Turbulence without Walls: Whither the Zeroth Law of Turbulence?

Experimental and numerical studies of incompressible turbulence suggest that the mean dissipation rate of kinetic energy remains constant as the Reynolds number tends to infinity (or the nondimensional viscosity tends to zero). This anomalous behavior is central to many theories of high-Reynolds-number turbulence and has been termed the "zeroth law" for this reason. Here, we report a sequence of direct numerical simulations of incompressible Navier-Stokes in a box with periodic boundary conditions, which indicate the likelihood that the anomaly vanishes at a rate that agrees with the scaling of third moment of absolute velocity increments. Our results suggest that turbulence without solid boundaries or walls may not develop strong enough singularities to sustain the strict version of the zeroth law.