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The paper considers the possibility of using different approaches to modeling turbulence under conditions of highly developed convection at high Rayleigh numbers. A number of industrially oriented problems with experimental data have been chosen for the study. It is shown that, at Rayleigh numbers from 10^{9} to 10^{17}, the application of the eddy-resolving LES model makes it possible to substantially increase the accuracy of modeling natural convection in comparison with the linear vortex viscosity model SST. This advantage is most pronounced for cases of a vertical temperature difference with the formation of a large zone of convection of strong intensity. The use of the Reynolds stress model EARSM is shown for cases of natural convective flow in domains with dihedral angles in the simulated region and the predominance of secondary currents. When simulating a less intense convective flow, when the temperature difference is reached at one boundary, the differences in the approaches used to model turbulence are less significant. It is shown that, with increasing values of Rayleigh numbers, errors in the determination of thermohydraulic characteristics increase and, for more accurate determination of them, it is expedient to use eddy-resolving approaches to the modeling of turbulence.

Flows of liquids with developed turbulent convection, characterized by a strong influence of gravitational forces, are of particular interest because this phenomenon still is poorly understood. The need to study such currents arises in many branches of science and technology, such as astrophysics, geophysics, geodynamics, atomic energy, and others. The most important tasks, where turbulent convection is necessary, include large-scale currents in the atmosphere and liquid cores of planets, the flow of liquid-metal coolants in reactor installations, convection in microgravity in spacecraft and fuel tanks, and other diverse flows in technical products. Particular interest in the study of turbulent convection is manifested in the nuclear energy industry. Along with the increase of technical and economic parameters of nuclear power plants (NPPs), the security of reactor facilities during severe accidents and minimizing their consequences have been of particular importance [

Experiments in nuclear power engineering, where the mechanism of convective heat transfer of a liquid in cavities with volume heating was studied at Rayleigh numbers of high orders of 10^{14} −10^{17}, include the COPO [^{7}). Interest in the study of turbulent convection is the high Rayleigh numbers. For technical applications, these are of orders 10^{14}–10^{17}. For compulsory convection problems, such as mixing high-speed multitemperature flows and jets characterized by strong buoyancy forces, numerical calculations using DNS are limited to small values of Reynolds numbers (of the order of 10^{5}), which leads to constraints on simulated flow regimes. Of interest in scientific and technical applications are problems with Reynolds numbers of the order of 10^{7} and higher. In most cases, to simulate the problems with such values of the Reynolds number an approach based on the solution of the Navier-Stokes equations averaged over the Reynolds numbers (RANS) is used, supplemented by one of the turbulence models to close the system [

The purpose of this paper is to numerically simulate the processes of highly developed convection of a fuel liquid at high Rayleigh numbers and to study the application of approaches to the description of turbulence for freely convective currents. The results of modeling based on RANS, RSM, and LES approaches are presented.

The first part of the paper presents the results of modeling a freely convective flow between two walls of different temperatures in a closed cubic cavity, which represents the most common class of natural convection problems. In the main part of the paper, we present the results of a numerical study of the problems of natural convection at high Rayleigh numbers using the example of simulation of the COPO and BALI experiments. Results of application of various approaches to modeling of turbulence in comparison with results of experimental researches are shown.

The modeling of turbulence is based on several basic approaches. RANS models of turbulence are currently the most popular; however, they are not universal and suitable for a wide range of applications, since they only describe the averaged characteristics, which imposes certain requirements on their applicability in practice [

The method of large eddy LES (Large Eddy Simulation) makes it possible to obtain good results for both attached and detached flows and to substantially refine the prediction of a number of basic physical processes [

All calculations in the present work were carried out with the help of the LOGOS package. This package is the Russian software product of engineering analysis intended for solving conjugate three-dimensional problems of convective heat and mass transfer, aerodynamics, and hydrodynamics on parallel computers. The LOGOS software package has successfully passed the verification and has shown quite good results on a series of various hydrodynamic problems [

Simulation of free-convective fluid flow in a cubic cavity between two different-temperature walls is of fundamental interest, since currents with a superimposed vertical temperature difference are characterized by the presence of different-scale vortex structures considering the main features of naturally convective flow and are widely used in engineering applications. The paper presents the results of numerical simulation of the circulation of distilled water in a domain heated from below and cooled from above. The basic research of this issue is carried out for convection in cylindrical and cubic cavities. Experimental studies of convection in a cubic cavity with a vertical temperature difference were performed in a wide range of Rayleigh numbers 10^{3}–10^{9}.

