Physics of Homogeneous Isotropic Turbulence

The approach taken here is to build on the success of the Dissipation Theorem used to elucidate steady turbulent shear flows and apply it to homogeneous isotropic turbulence where the various physical mechanisms can be treated simultaneously by the development of a predictive method. For some time, it has been assumed that small eddies dissipate faster than large eddies and that energy is transferred from larger to smaller eddies to facilitate the overall process. In 1895, Osborne Reynolds [1] described in detail how velocity fluctuations generate turbulent dissipation and stress which are significantly larger than those due to local averaged velocity and pressure gradients. Here it is shown how the decay process can be explained by the turbulent-dissipation mechanism, with little or no inter-eddy transfer. In a homogeneous system, momentum transfer is not important, and a mechanical-energy balance is used instead. With no energy source, both volumetric dissipation and mechanical energy decrease with time. Apparent energy transfer among eddies of different size can be due to diffusion of spectral energy in wave-number space or due to elongation of eddies or due to eddies breaking into smaller eddies. The assumed initial energy spectrum continues to be reflected in the energy profile for a substantial time. This indicates that, in experimental systems, the total initial energy is concentrated near a wave number corresponding to the mesh spacing of the grid. The total mechanical energy is inversely proportional to time except for very early times and very late times. Experimental data of Comte-Bellot and Corrsin [2] are shown for comparison. The use of modified variables to eliminate the density and viscosity allows results to be applied to any Newtonian fluid and clarifies thinking about dimensional issues.