7.9 Low-Re k-epsilon models

There are many low-eqn 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-eqn k models based on Eq. (7.1 ) and Eq. (7.2 ) with additional corrections eqn, eqn, and eqn:

@k ---+ r (uk) r (Dkrk) = @t 2 G -k(r u) (
The calculation of eqn also includes a correction eqn:
 k2 t = c f --:
This model was first presented by Jones and Launder in their seminal k publication.12 They proposed functions for eqn, eqn, eqn, and eqn, as well as the coefficients eqn, eqn and eqn.

Launder and Sharma subsequently presented13 the model with a modified eqn 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-eqn model today.

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

 p --
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 eqn:
  • fixed value ;
  • fixed value eqn.

The next significant modification is the eqn function

 h i f = exp 3:4=(1+ Re =50)2 : \relax \special {t4ht=
This modification recognises that eqn so decreases eqn through the buffer and viscous sub-layer to the wall, consistent with the decrease in eqn according to the van Driest model Eq. (7.12 ).

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

 2 E = 2 tjrruj : \relax \special {t4ht=
Finally, eqn and eqn provide damping of the production and dissipation terms close to the wall in Eq. (7.30 ). The standard functions are eqn (i.e. no damping), and
 2 f2 = 1 0:3exp Re ; \relax \special {t4ht=
which gives eqn 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