General Form of Turbulence Models

The general formation of turbulence models can be written as (1)

\underset{Unsteady term}{\underset{}{\frac{\partial (ρ ϕ)}{\partial t}}} + \underset{Convection term}{\underset{}{\frac{\partial (ρ {\bar{u}}_{j} ϕ)}{\partial x_{j}}}} = \underset{Diffusion term}{\underset{}{\frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ}) \frac{\partial ϕ}{\partial x_{j}}]}} + \underset{Production term}{\underset{}{P_{ϕ}}} + \underset{Dissipation term}{\underset{}{D_{ϕ}}}

where $ϕ$ is a turbulence model variable, $μ_{t}$ is the eddy viscosity, $μ$ is the material viscosity and $σ$ is a constant.

The left-hand side of the equation describes advection, which includes two terms, the acceleration term and convective term. The right-hand side of the equation represents the summation of the diffusion, production and dissipation terms. The unsteady term represents the time dependence of turbulence model variables, while the convective term accounts for the rate of change of variables due to convection by the mean flow. The diffusion term on the right side of the equation describes the transport of turbulent variables due to the summation of material viscosity and eddy viscosity. The production term indicates the production rate of turbulent variables from the mean flow gradient, while the dissipation term represents the dissipation rate of turbulent variables due to viscous stresses.