Page 311 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Turbulent heat transport 281

have some limitations. Nonetheless, one such model was developed within the frame-
work of the THINS project; for details, see Manservisi and Menghini (2014). This
model was validated for two simple geometries (plane and cylindrical). Accordingly,
it was further tested for liquid-metal flow in triangular rod bundles with different P/D
ratios. The simulations were performed for a Peclet number in the range of 300–2500.
The obtained results were compared with the experimental correlations for the predic-
tion of the Nusselt number and have shown good agreement. Nevertheless, further
validation of this model for other type of geometries needs to be done. In addition,
an extension of this model toward the natural and mixed convection regimes is
required. Recently, the von Karman Institute (VKI) has implemented this model in
OpenFOAM, and its further validation is foreseen to be performed in the near future.

6.2.1.2.2.2 Implicit AHFM
In an implicit formulation, the turbulent heat flux can be computed directly from an
algebraic solution of the turbulent heat flux transport equation, by solving a nonlinear
algebraic equation. In this regard, second-moment closures can serve as a basis for
derivingalgebraic models.By suitable elimination of differential terms, the differential
equations for the second moments can be truncated to yield algebraic expressions for
the turbulent heat flux (Dol et al., 1997; Gibson and Launder, 1978). Depending on the
levelof truncation,different forms ofalgebraic models can be derived (Hanjalic, 2002).
2
By keeping production and dissipation of k and θ locally in balance, the following
expression of the turbulence heat flux is obtained (Kenjeres and Hanjalic, 2000):

θ k ∂T ∂U i 2
θu i ¼ C u i u j + ξθu j + ηβg i θ + ε θi (6.2.1.9)
E ∂x j ∂x j
The above equation retains all three production terms from a second-order moment
closure (see Eqs. 6.2.1.5–6.2.1.8) and represents the physical mechanism that gener-
ates the turbulent heat flux:

1. Due to the nonuniformity of the mean thermal field ∂T
∂x j
2. Mechanical deformation (mean rate of stain ∂U i )
∂x j
3. Amplification and attenuation of turbulence fluctuations due to the buoyancy effect βg i θ 2
The last term in the Eq. (6.2.1.9), that is, ε θi , represents the molecular dissipation. For
details regarding the models coefficients, readers are referred to Kenjeres and Hanjalic
(2000). In Eq. (6.2.1.9), k and ε can be obtained from a two-equation shear stress
model, whereas a closure of the algebraic expression requires the basic variables
2
θ and ε θ to be provided from the separate modeled equations.

AHFM-2000 (Kenjeres and Hanjalic, 2000)
One of the approaches used for the aforementioned closure would be to solve separate
2
transport equations for θ and ε θ like was done by Hanjalic et al. (1996a,b), Kenjeres
and Hanjalic (2000), and Peeters and Henkes (1992). The closure of the latter equation