The two-equation turbulence models calculate turbulent length and time scales by solving two separate transport equations.
The standard κ-ε model is part of these two-equation models and has been widely used in the engineering world concerning fluid flow since its inception by Launder and Spalding.
Its capability to handle a wide range of cases, moderate computational effort, and reasonably good accuracy across a broad range of turbulent flows have made this model quite renowned.
This model is semi-empirical, derived based on empirical considerations of physical phenomena.
The equations of the standard κ-ε model assume the flow to be fully turbulent, disregarding the effects of molecular viscosity.
Here are the transport equations for κ and ε:
(6.6)
And
(6.7)
With Gk representing the formation of turbulence kinetic energy from the mean velocity gradient, Gb is the creation of turbulence kinetic energy due to buoyancy effects, and YM represents the contribution of compressible turbulent dilatation to the overall dissipation rate. C1,2,3,ε are constants, and sigma,κ and sigma,epsilon are respectively the turbulent Prandtl numbers for κ and ε. S,κ,ε stands for the source term.
These constants are obtained experimentally and often have defaults that can vary depending on the method and software used.
Reference: