Page 291 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Large-eddy simulation: Application to liquid metal fluid flow and heat transfer 261

Fig. 6.1.2.4 Kinetic and
temperature spectra for the case
of liquid metals (low Prandtl
number).
(Adapted from Sagaut, P., 2006.
Large Eddy Simulation for
Incompressible Flows. Springer,
Berlin.)

which shows that k T 0, 03k η for liquid metals (Pr 0.01). This results in two dif-
ferent regimes for the temperature in the inertial regime of the velocity (see
Fig. 6.1.2.4):
l for k ≪ k T ≪ k η : Inertial range for both the temperature and kinetic energy. Here the tem-
perature fluctuations are driven by the mixing induced by fluctuation of velocity. In this
range the temperature spectrum is

E T ðkÞ¼ βE T E 1=3 5=3 , (6.1.2.56)
k
where the Obukhov-Corrsin constant β 0.68 0.83. This range is called the inertial-
convective range.
for k T ≪ k ≪ k η : Here the velocity fluctuations are not sensitive to the molecular viscosity,
l
but temperature fluctuations are influenced by molecular heat diffusivity. This is the inertial-
diffusive range, which is characterized by a spectrum:

K 0 2=3 3 17=3
E T ðkÞ¼ E T E α k : (6.1.2.57)
3
For higher wavenumbers, both the viscosity and heat diffusivity are dominant
(dissipation range).
In Fig. 6.1.2.4, the cutoff wavenumber of an LES is indicated as k c and illustrates an
interesting feature. Indeed because in the offset of both spectra, there is a range of
cutoff wavenumbers to achieve an LES for the velocity field while resolving the
smallest scales of the temperature field. In other words, it is possible to achieve an
hybrid simulation, which is an LES for the velocity and a DNS for the temperature
(V-LES/T-DNS). In Gr€ otzbach (2011), the author indicates the grid requirement
for achieving a DNS of a temperature field, for example, h/η T < 1 for close to unity
Prandtl number fluids, could give h/η T < 3, 45 for a liquid metal. This condition could
be used a posteriori to check that the LES was indeed a T-DNS.
For coarser LES, when the cutoff wavenumber lies in the inertial convective range,
the temperature eddies are not fully resolved and it is necessary to introduce a subgrid