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

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εθ = 2α ∂θ
Basic Differential Stress Model Closure
∂θ
∂xk ∂xk
(3.47)
As mentioned earlier the pressure-strain φij and dissipation rateεij require
modelling. There are two different modelling alternatives for these terms used
in this study. One is called the basic second-moment-closure model and the
other one is called the two-component-limit model. First, we discuss the basic
model.
Modelling of the pressure-strain term is very important. This term can be
split into three parts:
φij = φij1 +φij2 +φij3
(3.48)
φij1 represents turbulence-turbulence interactions and is called the slow
pressure-strain term. φij2 is associated with mean strain terms and is called
the rapid pressure-strain term. φij3 is associated with the force fields such as
buoyancy.
The φij1 term is also called the “return to isotropy” term and it is responsi-
ble for the redistribution of the energy in order to decrease the anisotropy of
the Reynolds stresses. Its modelling has been suggested by Rotta [46]:
where aij is:
φij1 = −c1εaij
aij = uiuj
k
− 2
3 δij
(3.49)
(3.50)
The φij2 term is often modelled as acting to reduce the anisotropy of the
stress production. Its modelling has been suggested by Naot et al [47] as:
φij2 = −c2
Pij − 1
3 Pkkδij
(3.51)