Notes on Computational Fluid Dynamics: General Principles

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7.11 Enhancements to the k-omega model

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

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

Cross-diﬀusion

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

The cross-diﬀusion term makes the $\text{[math]}$ equation more equivalent to $\text{[math]}$ , and thus independent of freestream values.

Stress limiter

The original $\text{[math]}$ 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 $\text{[math]}$ can be signiﬁcantly higher than unity. The calculated $\text{[math]}$ from Eq. (7.39 ) is excessively high, causing an over-prediction of shear stress $\text{[math]}$ .

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

k t =-------------; max (!; =a1) \relax \special {t4ht=

(7.43)

where $\text{[math]}$ is a 3D representation of $\text{[math]}$ .

Standard models

Diﬀerent versions of the $\text{[math]}$ model are well catalogued in the Turbulence modeling resource, NASA Langley Research Center, https://turbmodels.larc.nasa.gov.

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

It combines the $\text{[math]}$ model near the wall with $k$ (expressed in terms of $\text{[math]}$ ) in the freestream, by applying blending functions to model coeﬃcients, the cross-diﬀusion term and the stress limiter.

Secondly, the $\text{[math]}$ - $\text{[math]}$ 2006 model¹⁹ applies the cross-diﬀusion term and stress limiting to the original $\text{[math]}$ 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 $\text{[math]}$ 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