Page 259 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

230 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

“statistical steady state,” meaning that long-term statistics are independent of time.
The assessment of such a condition is all but trivial. In principle, every flow
parameter—velocity, pressure, or temperature at a given point, average value in a
given line, on a selected surface or in a selected volume—should exhibit constant
mean values with fluctuations around the means. In the turbulent channel flow, typical
parameters used to identify statistical steady state are usually global variables, such as
mean velocity, mean temperature, turbulence kinetic energy, friction velocities and
friction temperatures computed at the walls, etc. Typically, statistical independence
must be observed over several tenths of flow-through times and several thousands
of viscous time units in order to recognize the statistical steady state in a running
DNS. These runs in most cases require several 10–100 thousands of time steps.
Once the statistical steady state is recognized, the spatial and temporal averaging of
the results can start. Typical averaging procedures include computations of mean
values in a point, over the line, surface, or in the selected volume. Flow statistics
can be computed upon runtime or a posteriori by using snapshots of the field variables.
Both approaches have obvious advantages and drawbacks. Because the number of sta-
tistical quantities to be computed can reach an order of a hundred parameters (when
various budget terms must be evaluated), the choice of the postprocessing approach is
influenced by the available computational and storage resources.
Instantaneous fields can offer certain information; however, statistical averaging is
necessary to compare results with experiments or other simulations. It is shown in the
channel flow section, which is a geometry with two homogeneous directions, that the
number of the time steps needed for the acceptable statistical uncertainty is rather large.
Thecorrespondingnumberof time steps fortheflowswitha singlehomogeneous direc-
tion is larger, and must be even larger in the geometries without homogeneous direc-
tions, where the only available type of averaging is averaging in time or ensemble
averaging with independent DNS runs performed in the same geometry. The paper
of Oliver et al. (2014) offers a methodology and a tool for analyses of uncertainties
in DNS simulations. Their tool, which has been recently used in the work by Flageul
and Tiselj (2017) to analyze statistical uncertainty in channel flow heat transfer, is rec-
ommended for quantification of the statistical uncertainties in the DNS studies.

6.1.1.4 Results: Channel flow

Overview of the turbulent heat transfer simulations performed in channel geometry at
low Prandtl number is collected in Tables 6.1.1.1 and 6.1.1.2. The fourth column of
Table 6.1.1.1 specifies thermal BCs used in the simulations: simulations denoted by
F and NF were performed only with the two limiting thermal BCs: fluctuating and
nonfluctuating temperature BC. Simulations denoted by C were conjugate heat trans-
fer simulations. Simulations of Tiselj and Cizelj (2012) and Tiselj (2014) were per-
formed at very low Prandtl numbers as benchmarks for the codes that will be used
for heat transfer analyses in liquid sodium fast reactors. The latest publication by
Oder et al. (2015) summarizes results of channel flow conjugate heat transfer simu-
lations using the spectral element code Nek5000.