Standard K-Epsilon Model

Basic Transport Equations

The transport equations for the standard K-Epsilon model are:

(187)

(188)

Turbulent Production

The production is evaluated as:

(189)

where is the velocity divergence and is the modulus of the mean strain rate tensor:

(190)

and:

(191)

Non-linear Production

If a non-linear model constitutive relation is used, then the production due to the non-linear parts of the stress must be included. This is termed the "non-linear" production . It is expressed as:

(192)

where , and are given by Eqns. (239), (240) and (241), respectively.

Buoyancy Production

The production due to buoyancy is evaluated as:

(193)

where is the coefficient of thermal expansion, is the gravitational vector, is the temperature gradient vector and is the turbulent Prandtl number.

For constant density flows using the Boussinesq approximation, you specify . For ideal gases, the following relation is used:

(194)

The available literature is not clear as to the specification of the coefficient . By default, it is computed according to [32] as:

(195)

where is the velocity component parallel to , and is the velocity component perpendicular to .

This formulation tends to set the coefficient to zero outside natural convection boundary layers. Alternatively, may be taken as zero everywhere, or specified as follows:

(196)

Compressibility Modification

The dilatation dissipation is modeled according to Sarkar as:

(197)

where c is the speed of sound and .

Yap Correction

The Yap correction requires the computation of the wall distance. Therefore, it is only available when the standard K-Epsilon model is used together with the two-layer model.

It is defined as:

(198)

and the length scales are defined:

(199)

(200)

where is the distance to the wall. The coefficients and are defined as:

(201)

Relation for Turbulent Viscosity

The turbulent viscosity is computed as:

(202)

Turbulent Time Scale

The turbulent time scale is computed as:

(203)

Model Coefficients

(204)