The k-epsilon (k-ε) turbulence model is one of the most common models used in computational fluid dynamics (CFD) to simulate mean flow characteristics for turbulent flow conditions. It is a two-equation model that gives a general description of turbulence by means of two transport equations (partial differential equations, PDEs). The original impetus for the k-epsilon model was to improve the mixing-length model, as well as to find an alternative to algebraically prescribing turbulent length scales in moderate- to high-complexity flows.
The first transported variable is the turbulence kinetic energy (k). The second transported variable is the rate of dissipation of turbulence kinetic energy (ε).
Principle Unlike earlier turbulence models, the k-ε model focuses on the mechanisms that affect the turbulence kinetic energy. The mixing length model lacks this kind of generality. The underlying assumption of this model is that the turbulent viscosity is isotropic, in other words, the ratio between Reynolds stress and mean rate of deformations is the same in all directions.
Standard k-ε turbulence model The exact k-ε equations contain many unknown and unmeasurable terms. For a much more practical approach, the standard k-ε turbulence model (Launder and Spalding, 1974) is used which is based on our best understanding of the relevant processes, thus minimizing unknowns and presenting a set of equations which can be applied to a large number of turbulent applications. For turbulence kinetic energy k
∂ ( ρ k ) ∂ t + ∂ ( ρ k u i ) ∂ x i = ∂ ∂ x j [ ( μ + μ t σ k ) ∂ k ∂ x j ] + 2 μ t E i j E i j − ρ ε {\displaystyle {\frac {\partial (\rho k)}{\partial t}}+{\frac {\partial (\rho ku_{i})}{\partial x_{i}}}={\frac {\partial }{\partial x_{j}}}\left[\left(\mu +{\frac {\mu _{t}}{\sigma _{k}}}\right){\frac {\partial k}{\partial x_{j}}}\right]+2{\mu _{t}}{E_{ij}}{E_{ij}}-\rho \varepsilon }
For dissipation ε {\displaystyle \varepsilon }
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