Page 255 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

226 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Finite difference method vs. Finite volume method
Structured grids Unstructured grids
Noninherently conservative Inherently conservative
Close to the known equations Physically meaningful
Simple geometries Suitable for complex geometries
Derivation and interpolation Also integration and deconvolution
Higher-order spatial accuracy Lower-order spatial accuracy
High-order methods Difficult for high-order methods

Probably the most frequently used DNS database obtained with finite volume
schemes is being maintained since 1998 by Kawamura (2017). His team performed
simulations of turbulent channel flow at various friction Reynolds numbers up to
Re τ ¼ 1000 and Prandtl numbers between Pr ¼ 0.025 and 10.
The recent DNS study of Vreman and Kuerten (2014) shows a detailed compar-
ison of DNS results obtained with spectral and finite difference schemes.
They found that their finite difference code (second-order spatial accuracy in peri-
odic, fourth order in wall-normal direction, second order in time) required 3/4
smaller grid spacing than their spectral code to achieve the same accuracy.
A number of other approaches have been also applied to DNS. We do not address
the lattice Boltzmann method in this chapter, which has been shown to be capable of
DNS and is known to be extremely efficient on massive parallel computers with GPUs
due to its simplicity (Mayer and Hazi, 2006). However, the method, which is based on
the Boltzmann transport equations, has not been widely used and thoroughly verified
in single-phase heat transfer studies.
It is instead worth to mention the spectral element method, developed in 1984 by
Patera (1984). The method employs finite elements to discretize the domain and high-
order Chebyshev or Legendre polynomials as basis functions within each element. It
has gained a significant attention as a DNS tool thanks to the Open Source Code
Nek5000, developed by the Argonne National Laboratory (ANL, 2017). It is known
to be of similar accuracy as spectral schemes, with the advantage of being applicable
to rather complex geometries.
Practically speaking, when setting up a DNS from scratch, it is advisable to refer to
previous works with similar geometries, for comparable flow regimes, conducted with
similar numerical schemes, for the choice of spatial and temporal resolution.
A simulation with a, for example, 2 times coarser grid and longer time steps is often
run first in order to reach statistically steady conditions, saving computational
resources. The computed fields can then be used as initial conditions for the resolved
DNS, by interpolation on the final mesh.