Page 245 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

216 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

RANS-LES coupling methods since there is no mechanism to transfer the modeled turbulence
energy into resolved turbulence energy. Despite a wide number of approaches (and acronyms),
these methods are quite similar and can very often be rewritten as variants of a small group of
generic approaches. In practice, they tend to decrease the level of RANS eddy viscosity, thus
permitting strong instabilities to develop. It is now commonly accepted that hybrid RANS-LES
is the main strategy to drastically reduce computational cost (compared with LES) in a wide
range of complex industrial applications if attached boundary layers have a significant impact
on the global flow dynamics. In the frame of this book, however, hybrid methods are not dis-
cussed in further detail.

In order to close the system of equations, a model is needed to express the Reynolds
stresses in terms of mean flow quantities. A widely adopted approach of modeling the
Reynolds stresses exploits the Boussinesq hypothesis. Based on an analogy with lam-
inar flow, it states that the Reynolds stress tensor can be related to the mean velocity
gradients via the turbulent “eddy” viscosity. In this approach of modeling, the effect of
turbulence as an added viscosity is widely used for turbulent flow simulations. It is
very useful as an engineering method, because the computational time is only weakly
dependent on the Reynolds number. However, the validity of the Boussinesq hypoth-
esis is limited. The turbulent eddy viscosity is not a property of the fluid but rather a
property of the type of flow in question. The Boussinesq hypothesis is therefore inad-
equate in many situations, for example, for flows with sudden changes in mean strain
rate (e.g., the shear layer of the wake), anisotropic flows, and three-dimensional (3-D)
flows as explained by Davidson (2004) and Wilcox (2006). A number of methods have
been suggested to calculate the turbulent eddy viscosity, such as zero-equation (alge-
braic closure, such as mixing length), one-equation, and two-equation models (see
e.g., Wilcox (2006) and the references therein). The k-ε model and k-ω model are
examples of two-equation models often encountered in nuclear engineering
applications.
A fundamentally different approach is the Reynolds stress model (RSM) as
described by Launder et al. (1975), also called differential second-moment closure
model, which does not rely on the Boussinesq hypothesis. In the RSM, all the com-
ponents of the Reynolds stress tensor are modeled, which makes it suitable for aniso-
tropic flows. However, it leads to six additional partial differential equations, making
the approach computationally expensive. Moreover, these equations contain terms
that have to be modeled again, and often, closure relations resembling the Boussinesq
hypothesis are still employed.
Similarly, the energy conservation equations need to be closed. However, the often
applied analogy between the temperature and the velocity field is not valid in the case
of liquid metals. Therefore, special attention is needed to model the closure of the
energy equations for such applications. This is the topic of Section 6.2.1. The remain-
der of Section 6.2 deals with various applications of RANS modeling techniques.