## 7.10Speciﬁc dissipation rate

The model is one of a family of two-equation models for turbulence. With two equations, the models can represent each of the scales, and , which characterise . Most often, is used to represent .

The other variable must represent and so far we have used with SI units . The speciﬁc dissipation rate , with SI units of , is a popular alternative for this variable in turbulence modelling.

While Kolmogorov ﬁrst proposed a two-equation model,15 the models used in CFD originate from Wilcox.16 Here, “models” is plural since there are several versions of model with modiﬁcations and additions from its original form.

The original model is presented below (with some changes to the original variable names), assuming = constant for direct comparison with in Sec. 7.1 . (7.36) (7.37)

The standard model coeﬃcients are (7.38)
Comparing dissipation terms in Eq. (7.1 ) and Eq. (7.36) gives the relation . Substituting in Eq. (6.31 ) leads to a simple relation for turbulent viscosity, given by (7.39)
Inlet and initial estimates for can be calculated by (7.40)
using , in a manner similar to in Eq. (7.4 ).

With wall functions, the boundary condition applied to set a near-wall cell value according to (7.41)
The expression for ( ) is a solution to the following equation for the viscous sub-layer where diﬀusion and dissipation terms dominate in Eq. (7.37 ): (7.42)
The equivalent dissipation terms for in Eq. (7.37) and in Eq. (7.2 ) are and respectively. The former is more stable in a numerical solution since it is insensitive to variations in .
15Andrey Nikolaevich Kolmogorov, Equations of turbulent motion in an incompressible ﬂuid, ﬁrst published in Russian in Izv. Akad. Nauk SSSR 6, 1941.
16David Wilcox, Reassessment of the scale-determining equation for advanced turbulence models, 1988.

Notes on CFD: General Principles - 7.10 Speciﬁc dissipation rate 