Notes on Computational Fluid Dynamics: General Principles

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7.9 Low-Re k-epsilon models

There are many low- $\text{[math]}$ 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- $\text{[math]}$ $k$ models based on Eq. (7.1 ) and Eq. (7.2 ) with additional corrections $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ :

(7.29)

(7.30)

The calculation of $\text{[math]}$ also includes a correction $\text{[math]}$ :

(7.31)

This model was ﬁrst presented by Jones and Launder in their seminal $k$ publication.¹² They proposed functions for $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ , as well as the coeﬃcients $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ .

Launder and Sharma subsequently presented¹³ the model with a modiﬁed $\text{[math]}$ function and the more established coeﬃcients listed in Eq. (7.3 ).

The resulting model became known as the Launder-Sharma $k$ model.¹⁴ It is arguably the most popular low- $\text{[math]}$ model today.

The ﬁrst notable modiﬁcation to the standard $k$ model is $\text{[math]}$ (sometimes denoted “ $\text{[math]}$ ”) in Eq. (7.29 ). It is the dissipation rate at the wall ( $\text{[math]}$ ), 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 $\text{[math]}$ is that the boundary conditions at a wall for the Launder-Sharma model are the same for $\text{[math]}$ and $\text{[math]}$ :

ﬁxed value $\text{[math]}$ ;
ﬁxed value $\text{[math]}$ .

The next signiﬁcant modiﬁcation is the $\text{[math]}$ function

h i f = exp 3:4=(1+ Re =50)2 : \relax \special {t4ht=

(7.33)

This modiﬁcation recognises that $\text{[math]}$ so decreases $\text{[math]}$ through the buﬀer and viscous sub-layer to the wall, consistent with the decrease in $\text{[math]}$ according to the van Driest model Eq. (7.12 ).

The extra term $\text{[math]}$ in Eq. (7.30) is a follows, designed so that $\text{[math]}$ matches its recognised peak value within the buﬀer layer:

2 E = 2 tjrruj : \relax \special {t4ht=

(7.34)

Finally, $\text{[math]}$ and $\text{[math]}$ provide damping of the production and dissipation terms close to the wall in Eq. (7.30 ). The standard functions are $\text{[math]}$ (i.e. no damping), and

2 f2 = 1 0:3exp Re ; \relax \special {t4ht=

(7.35)

which gives $\text{[math]}$ 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 $k$ model would seem more equitable.

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