Numerical
simulation of temperature and velocity fields of turbulent non-isothermal flows
is an important and usefull in the studies of processes in power plants,
including nuclear (NPP). Temperature fluctuations arising in turbulent
non-isothermal flows (in comparison with laminar), lead to additional cyclic
thermal strains on the walls of the equipment and in some cases significantly
reduce the service period of both individual equipment and the facilities in a
whole. In addition, non-stationary non-isothermal flows have a significant
impact on the output of temperature gauges, which are used to monitor and
control processes on the NPP.

We
consider as the reference the International Working Group of Experts CFD4NRS
(uniting experts in the field of NPP safety studies) experiment to verify
numerical modeling recommended the studying natural convection inside a cubic
cavity with a given vertical temperature gradient. Here, it should also be
noted that natural convection in closed cavities is a traditional object of
basic research, due to the possibility of implementing various flow regimes
under similar conditions [1, 2].

The
studying of natural convection in closed cavities of various configurations is
one of the main procedures in modeling various convective processes in
engineering applications. The main part of the research in this area is devoted
to the study of the convection in the rectangular cavities with the temperature
differences in the horizontal plane (heating and cooling on the side walls).

The
studies with a vertical temperature gradient are more interesting when the
complex process-time structure of large-scale flows are considered. Those flows
can be obtained under these conditions. Most of these studies were performed
for cylindrical [3] and cubic cavities [5, 6]. Numerical calculations of convection
in a cubic cavity with a vertical temperature difference were performed for small
and moderate values of the Rayleigh number (3.5⋅10^{3}
≤ Ra ≤ 6⋅10^{4})
[3-6]. It was demonstrated that even for small values of supercriticality in
laminar regimes, the formation of turbulent flows of various spatial structures
is possible.

Of
great interest for the above-mentioned practical applications associated with
modeling processes at NPP are the convective flows with developed turbulence at
10^{8} ≤ Ra ≤ 101^{3} and higher . However, such
regimes have been studied quite a little. The reason for this is that the
evolution of large-scale flows in turbulent regimes proceeds rather slowly,
and, as experiments show, the describtion of the evolution of large-scale
structures takes from several to tens of hours. Performing numerical
simulations on such long time intervals is a non-trivial task and requires
careful selection of a numerical model and the use of a high-performance
computing system. Thus, the procedure for verifying numerical calculations
becomes important.

It
is necessary to provide the most representative comparison of its results with
the available experimental data within verifying the results of numerical
simulation. The comparison of instantaneous or in contrast average flow
characteristics, including velocity fields, does not allow one to properly
verify the results of numerical calculations in terms of time evolution. On the
other hand, a comparison of the time spectra obtained at control points may also
not to be comprehensive. The representativeness of the choice of their position
can be obtained only taking into account the spatial structure of the flow. In
this case, the use of the Proper Orthogonal Decomposition (POD) method can
help, which, on the one hand, makes it possible to show the main spatial
characteristics of the flow, and on the other, to obtain their temporal
spectral characteristics.

The
benchmark was a cube with a side of 25 cm made of optical glass with a
thickness of 5 mm, completely filled with fresh water. , We provided a special
stabilized system of electrical heating of the bottom with feedback on
temperature gauges in order to maintain a constant temperature difference of up
to 20 degrees between the upper and lower bound. The corresponding Rayleigh
number in the experiment is Ra = 4.4⋅10^{9}
(which corresponds approximately to the middle of the range of values of
interest in modeling real conditions, see above). Fig. 1 shows the general
scheme of the experiment and a separate illustrative frame obtained with a long
exposure (3 seconds).

Fig. 1. Left: principal
scheme of experiment: 1- laser, 2 – laser sheet, 3 – cub with water, 4 – camera.
Right: The obtained with long exposition image,
that demonstrates large scale structures in the flow

To
measure the fields of the flow rate of natural convection due to the vertical
temperature gradient, the PIV method was used. The studies were carried out
according to the simplest scheme with a continuous laser illumination of a
LCS-DTL-413 diode-pumped green laser (1.5 W, 527 nm). A vertical laser sheet
was formed from a beam using a defocusing cylindrical lens (2.5 mm radius).
Microparticles (HGS hollow glass beads) with a diameter of 10 μm
were used to visualize the flow. The filming was performed with a Canon EOS 5D
Mark II serial camera with a CMOS matrix of 21.1 Mpx (the scale of the
resulting image is 240 µm / px). Frame resolution 1920 1080 with a frequency of
30 frames / second, and shutter time of 30 ms. Comparison with the results obtained
for the Dantec serial 2D-PIV system under the same conditions demonstrated that
this simplified system allow to measure velocity fluctuations with a time resolution
from 0.001 to 15 Hz and accuracy not worse than 0.1 mm/s in the range of
velocities up to 2 cm/s at the selected filming scaling.

