Page 279 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Large-eddy simulation: Application to liquid metal fluid flow and heat transfer 249

2h, that is, the highest (cutoff ) wave number seen by the grid is π/h. This filtering
option is thus natural as far as the numerical simulation approach is solving the
governing equations on a computational grid. The second approach is more mathe-
matic and related to applying a filter. This filtering option consists in applying an
explicit filter such as in signal processing, for example, a Gaussian filter. The purpose
of such filters is to keep the largest scales while damping the smallest one.
The major difference between these two options is crucial. For the first one
(implicit filtering) there is no information beyond the filter cutoff k c ¼ π/h. Practically,
this means that once a real or DNS field has been projected on a grid, all the infor-
mation contained below the grid size is definitively lost with no possibility to recover
it. For the explicit filtering, the information has been damped but not lost because it is
theoretically possible to fully (if the filter is invertible) or partially (by deconvolution)
recover it. This chapter will address grid-LES only. The explicit filtering approach is
largely covered in the books of Sagaut (2006) and Pope (2000).

6.1.2.2.1 Governing equations
As this chapter concerns the application of LES to heat transfer in liquid metals, we
will limit ourselves to the incompressible formulation stricto sensu, that is, the density
will be considered constant. In addition, the variation of any other thermophysical
property with temperature will be neglected. In this frame, the mass and momentum
conservation equations can be written as in Eqs. (6.1.2.6), (6.1.2.7), respectively. The
energy equation will be addressed in Section 6.1.2.2.4.

∂u i
¼ 0, (6.1.2.6)
∂x i
∂P
∂u i ∂u i u j ∂
+ ¼ + 2νS ij , (6.1.2.7)
∂t ∂x j ∂x i ∂x j
where ν is the kinematic viscosity; the reduced pressure and the strain rate tensor are
defined as:
p
P¼ , (6.1.2.8)
ρ

1 ∂u i ∂u j
S ij ¼ + , (6.1.2.9)
2 ∂x j ∂x i
where ρ is the density. The LES equations are then obtained by applying a filter oper-
ator on the Navier-Stokes equations. Because we are interested in numerical simula-
tion using discretized equations on a grid, the simplest filter we could imagine is the
LES grid itself which is equivalent to a sharp Fourier cutoff. This operation gives rise
to the implicitly filtered LES formulation (Section 6.1.2.2.2).