Foundation for an Energy-Vorticity Turbulence Model

Abstract

Based on a more direct analogy between turbulent and molecular transport, a foundation is presented for an energy-vorticity turbulence model. Whereas traditional k-e , k-ω , and k-ζ turbulence models relate the eddy viscosity to a dissipation length scale associated with the smaller eddies having the highest strain rates per unit kinetic energy, the proposed model relates the eddy viscosity to a mean vortex wavelength associated with the larger energybearing eddies primarily responsible for turbulent transport. The hypothesized kinematiceddy-viscosity model depends on only the turbulent velocity fluctuations, just as the molecular viscosity depends on only the molecular velocity fluctuations. In contrast, the eddy-viscosity model used in traditional dissipation-based turbulence models results in a kinematic eddy viscosity that is inversely proportional to the molecular viscosity, which violates a fundamental requirement for a Boussinesq model of turbulent transport that is consistent with the definition of the specific Reynolds stress tensor. A rigorous development of the turbulent-energy-transport equation from the Navier-Stokes equations includes exact relations for the viscous dissipation and molecular transport of turbulent kinetic energy. Application of Boussinesq’s analogy between turbulent and molecular transport leads to a transport equation, which shows neither molecular nor turbulent transport of turbulent energy to be simple gradient diffusion.