Numerical evidence of anomalous energy dissipation in incompressible Euler flows: Towards grid-converged results for the inviscid Taylor-Green problem

Abstract

Providing evidence of finite-time singularities of the incompressible Euler equations in three space dimensions is still an unsolved problem. Likewise, the zeroth law of turbulence has not been proven to date by numerical experiments. We address this issue by high-resolution numerical simulations of the inviscid three-dimensional Taylor-Green vortex problem using a novel high-order discontinuous Galerkin discretization approach. Our main finding is that the kinetic energy evolution does not tend towards exact energy conservation for increasing spatial resolution of the numerical scheme, but instead converges to a solution with nonzero kinetic energy dissipation rate. This implies an energy dissipation anomaly in the absense of viscous dissipation according to Onsager's conjecture, and serves as an indication of finite-time singularities in incompressible inviscid flows. We demonstrate convergence to a dissipative solution for the three-dimensional inviscid Taylor-Green problem with a measured relative $L_2$-error of $0.27 \%$ for the temporal evolution of the kinetic energy and $3.52 \%$ for the kinetic energy dissipation rate.