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

Turbulence modelling 78
uiθ. The temperature variance θ 2 appears inside this equation. Thus, it is nec-
essary to solve a transport equation for θ 2 (equation 3.21). Then there are two
methods to calculate dissipation rate of temperature variance, εθ which are
described in the section 3.3.5.
3.3.5 Low-Reynolds-Number Three and Four Equations model
k-ε-θ 2 and k-ε-θ 2 -εθ
It is worth noting that even flow in undivided two-dimensional enclosures
shows some characteristics related to complex buoyant flows. There are two
distinct regions in buoyant cavity flows. One is the boundary layer region
along the walls and the other is the circulation region of fluid in the core. They
interact at their interface. They possess different turbulent structures and tur-
bulent scales. The circulating flow in the core can be a single cell or a multi-cell
structure, depending on thermal boundary conditions, enclosure aspect ratio
and Ra value.
Another complexity of buoyant cavity flows is the transition to turbulence.
In practical cases, such as flow in building structures and the cooling of nu-
clear reactors, etc., there is significant turbulent flow only in some parts of the
domain. Therefore prediction of transition locations, and also the decay of tur-
bulence away from the wall, is another challenge of turbulence modelling for
buoyant flows.
Three- and four-equation models, k-ε-θ 2 and k-ε-θ 2 -εθ schemes, have been
proposed by Hanjalic et al.[25] which are presented in this section. The equa-
tions for k and ε are the same as those presented in Section 3.3.3, whilst trans-
port equations were also introduced for θ 2 and εθ.
∂
ρθ2
∂t
∂
+
ρUjθ2
∂xj
= ∂
µ+
∂xj
µt
∂θ2 σk ∂xj
+2Pθ −2ρεθ
(3.21)