Page 257 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

228 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

As outflow BCs have to represent the continuation of the physical domain beyond
the edge of the computational domain, transport equations extrapolated from the
domain interior are often enforced in the form of

∂Φ ∂Φ
+ U c ¼ 0 (6.1.1.12)
∂t ∂x
InmanycasesauniformvalueforU c ensuring massconservationisused.Anexpression
for the convective velocity U c can be imposed in Eq. (6.1.1.12) also based on the spe-
cific flow configuration. For example, Craske and van Reeuwijk (2013), which deal
with turbulent jets and plumes, suggest the use of an exponential function for U c (y).
In the review on open outflow conditions by Hattori et al. (2013), the method by
Stevens (1990) is described as a promising approach for buoyant, turbulent plumes.
The method proposed by Stevens (1990) is based on a one-dimensional advection-
diffusion equation at the outflow, where a diffusion term is added to Eq. (6.1.1.12)
and a phase velocity is introduced in addition to U c .

6.1.1.3.4 Boundary conditions for the thermal field

Turbulent channel geometry is also the most frequently used geometry for studies of
the near-wall heat transfer. The most accurate approach is conjugate heat transfer,
where heat conduction inside the realistic heated walls is taken into account.
A relevant study for liquid metal applications is the DNS of Tiselj and Cizelj
(2012), who performed a conjugate heat transfer DNS at a low Prandtl number,
Pr ¼ 0.01. Most of the DNS heat transfer studies (Kasagi et al., 1992; Kawamura
et al., 1998; Na and Hanratty, 2000) were performed with simplified thermal BCs
without solid walls and the dimensionless temperature at the fluid-solid contact plane
that was fixed to zero:

θðy ¼ hÞ¼ 0 ðnonfluctuating thermal BCÞ (6.1.1.13)

This type of thermal BC is a very good approximation of reality when the thermal activ-
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi +
ityratioK ¼ ðλρc p Þ =ðλρc p Þ goestozero,whiletheheatedwallthicknessd ,andthe
f w
parameter G ¼ α f /α w remains finite (α ¼ λ/ρc p ). In that case, temperature fluctuations
generated in the fluid do not penetrate into the solid wall. This type of ideal thermal BC
was denoted as “nonfluctuating BC” by Tiselj and Cizelj (2012). An example of such
systemisair/metalcombinationoffluidand solid.Forwater/steelcombinationthether-
mal BC is still close to nonfluctuating BC if the heated wall is thick enough; however,
the approximation fails when the metal wall thickness is small.
The thermal BC, which permits maximum penetration of the turbulent temperature
fluctuations from the fluid into the solid, is defined as:

!
∂θ 0
hθðy ¼ hÞi ¼ 0 ¼ 0 ðfluctuating thermal BCÞ (6.1.1.14)
x,z,t
∂y
y¼ h