Page 309 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Turbulent heat transport 279

6.2.1.2.1 Second-moment closures

In second-moment closures, the system of the equations can be closed by the transport
equations for the turbulent heat fluxes, the temperature variance, and its respective
dissipation. There are a large number of second-moment closure models that appear
in the literature, for example, Carteciano et al. (1997), Donaldson (1973), Gr€ otzbach
(2007), Launder (1989), and Rodi (1993). One such model has also been developed in
the nuclear field with a special focus on the liquid-metal flows and is known as Tur-
bulence Model for Buoyant Flows (TMBF) (Carteciano et al., 1997). It is a full
second-order heat flux model involving the temperature variance equation and also
has an option for the dissipation equation for the thermal variance. This model was
first implemented in FLUTON code (Willerding and Baumann, 1996; Gr€ otzbach
et al., 2002), in combination with a low Reynolds k-ε model. The model has been
adopted for liquid-metal convection cases, and its latest version is given in
Carteciano and Gr€ otzbach (2003). The transport equation for the turbulent heat flux
is given as

Du i θ 0 ∂ k 2 κ + ν ∂u i θ 0 ∂T ∂u i

0
0
0 0
0 0
¼ C TD + u u + u θ
Dt ∂x j ε 2 ∂x j i j ∂x j j ∂x j
(6.2.1.5)
u θ u i
G 0 0 + π i + ε 0 θ 0
i
λ
where κ ¼ is the thermal diffusivity:
ρC p
G 0 0 ¼ βg i θ 0 2 (6.2.1.6)
u θ
i
3
ε ε k 2
0 2
π i ¼ C T1 u θ + C T2 u θ ∂u i + C T3 βg i θ C T4 u θ n i n k (6.2.1.7)
0 0
0 0
0 0
k i j ∂x j k k d w ε
where n i is normal to the wall and d w is the normal distance to the wall:
ε
1+ Pr
0 0
p
ε 0 θ 0 ¼ p ffiffiffiffiffi ffiffiffi exp C T5 Re t + Pe t Þ½ ð Šu θ (6.2.1.8)
i
u i 2 Pr R k
where Re t is the turbulence Reynolds number and Pe t ¼Re t Pr is the turbulent Peclet
number. For details regarding the TMBF formulation, see Carteciano and Gr€ otzbach
(2003). The performance of the TMBF was assessed in two benchmarks; for details,
see Baumann et al. (1997) and Arien et al. (2004). The simulations were performed for
the TEFLU experiment as described by Knebel (1993 and 1998), in which the mixing
of a hot sodium jet was analyzed in a highly turbulent surrounding formed by a mul-
tijet environment. Fig. 6.2.1.4 displays a comparison of the TMBF along with an iso-
tropic k-ε model with a constant Pr t value. The figure highlights the axial development
of the radial temperature profile. It is clearly noticeable that the anisotropic TMBF
shows better results compare with the isotropic k-ε-Pr t model. Nevertheless, despite
showing good results for the TEFLU case (Knebel, 1993, 1998), further testing of the