Page 254 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Direct numerical simulations for liquid metal applications 225

turbulent statistics for most of the relevant quantities. Nevertheless, a detailed reso-
lution study performed by Vreman and Kuerten (2014) has shown that some
higher-order statistics of turbulent flow (e.g., turbulent dissipation) require a finer res-
+
+
olution, namely Δx ¼ 6 and Δz ¼ 4, in the streamwise and spanwise directions, while
+
+
they recommend Δy ¼ 3 in the center for the channel and Δy ¼ 1 in the near-wall
+
region at y ¼ 12 in wall-normal direction. The resolution requirements proposed by
Vreman and Kuerten are actually close to the theoretical prediction from Eq. (6.1.1.5),
+
+
+
for which Δx , Δy , Δz η.
For the choice of the time step size in DNS, the physical constraints and numerical
accuracy and stability constraints must be met. In order to resolve the smallest time
scales, the time step size should not exceed a fraction of the Kolmogorov time scale.
However, numerical stability and accuracy criteria are often more stringent. For
instance, in DNS of Tiselj et al. (2001b), with second-order accurate time scheme,
a time step size is typically chosen such that the Courant number is around 0.1 to
ensure the sufficient accuracy. The scheme is stable (but with poorer accuracy) for
Courant numbers up to 0.5.
Various finite volume and finite difference schemes were later developed for DNS
studies. The reason why these methods are sometimes preferred to spectral schemes is
that different types of BCs can be more easily implemented than in a spectral method
framework.
The finite difference method is thoroughly explained and documented in a couple
of books (Fletcher, 1991; Anderson, 1995). In the framework of the finite difference
method, compact schemes are currently used as high-order spatial schemes that
increase accuracy and resolution without widening the computational stencil. The
properties and techniques of the finite difference compact schemes are explained in
a paper by Lele (1992). Starting in the 1990s, compact finite difference schemes have
been used for DNSs. More recently, these schemes were implemented in
Incompact3d, which is one of the most important Open Source codes for performing
DNSs (Laizet et al., 2010).
In a finite volume framework, balance equations for mass, momentum, and energy
are expressed in their integral form over each of the nonoverlapping control volumes
n
Ω i , which form the computational domain [ Ω i ¼ Ω. Applicability of finite volume
i¼1
schemes on unstructured grids in complex geometries is one of the reasons why many
commercial codes for industrial applications are based on a finite volume approach.
The finite volume methods are usually presented in the literature restricted to the
second-order accuracy form. Implementing high-order finite volume methods
requires devising specific schemes for evaluating space averaged derivatives, interpo-
lations, and fluxes, while deconvolution schemes are needed to extract pointwise
values from spatial averages (Piller and Stalio, 2004). In summary, development of
high-order finite volume algorithms is more demanding than development of such
schemes for finite difference methods especially in complex domains (Piller and
Stalio, 2008).