The experimental setup is a cubic region with a side ^{9}, which is determined by the following formula:

For calculations, an isotropic block-structured grid consisting of 3.375 million cells of regular hexahedral shape with a size of 0.0016 m was used. This grid model corresponds to the requirement of applicability of standard wall functions with parameter

A detailed and isotropic grid allows for RANS calculations to use a countercurrent scheme of the first order. For the LES calculation, a fully implicit scheme was used, increasing the accuracy of the calculation and allowing in a stable mode using a time step corresponding to the Courant number equal to unity. A description of the scheme is given in [

The nonstationary calculation was carried out up to the time point of 4200 seconds, which corresponds to the estimated time of the main characteristics of the physical fields in the experiment. The averaging of the basic physical fields was carried out from the instant of time of 100 seconds. Figure

The velocity field distribution: instantaneous velocity field, mm/s (a); averaged velocity field, mm/s (b) (experimental data are presented on the left panel, and calculations on the right panel).

The maximum value of the velocity in the picture of the experimental instantaneous field is reached near the walls; vortices appear at opposite angles of the region; the numerical field of instantaneous velocity represents a similar picture, describing small-sized vortices localized in the corners. In the experiment and in the calculation, the structure of the average velocity fields is very close. It is clearly seen that a large-scale vortex appears in the region commensurate with its size. In addition to the main vortex in the middle fields, there are also secondary vortices of smaller scale localized in opposite corners of the cubic region.

For qualitative and quantitative evaluation, Figure

Profiles of the time-average profile of the horizontal (a) and vertical (b) velocity components for a fixed coordinate

The velocity profiles obtained with the SST linear vortex viscosity model deviate from the experiment. The root-mean-square deviation for this model was about 6%. Differences in the calculated data can be related to the geometric feature of the region characterized by a large number of dihedral angles. Thus, in [

Using the LES model, the best agreement was obtained with the experimental data. The calculated profiles of the averaged velocity according to LES reproduce the general character of the convective heat transfer in a cubic cavity heated from below and cooled from above with a high accuracy; the error of the results was ~1.5%. Note here that the results for all models are obtained on the same calculation grid.

The numerical solution of this problem shows that the simulation of convective motion for the numbers Ra ~10^{9} over all models of turbulence yields an acceptable result. However, practically important currents for the safety of nuclear power plants are characterized by higher Rayleigh numbers, up to 10^{17}. The applicability of the presented approaches to turbulence modeling for such Rayleigh numbers can be estimated in the experiments of COPO and BALI, which are presented below.

In the COPO experiment, the problem of developed turbulent convection of an aqueous solution of salt in a closed region is considered. The COPO experiment simulates an emergency in a water-water power reactor (WWPR) at which the core is destroyed, which leads to the formation of a high-temperature fuel melt in the lower part of the reactor vessel [

The experiment was carried out at a scale of 1 : 2. In slice geometry, the installation length was 1.77 m, the thickness of the layer was 0.094, the large horizontal axis of the half-ellipse was

COPO setup scheme.

The liquid simulating the melt was a solution of salt in water. The volumetric heating of which provided distributed electrodes throughout the all domain. Vertical boundaries were thermally insulated. The lateral and upper boundaries were cooled by liquid nitrogen through the interlayer of the heat insulator, which led to the formation of a relatively thin crust of ice and provided isothermal boundary conditions. In one of the modes, the upper bound of the experimental setup was thermally insulated, which is analogous to the case in which the upper oxidized basin is covered in the melt localization device and the heat release is not large. For the remaining series of experiments, the upper bound was cooled similarly to a cylindrical one, which provided a constant temperature. This mode corresponds to the strongest heat sink from the upper boundary of the melt.

The main characteristics of the flow with internal heat sources are the parameters of the heat flux density at the boundary and the maximum excess of the liquid temperature in the volume above the boundary temperature. The dimensionless numbers characterizing these quantities are the Nusselt and Rayleigh criteria, respectively. For a heat-generating melt, a modified Rayleigh number is used:^{15}.

The indicator of effective heat transfer through the volume boundary is the Nusselt number:

The all thermophilically quantities are given by temperature-dependent (

In numerical experiments, we consider a mode with a heat-insulated top wall (mode 1) and also three regimes with different energy release and different height of the region (aspect ratio). Table

Parameters of experimental regimes.

