The computation of turbulent natural convection flows - Turbulence ...

Turbulence modelling 74
The turbulent kinetic energy, k, is related to the normal Reynolds stresses by
k = u2 +v 2 +w 2
2
(3.12)
In table 3.1, all the constants and functions of the high-Reynolds-number
k-ε model are listed.
cµ cε1 cε2 σk σε
0.09 1.44 1.92 1.0 1.3
Table 3.1 – Values of constants in High-Rek-ε, model.
Effect of molecular viscosity is not considered in the high-Reynolds-number
k-ε model therefore can not be integrated right across the wall sub-layer. The
wall-function method is used to provide wall boundary conditions and in-
clude the effect of molecular viscosity. These will be discussed in the next
Chapter in detail.
3.3.2 Further modification in High-Re k-ε model for 3D simu-
lations
In cases where there are large strain rates equation 3.4 can become non-
realizable, for example producing negative value for normal stresses. It can
also return a stress field that violates the Schwarz inequality of equation 3.13:
uiuj 2 ≤ u 2 i u2 j
(3.13)
when i = j. To prevent the High-Rek-ε from producing non-realizable results
in circumstances that strong shear flow exists, the following constraint can be
used for calculation of the turbulent viscosity in 3D simulations[43]:
⎧
⎨
µt = min
⎩ cµ
k 2
ε ,
∂U max ∂y
∂V + ∂x
2/3k
, ∂U
∂z
+ ∂W
∂x
,
∂V
∂z
+ ∂W
∂y
⎫
⎬
⎭
(3.14)