Standard k-ε Model

Launder and Spalding (1974) proposed the standard k-ε turbulence model utilizing the relationships described below.

The model has been shown to be relatively accurate for high Reynolds number flows in which the turbulence behavior is close to homogeneous, and the turbulence production is nearly balanced by dissipation. Nevertheless, its performance deteriorates when predicting boundary layers under adverse pressure gradients (Bradshaw, 1997). It also has difficulty in predicting the viscous sublayer. To resolve this issue, it is suggested that a correction be made to reproduce the law of the wall for incompressible flat-plate boundary layers (Wilcox, 2000). For low Reynolds number flows, the difference between the turbulent kinetic energy production and the dissipation rates may depart from their equilibrium value of zero, thus, ad-hoc adjustments of empirical parameters are inevitable. Furthermore, the standard k-ε turbulence model does not perform well in shear layers and jets, where the turbulent kinetic energy is not balanced with the dissipation rates (Versteeg and Malalasekera, 2007). Finally, this model is not recommended for high swirling/curvature flows, diverging passage flows as well as flows with a body force under the influence of a rotating reference frame.

Transport Equations

Turbulent Kinetic Energy k (1)
∂ ( ρ k ) ∂ t + ∂ ( ρ u j ¯ k ) ∂ x j   = ∂ ∂ x j [ ( μ + μ t σ k ) ∂ k ∂ x j ] + P k + D k
Turbulent Dissipation Rate ε (2)
∂ ( ρ ε ) ∂ t + ∂ ( ρ u j ¯ ε ) ∂ x j   = ∂ ∂ x j [ ( μ + μ t σ ε ) ∂ ε ∂ x j ] + P ε + D ε

Production Modeling

Turbulent Kinetic Energy k (3)
P k = μ t S 2
where
  • S = 2 S i j S i j is the strain rate magnitude.
  • μ t is dynamic eddy viscosity.
Turbulent Dissipation Rate ε (4)
P ε = C ε 1 ε k μ t S 2 = C ε 1 ε k P k

Dissipation Modeling

Turbulent Kinetic Energy k (5)
D k = − ρ ε
Turbulent Dissipation Rate ε (6)
D ε = − C ε 2 ρ ε 2 k

Modeling of Turbulent Viscosity μ t

(7)
μ t = C μ k 2 ε

Model Coefficients

C ε 1 =1.44, C ε 2 =1.92, C μ =0.09, σ k =1.0, σ ε =1.3.