Page 250 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Direct numerical simulations for liquid metal applications 221

The dimensionless temperature T in Eq. (6.1.1.2) may be interpreted also as the con-
centration of a specific species and the Prandtl number is in that case replaced with the
Schmidt number. The mass transfer studies are frequently met in the publications in
the field of chemical engineering.
In the literature, two archetypal cases of turbulent transport of thermal energy
stand out, namely infinite channel flow and boundary layer flow (Li et al., 2009). This
chapter focuses primarily on channel or duct flows, given their higher relevance to the
specific field of application. Fully developed flows in ducts of constant or periodic
cross-sections are often studied with the DNS approach. In such cases, it is convenient
to assume the homogeneity of the flow also in the streamwise direction, although it is
clear that some variables, such as pressure and temperature, are actually not homoge-
neous. However, it is possible to solve for a set of new variables, which are periodical
by definition and can be used to recover the original, nonperiodic variables (see
discussion in Section 6.1.1.3). Note that the enforcement of periodic boundary con-
ditions (BCs) to represent homogeneity is a physical model, which is more accurate
when the computational domain that corresponds to the periodic length imposed to the
numerical solution is large enough to contain the largest relevant turbulent eddies of
the physical case (see discussion of the characteristic scales in Section 6.1.1.1.3).
For a fully developed duct flow the dimensionless energy equation written in terms
of the fluid excess temperature θ ¼ T T m (x) becomes:

∂θ ! 1 ω U
2
¼ r ðU θÞ + r θ (6.1.1.3)
∂t Re τ Pr ΩU m

where U is the instantaneous x-component of the velocity vector, ω/Ω is the dimen-
sionless ratio between the normalized duct lateral surface and volume, and U m and T m
are the dimensionless bulk velocity and temperature, respectively. Equations are nor-
malized with the channel half width h, friction velocity u τ , kinematic viscosity ν, and
friction temperature T τ ¼ q w /(u τ ρ f c p )(q w is the heat flux from the wall into the fluid). It
is worthy to note that the last term in Eq. (6.1.1.3) is a source term arising from the
definition of θ, which closes the energy balance so as to enforce fully developed con-
ditions while maintaining the periodicity of θ.
When buoyancy effects are taken into account, the Boussinesq approximation is
usually enforced (Gray and Giorgini, 1976). A typical form of the momentum equa-
tion for fully developed mixed convection flows is:

!
∂ U ! ! 1 2 ! Gr ! ω !
+ U r U¼ rp + r U + θ g + ı (6.1.1.4)
0
m
∂t Re τ Re 2 τ Ω
!
The forcing term along ı is an imposed pressure gradient in the streamwise direction,
!
while g is the gravity unit vector. Note that Eq. (6.1.1.4) is written in terms of a mod-
ified pressure p m and the previously introduced fluid excess temperature θ, whose
governing equation remains Eq. (6.1.1.3).