Turbulent cascade, bottleneck, and thermalized spectrum in hyperviscous flows

In many simulations of turbulent flows, the viscous forces nu del(2)u are replaced by a hyperviscous term -nu(p)(-del(2))(p)u to suppress the effect of viscosity at the large scales. In this work we examine the effect of hyperviscosity on decaying turbulence for values of p ranging from p = 1 (ordinary 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 hyperviscosity used. However, as the order of the hyperviscosity 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) proportional to k(alpha) with the power-law exponent a approaching the value obtained in flows at thermal equilibrium a = 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 hyperviscous Navier-Stokes goes over to the TEE system and we discuss why thermalization appears at smaller values of p.