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

Turbulence modelling 82
gradient diffusion hypothesis (GGDH) of Daly and Harlow. In this approxi-
mation, the flux of quantity φ is modelled through:
ukφ ∝ − k
ε ukul
Applying this approximation to the triple moment correlation results in:
∂φ
∂xl
ukuiuj ∝ − k
ε ukul
∂uiuj
∂xl
(3.33)
(3.34)
Introduction of the above approximation to (3.31) and canceling the sec-
ond and third terms (assuming them to be negligible compared to the others)
results in the following equation for the diffusion process:
dij = ∂
∂xk
csρ k
ε ukul
∂uiuj
∂xl
the coefficient cs is taken equal to 0.22.
+µ ∂uiuj
∂xk
(3.35)
There are two major modelling practices for the pressure-strain term φij
and the dissipation tensor εij used in the present work that will be discussed
later.
To close the set of equations, it is required to solve a transport equation
for the energy dissipation rate ε, in addition to the transport equations for the
Reynolds stresses. The transport equation for the energy dissipation rate, ε, is
of the same general form as the one employed in the k-ε closure:
D(ρε)
Dt
= ρε2
k
cε1
P
ε −cε2
+dε
(3.36)
Its diffusion term is modelled using the generalized gradient diffusion hy-
pothesis (GGDH):
dε = ∂
∂xk
cερ k
ε ukul
∂ε
∂xl
+µ ∂ε
∂xk
where cε, cε1 and cε2 are taken 0.16, 1.45 and 1.90, respectively.
(3.37)
In order to model the turbulent heat fluxes,uiθ, there are two main options.