## 7.11Enhancements to the k-omega model

A comparison of the and models shows the dissipation term is in Eq. (7.37 ) for and in Eq. (7.2 ) for . While the term behaves well, the term is troublesome at a wall as , so requires damping in the low- formulation in Sec. 7.9 to resolve the viscous sub-layer.

Its better dissipation term gives the model an advantage over in the near-wall region. The disadvantage the original model is its sensitivity to the freestream values of , which is not present in in the model. Neither model, in their original form, performs well under adverse pressure gradients.

However, since its initial publication, many enhancements have been made to the original model, in particular to address the problems mentioned above.

### Cross-diﬀusion

The dependency on freestream values of is addressed by the inclusion of a cross-diﬀusion term in the equation.17 The term is derived when the equation, e.g. Eq. (7.2 ), is expressed in terms of by substituting . Its form is due to the expansion of in the diﬀusion term by Eq. (2.74b ).

The cross-diﬀusion term makes the equation more equivalent to , and thus independent of freestream values.

### Stress limiter

The original and models are known to delay or suppress ﬂow separation under adverse pressure gradients (described in Sec. 6.5 ). Under such conditions the ratio of the production to dissipation of can be signiﬁcantly higher than unity. The calculated from Eq. (7.39 ) is excessively high, causing an over-prediction of shear stress .

The problem is alleviated by limiting the shear stress, based on the assumption it is proportional to in the boundary layer, i.e. where is a constant. A stress limiter is implemented through a modiﬁcation to the calculation of :

 (7.43)
where is a 3D representation of .

### Standard models

Diﬀerent versions of the model are well catalogued in the Turbulence modeling resource, NASA Langley Research Center, https://turbmodels.larc.nasa.gov.

Today, there are arguably two “standard” models, First, the - SST model,18 (SST = shear stress transport) which emerged as a popular choice in CFD over recent decades.

It combines the model near the wall with (expressed in terms of ) in the freestream, by applying blending functions to model coeﬃcients, the cross-diﬀusion term and the stress limiter.

Secondly, the - 2006 model19 applies the cross-diﬀusion term and stress limiting to the original model. The terms are applied using switches so that the model maintains its simplicity, without the need for blending functions.

17Florian Menter, Zonal two equation models for aerodynamic ﬂows, 1993.
18Florian Menter, Two-equation eddy-viscosity turbulence models for engineering applications, 1994.
19David Wilcox, Turbulence modeling for CFD, 3rd ed., 2006.

Notes on CFD: General Principles - 7.11 Enhancements to the k-omega model