turbulent flows the viscous forces $\nu\nabla^2 {\bf u}$ are replaced by a hyper-viscous term $-\nu_p(-\nabla^2)^{p}{\bf u}$ in order to suppress the effect of viscosity at the large scales. In this work we examine the effect of hyper-viscosity on decaying turbulence for values of $p$ ranging from $p=1$ (regular viscosity) up to $p=100$. Our study is based on direct numerical simulations of the Taylor-Green vortex for resolutions from $512^3$ to $2048^3$. Our results demonstrate that the evolution of the total energy $E$ and the energy dissipation $\epsilon$ remain almost unaffected by the order of the hyper-viscosity used. However, as the order of the hyper-viscosity is increased ,the energy spectrum develops a more pronounced bottleneck that contaminates the inertial range. At the largest values of $p$ examined, the spectrum at the bottleneck range has a positive power-law behavior $E(k)\propto k^\alpha$ with the power-law exponent $\alpha$ approaching the value obtained in flows at thermal equilibrium $\alpha=2$. This agrees with the prediction of Frisch et al. [Phys. Rev. Lett. 101, 144501 (2008)] who suggested that at high values of $p$, the flow should behave like the truncated Euler equations (TEE). Nonetheless, despite the thermalization of the spectrum, the flow retains a finite dissipation rate up to the examined order, which disagrees with the predictions of the TEE system implying suppression of energy dissipation. We reconcile the two apparently contradictory results, predicting the value of $p$ for which the hyper-viscous Navier-Stokes goes over to the TEE system and we discuss why thermalization appears at smaller values of $p$.