3200
seconds of video were processed by PIV-method totally. Cross-correlation processing
for pairs of sequent frames was performed. The interrogation window was 32´32
pixels, overlapping 50%. Thus interval on coordinate grid of velocity field was
16 pixels or 3.8 mm. An adaptive scheme for searching for a cross-correlation
function with a Gaussian approximation by three points of its peak was used to
ensure sub-pixel accuracy, similar to that we used earlier in [6]. The
resulting velocity fields were filtered by a 10-fold downsampling sample in
time. This was done using averaging with a 10-frame sliding window. For the
window containing less than 30% of unfiltered velocity values the linear time
interpolation was performed instead of averaged value calculation. As a result,
all measured velocity fields were filtered and combined into a
three-dimensional array containing 61´62´9587
points, in each two velocity components were measured.

Based
on the results of processing this array, we obtained average velocity fields
and fields of deviation for each of the velocity components (see Fig. 2). As it
was expected, it has demonstrated the strongest variations in the near-wall
regions. Comparison of the velocity fields obtained by averaging over various
time intervals (see Fig. 3) demonstrates the absence of any significant changes
at times exceeding 60 seconds, which gives a lower estimate for the time during
which the numerical count, the results of which can be compared with the data
of the present experiment, if we consider it as a reference.

à) á) â)

Fig. 2. Average velocity field
(top) and a field of standard deviations for horizontal (middle) and vertical (bottom)
velocity components. Color scale indicate m/s.

Fig. 3. Velocity fields
for a set of 9 different averaging times from 1 to 150 seconds (bottom).

The
high spatial resolution of the obtained velocity fields made it possible to
successfully use the Proper orthogonal decomposition (POD) method to isolate
the basic restore modes of the flow.

The
results of using POD in processing velocity fields obtained by PIV-methods can
be found in the relatively recently published papers (see [8-11]). For example,
in [8], POD was used to obtain the phase-averaged statistical characteristics
of the turbulent wake behind the bodies. Interpretation of the data obtained in
the short series of PIV-measurements in a diesel model with a high temporal
resolution using POD method is described in [9]. In [10] POD was used to study
the relationship between the characteristics of power spectra with the observed
structures in a flow over an open cavity. In [11], POD was used to reconstruct
three-dimensional flow structures and study their temporal evolution in a
turbulent jet in a transverse flow. To analyze the velocity fields, we used the
so-called “shapshot POD” technique, in the variant described in [11]. The main
idea of the POD algorithm is to represent the instantaneous velocity fields as
a sum of modes with time-dependent coefficients. To find those modes from
ansamble of N velocity fields, the following representation in form of matrix *l*×*m*
of fluctuating parts of both velocity componens is
used:

,

*u*
and *v *here are fluctuating parts of the horizontal and vertical
components correspondingly, and the superscript of the velocity indicates the
number of the velocity field in the sequence. Than the
matrix is calculated and
corresponding eigenvalue problem is
solved. The solutions are arranged by eigenvalue as: . And
POD modes are
found as:

.

With
POD modes arranged as decomposition coefficients
can be found for the velocity
field as .
A fluctuating part
of velocity field can be
reconstructed as:

.

The obtained velocity fields were processed with POD method to
obtain typical structures (modes) in the flow (Fig. 4). It was shown that the
steady flow can be described by combination of the mean flow and the first
three modes (up to 95% of the kinetic energy of the flow is contained in them).
It was demonstrated that only starting with the third mode there is a
significant change in the spectra of the amplitude coefficients of the modes,
the peak shifts to the region of higher frequencies to 0.02 Hz in comparison
with the first two modes (0.006 Hz) (Fig. 5).

Fig.
4. Mean velocity field and 4 main POD modes, arranged by descending energy.

Fig.5. Spectra obtained from
time dependences coefficients for the four modes (are listed from top to bottom
in descending order of average kinetic energy of fluctuations).

Experimental
studies of natural convection processes on the benchmark “cube” were performed for
the verification problems of CFD processes in NPP. The experiments were
performed under a constant vertical temperature gradient with a Rayleigh number
of 4.4⋅10^{9},
which corresponds to turbulent flow regimes. PIV system based on the simplest
scheme with continuous illumination was used for measurements. For the measured
velocity field sequences, the POD method procedure was used to develop the most
representative verification methods for numerical CFD calculations. It is shown
that more than 95% of the kinetic energy of turbulent fluctuations is
determined by the combination of the first three modes. Thus, a comparison with
the results of CFD is reasonable to carry out on the average, as well as on the
spectral characteristics of these modes.

This
work was partial supported by the Russian Foundation of Basic Research No. 18-48-520023 (providing numerical simulations)
and Russian Science Foundation No. 18-19-00473
(providing measurements).

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