7.11 Enhancements to the k-omega model

A comparison of the eqn and k models shows the dissipation term is eqn in Eq. (7.37 ) for eqn and c2 in Eq. (7.2 ) for . While the eqn term behaves well, the term is troublesome at a wall as eqn, so requires damping in the low-eqn formulation in Sec. 7.9 to resolve the viscous sub-layer.

Its better dissipation term gives the eqn model an advantage over k in the near-wall region. The disadvantage the original eqn model is its sensitivity to the freestream values of eqn, which is not present in in the k 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 eqn model, in particular to address the problems mentioned above.


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

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

Stress limiter

The original eqn and k 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 eqn can be significantly higher than unity. The calculated eqn from Eq. (7.39 ) is excessively high, causing an over-prediction of shear stress eqn.

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

 k t =-------------; max (!; =a1) \relax \special {t4ht=
where eqn is a 3D representation of eqn.

Standard models

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

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

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

Secondly, the eqn-eqn 2006 model19 applies the cross-diffusion term and stress limiting to the original eqn 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 eqn 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