Mode | Height of the area, m | Volumetric energy release, kW/m^{2} |
---|---|---|

1 | 0.847 | 46.5 |

2 | 0.639 | 153.2 |

3 | 0.73 | 161.5 |

4 | 0.847 | 104.3 |

The calculated region corresponded to the complete model of the experimental setup. A block-structured grid consisting of 2.5 million cells of a hexahedral shape was used and the cell size in the main region was 0.004 m thick, with concentration to all walls (the size of the wall cell was 0.0002 m). The grid model corresponds to the requirement of applicability of standard wall functions with parameter

A detailed and isotropic grid allows for RANS calculations to use a countercurrent scheme of the first order. The LES as scheme discretization in space and the equation of conservation of momentum was used with the BCD scheme calibrated for this scheme constant Smagorinsky. The BCD scheme is a scheme of limited central differences, where a diagram of normalized variable is used to formulate the restrictive condition, similarly to the GAMMA scheme [

The time for setting the heat-hydraulic process in the calculations is about 5000 seconds. The averaging of the basic physical fields was carried out from the time point of 2000 seconds. A series of nonstationary calculations was performed; the time step for RANS models was 1 s, for LES – 0.1 s. It has been experimentally confirmed that, for regime number 1 with a heat-insulated top wall, a stratified temperature distribution is typical [

Temperature distribution.

The results of calculating the experimental regime 4 with a cooled upper boundary are below. For this regime, the stratified temperature distribution is observed only in the lower part of the calculated region, and most of the upper region is occupied by the Rayleigh-Bener convection zone, where large vortex structures are observed. Figures

The temperature distribution in the central section of the region: instantaneous temperature (a) and average temperature (b).

The velocity distribution in the central section of the region: instantaneous velocity, m/s (a) and average velocity, m/s (b).

For the experimental mode number 1 in Figure

Heat flow distribution: mode with a heat-insulated upper boundary (a) and mode with a cooled one (b).

For the experimental regime with a cooled upper boundary (Figure

The main integral indicator for tasks with internal energy release and heat dissipation from boundaries is the distribution of the boundary heat transfer, characterized by the Nusselt number. Based on the results of recent experimental studies, the following dependence of the Nu number on the modified Ra number was obtained (the dependence was established for Ra ~10^{13}–10^{17} [

In Figure

Dependence of the Nusselt number from the modified Rayleigh number.

Analyzing the results, we can assume that a sufficiently large deviation is associated with a understating of the heat flux at the upper boundary, while at the cylindrical boundary the heat flux is overestimated. The heat dissipation from the upper boundary is less intensive: in comparison with the experimental data, on the elliptical boundary it is overestimated by an average of 30%. The maximum deviation from the experimental dependence reaches 33%; for the LES, the maximum deviation was 25%. Thus, the use of the LES approach refines the forecasting, but the error remains substantial, which may be due to the overestimation of the heat flow through the upper boundary.

In the framework of the COPO experiment, a convective flow (with Rayleigh number ~10^{15} [^{17}. The experiment that covers the region most important for the safety problems of nuclear reactors is the BALI experiment.

Experiments of BALI were carried out in order to investigate the convection of the fuel at a serious accident of the PWR reactor in the second half of the 1990s. In contrast to the COPO experiment, BALI experiments simulate the processes occurring during a severe accident in a reactor with a spherical bottom. Four series of experiments were performed under different conditions and configurations on a 1 : 1 scale. The general view of the experimental setup is shown in Figure

BALI installation scheme.

The calculated region corresponded to the complete model of the experimental setup. To simulate the BALI experiment, a block-structured grid was constructed, consisting of 4.2 million cells of a hexahedral shape, the cell size in the main region 0.006 m thick with concentration to all walls, and the size of the wall cell 0.0001 m. As a space-sampling scheme and in the pulse-conservation equation for the LES model, we used the BCD scheme with the Smagorinsky constant. The time step was chosen in such a way as to ensure the value of the Courant number of the order of unity. The thermophilically properties of the coolant used in modeling the experiments are given by temperature-dependent (

It should be noted that in [

The output parameters for the heat flux and temperature are given in a dimensionless form. The physical picture for the first calculation case, with the thermally insulated top wall, is similar to the COPO problem, the temperature field is clearly stratified, the liquid with the higher temperature (minimum density) is distributed at the top of the region, there is practically no disturbance in the flow, and the greatest value of the velocity arises near the cooled cylindrical wall.

In calculations of the current with the adiabatic upper boundary, volumetric energy release was varied, it was selected in such a way as to provide the following values of the Rayleigh criterion: ^{16} (^{16} (

The distribution of the time-averaged dimensionless temperature along the vertical boundary (a) and the dimensionless density of the heat flux at the cylindrical boundary (b).

