Page 290 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

260 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

where ν is the kinematic viscosity and α is the heat diffusivity. This number actually
compares the momentum diffusivity and the heat diffusivity, thus governs the relative
thickness difference between the momentum and the thermal boundary layers (or shear
layer in a free shear layer). Hence, for a unity Pr, there is a perfect match between
momentum and thermal boundary layers development. This physically means that
the momentum transfer from the wall to the bulk flow is similar to the heat transfer. This
last remarks is actually the basis of the so-called Reynolds analogy. A detailed descrip-
tion of the Reynolds analogy can be found in Kays and Crawford (1993).
The Reynolds analogy is the oldest and simplest model to estimate the turbulent
Prandtl number Pr t when performing RANS simulations. It expresses a similarity
between turbulent momentum exchange and turbulent heat transfer in a fluid. When
momentum and heat transports are considered similar: ν T ¼ α T or Pr t ¼ 1. In practice,
this simple model seems to coincide well with experimental data for fluids with
Pr Oð1Þ. The turbulent Prandtl number used in most commercial CFD codes is
0.85. On the contrary, for low Prandtl number flows such as liquid metals, the mech-
anism described above is not valid anymore and Pr t appears to be higher than one.
As it is done in most RANS approaches, it is convenient to introduce a subgrid-
scale Prandtl number in LES defined as

ν sgs
SGS
Pr ¼ , (6.1.2.52)
α SGS
which could directly give a subgrid-scale heat diffusivity necessary to close the system
using an eddy heat diffusivity approach by analogy of the eddy-viscosity concept:

∂ T
σ SGS : (6.1.2.53)
T ¼ α SGS
∂x j
Obviously such an approach is not able to capture the effect of the molecular Prandtl,
especially if Pr SGS is given by a single value, and is therefore not appropriate. How-
ever, because of its simplicity, it is widely used in practice with values of Pr SGS rang-
ing from 0.1 to 1 with a common value of 0.6 (Sagaut, 2006).

6.1.2.3.4.2 The case of liquid metals

As for the spatial scales, we can introduce the scalar diffusion cutoff length also called
the Obukhov-Corrsin scale η T (Corrsin, 1951; Sagaut, 2006), which is related to the
Kolmogorov scale by

1=4 3=4
3
α 1
η ¼ ¼ η : (6.1.2.54)
T E Pr K
Hence it is possible to relate the cutoff wavenumber for the temperature to the
Kolmogorov cutoff:

k T ¼ Pr 3=4 k η , (6.1.2.55)