## 7.9Low-Re k-epsilon models

There are many low- turbulence models for CFD simulations where the cells near the solid walls are suﬃciently thin to resolve the ﬂow through the viscous sub-layer.

Among them are several low- models based on Eq. (7.1 ) and Eq. (7.2 ) with additional corrections , , and :

 (7.29)
 (7.30)
The calculation of also includes a correction :
 (7.31)
This model was ﬁrst presented by Jones and Launder in their seminal publication.12 They proposed functions for , , , and , as well as the coeﬃcients , and .

Launder and Sharma subsequently presented13 the model with a modiﬁed function and the more established coeﬃcients listed in Eq. (7.3 ).

The resulting model became known as the Launder-Sharma model.14 It is arguably the most popular low- model today.

The ﬁrst notable modiﬁcation to the standard model is (sometimes denoted “”) in Eq. (7.29 ). It is the dissipation rate at the wall (), see ﬁgure, Sec. 7.7 , calculated by

 (7.32)
The term equates to in the boundary layer which is consistent with Eq. (7.28 ). The beneﬁt of redeﬁning the dissipation rate as is that the boundary conditions at a wall for the Launder-Sharma model are the same for and :
• ﬁxed value ;
• ﬁxed value .

The next signiﬁcant modiﬁcation is the function

 (7.33)
This modiﬁcation recognises that so decreases through the buﬀer and viscous sub-layer to the wall, consistent with the decrease in according to the van Driest model Eq. (7.12 ).

The extra term in Eq. (7.30) is a follows, designed so that matches its recognised peak value within the buﬀer layer:

 (7.34)
Finally, and provide damping of the production and dissipation terms close to the wall in Eq. (7.30 ). The standard functions are (i.e. no damping), and
 (7.35)
which gives at the wall.
12William Jones and Brian Launder, The prediction of laminarization with a two-equation model of turbulence, 1972.
13Brian Launder and B.I. Sharma, Application of the energy-dissipation model of turbulence to the calculation of ﬂow near a spinning disc, 1974.
14although Jones-Launder-Sharma model would seem more equitable.

Notes on CFD: General Principles - 7.9 Low-Re k-epsilon models