The SST and EARSM models represented a similar result of the distribution of heat flow and temperature; the root-mean-square error for the heat flux is about 6%. The use of EARSM gives a slight advantage, reducing the error by 1%. The LES model refines the prediction; the rms error of the results is about 4%.

The results of the numerical simulation of the problem of BALI for the regime with a cooled upper boundary are below. Several flow zones are observed: the upper unstable layer of a homogeneous temperature, where the Rayleigh-Boehner convection zone occupies the most part; a lower zone with a temperature stratification at which the liquid rises to the center at a low velocity; boundary layer where the liquid in contact with ice cools and returned at a high speed to the bottom of the basin. A similar picture is observed in numerical simulation; a large part of the domain is occupied by a zone of large-scale convection where large vortex structures are localized; heat exchange occurs in the regime of highly developed turbulent convection. For this mode, Figure ^{16} for the LES turbulence simulation.

Instantaneous temperature field (a) and instantaneous velocity field (b).

The stratified temperature distribution is observed only in the lower part of the calculation area. The greater part of the upper domain is occupied by the convection zone of Rayleigh-Bener and it is much larger than the zone observed in the COPO simulation. This may be due to both the geometry of the installation and the increased value of the number Rayleigh.

The experiment contains data on the distribution of temperature and heat flux for the flow regime with cooling walls. The results are presented for the averaged density of the heat flux (Figure

The distribution of the time-averaged temperature along the vertical boundary (a) and the distribution of the heat flux density at the cylindrical boundary (b).

Analysis of the calculation results shows that the RANS models underestimate the heat flow in the upper part of the calculation area. The LES results are closer to the experiment. The root-mean-square deviation of the EARSM and SST results is 11% and 13%, respectively. The LES results are 6%. The results using the SST and EARSM models correlated, giving approximately the same result, but the EARSM model lies closer to the experimental data.

Figure ^{15}, ^{16}, and ^{17} in comparison with the obtained Nusselt number dependence on the modified Rayleigh number (

Dependencies of the number Nu on the modified number Ra.

The maximum error for the Nusselt number at the upper boundary was about 36%, the vortex-resolving LES Smagorinsky model refines the prediction, and the maximum error using this model is about 33% in the case of the maximum Rayleigh number. Numerical modeling qualitatively corresponds to the physical picture of natural convection. Due to the high-temperature difference, large vortex flows occur at the upper wall. In the lower part of the basin, there is a stable zone of temperature stratification. This picture corresponds to the flow in the experimental setup. The distribution of the dimensionless temperature at the vertical boundary is described with sufficient accuracy, with the greatest deviations being observed in the central area. The calculated density distribution of the dimensionless heat flux at the cooled upper boundary is described with some deviation from the experimental data.

Numerical simulation of the processes of turbulent natural convection is demonstrated for different values of the Rayleigh number. The application of different approaches to the description of turbulence is provided. Analysis of the experimental data and the results of calculations show that the qualitative picture of the flow with the values of the Rayleigh criterion to 10^{17} is modeled correctly. In the case that modeling convective heat transfer in a cubic cavity at

It is shown that the distribution of the dimensionless temperature and the heat flux density along the boundaries does not depend on the turbulence model and is not dependent on Rayleigh criterion. This is in good agreement with the experimental data, but the quantitative characteristics for some regimes have significant differences. For experiments with a heat-generating liquid (COPO and BALI) with a heat-insulated upper boundary, the effect of applying approaches to turbulence modeling is not particularly significant. This may be due to a less intense internal flow, since heat exchange occurs with only one boundary. Within the framework of these regimes, the results of numerical simulation are in good agreement with the results of experimental studies. For problems with two cooled walls, where there is intense circulation of liquid in the closed region, the value of the heat flux at the lower boundary is underestimated and the heat transfer at the upper boundary is increased. In various sources [

The LOGOS software at a sufficiently high level reproduces the heat exchange in the heat-generating fluid with natural convection under conditions of high Rayleigh numbers

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This study was initiated in the framework of the state task programme in the sphere of scientific activity of the Ministry of Education and Science of the Russian Federation (Projects no. 5.4568.2017/6.7 and no. 13.8823.2017/8.9), the President of the Russian Federation Grant NSh-6637.2016.5, and the Russian Foundation for Basic Research Grant no. 16-31-00080_mol_a.

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