Page 277 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Large-eddy simulation: Application to liquid metal fluid flow and heat transfer 247

t η 1=2
¼ Re : (6.1.2.3)
t 0

where η,u η , and t η are, respectively, the Kolmogorov length scale, the Kolmogorov
velocity scale, and the Kolmogorov time scale. Typically, the end of the inertial range
and the beginning of the dissipation range are for k η ¼ 0.1 0.2.
3 2
The dimension of E(k)is(m s ). In the inertial range we have E(k) ¼ f(k, E). As k
1
2 3
has the dimension (m ), and the dissipation E as the dimension (m s ), we obtain:
EðkÞ E 2=3 5=3 : (6.1.2.4)
k
2/3 5/3
This finally gives the so-called Kolmogorov law E(k) ¼ C K E k with C K ¼ 1.6 and
explains the 5/3 slope in Fig. 6.1.2.1 for the inertial range. From the CFD point of
view, where space is discretized in a finite number of cells in each direction,
Eq. (6.1.2.1) has a crucial consequence. Indeed, l 0 is proportional to L, the physical
geometry length scale, which can be discretized by n points in each direction of space,
L ¼ nΔh, Δh being the grid size. This implies in 3D:

3
n Re 9=4 : (6.1.2.5)
As shown in Fig. 6.1.2.1, the DNS approach explicitly simulates all the range of pos-
sible wavenumbers and time scales in a turbulent flow. From Eq. (6.1.2.5), the number
9/4
of grid points to correctly capture the flow scales as Re . For instance, describing a
6
common flow at Re ¼ 10 by a DNS would require of the order of 20 10 11 grid
points (200 billion points)!! More precisely, as a grid of size h could capture an eddy
of size 2h (Nyquist theorem), an ideal DNS grid would require h DNS 2η.
The opposite approach (e.g., the RANS) is based on the modeling of the full spec-
trum presented in Fig. 6.1.2.1. It means that (3D) turbulent fluctuations are not explic-
itly simulated, the effect of turbulence is simply mimicked by adding an extra
diffusion to any transported variable. The grid has to be fine enough to capture gra-
dients of the mean flow, which is far less demanding than DNS.
A good compromise between these two extreme approaches (DNS and RANS) is
the LES technique. It allows to have a good representation of the turbulence, yet
requiring much less computation resources than a DNS. The spirit of LES is to explic-
itly simulate or resolve the large scales responsible for the mixing and for determining
the dissipation rate, while modeling the smallest eddies which are more universal and
responsible for the final energy dissipation into heat. Based on Fig. 6.1.2.1, this
method relies on a scale separation at a certain cutoff wavenumber k c . All the scales
below k c will not be explicitly simulated but rather modeled, using the so-called
subgrid-scale model. This model will account for the effect of the small scales on
the resolved ones. As the smallest scales are thought to be universal and because most
of the spectrum is resolved, it is expected that the results of a good LES simulation will
be not so drastically sensitive to the subgrid-scale model (much less than an RANS is
sensitive to the turbulence model). A widely used technique to achieve this scale sep-
aration is to use the projection operator related to the computational grid. Hence the