Modelling and Simulation in Engineering

Volume 2018 (2018), Article ID 5781602, 12 pages

https://doi.org/10.1155/2018/5781602

## Simulation of Turbulent Convection at High Rayleigh Numbers

^{1}Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia^{2}Russian Federal Nuclear Center-All-Russia Institute of Experimental Physics, Sarov, Russia^{3}Shipunov Instrument Design Bureau, Tula 300001, Russia^{4}Institute of Space Technologies, Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia

Correspondence should be addressed to Andrey Kurkin

Received 19 September 2017; Accepted 12 November 2017; Published 9 January 2018

Academic Editor: Azah Mohamed

Copyright © 2018 Sergey Dmitriev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1]. In severe accidents accompanied by loss of the coolant, the reactor core is destroyed. As a result, the molten elements of the reactor and reactor core structure are moved to the bottom of the reactor vessel, which results in the formation of a high-temperature fuel melt in its lower part, where it must be localized. The measure to ensure the safety of the reactor facility is to ensure the necessary heat removal from the boundaries of the melt localization device, where heat exchange takes place in the natural convection mode.

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 [2–5] and BALI [6, 7]. The complexity of experimental studies of this class of currents actualizes the development of numerical simulation. Since in the course of experimental studies the pronounced turbulent nature of the flow is confirmed, one of the main problems in the numerical modeling of these flows is turbulence. The question of the influence of turbulence on the nature of the flow and the magnitude of the boundary heat transfer is of interest. The existing level of computing power allows modeling convection, at an acceptable time, using the Reynolds-averaged Navier-Stokes equations (RANS approach) [8, 9]. This imposes limitations on the description of flow turbulence, for both natural and forced convection, due to the presence of gravitational forces modeled in the RANS approach by empirical relationships, which leads to a significant error in describing the process as a whole. This problem can be solved by using direct numerical simulation (DNS). However, the currently conducted numerical experiments using DNS for natural convection problems are limited to modest Rayleigh numbers [10] (maximum 10^{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 [11]; however, in [12], it is noted that the standard model of RANS models is not always applicable for solving problems of turbulent convection. Numerical modeling of the BALI experiment for the purpose of verification and calibration of various models of turbulence was carried out in [13]. It was shown in [14] that the use AKN turbulence model (the AKN model is described in [15–17]) improves the prediction of the heat flux distribution. Modeling using the standard - model of turbulence shows that its application leads to an underestimated heat flux distribution. At the same time, an increase in the accuracy and informativeness of the results for the problems of natural convection can be achieved by using eddy-resolving turbulence models, for example, LES, the successful application of which can be found in [8, 9].

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.

#### 2. Approaches to Modeling Turbulence

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 [8, 18]. The highest rating of applicability from RANS models is model SST (Shear Stress Transport) [7]. However, as in all RANS models, SST uses the Boussinesq hypothesis on turbulent viscosity [19], which is valid only in the case of isotropic turbulence. For turbulent convection, the anisotropic properties of the flow have a significant effect; this is due to the complex nature of the flow and the presence of secondary currents. To correctly predict the structure of such flows, it is necessary to use RSM models (Reynolds Stress Modeling) or alternative EARSMs (Explicit Algebraic Reynolds Stress Modeling) that take into account the influence on the main flow of all components of the Reynolds stress tensor with the help of nonlinear relationships and thus can increase the accuracy of turbulent calculations, anisotropic flows [19]. In this paper, as an alternative to the SST model, an explicit algebraic EARSM model is considered, which by its qualities is not inferior to differential RSM and allows significantly reducing computational costs [20].

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 [8, 9, 21]. This approach can provide detailed information about nonstationary fields of fluctuations of velocity, temperature, and pressure, which in turn can influence the magnitude and nature of the boundary heat transfer. However, the LES method imposes great demands on the quality of discrete models and significantly increases, in comparison with RANS, the amount of required computational resources. Thus, for convective currents, there arises the question of studying the structure of the turbulent flow and the application of eddy-resolving turbulence models for establishing the connection between small-scale oscillations and heat transfer in the zone under consideration.

#### 3. Numerical Experiments

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 [8, 9, 17, 22, 23], including calculations of turbulent and nonstationary flows [20, 21] and geophysical phenomena [24–26].

##### 3.1. The Problem of Developed Turbulent Convection

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 = 250 mm [10]. The horizontal walls are made of copper and they act as heat exchangers, and the vertical walls are made of plexiglass and provide isothermal boundary conditions. With the help of thermostats, a thermostatic fluid is passed through the heat exchangers, which heats (cools) the heat exchangers with respect to the average temperature of the liquid. The experiment is described in detail in [10]. In this paper we consider a flow characterized by a difference of temperature of 20°C, which corresponds to the Rayleigh number 6,1 × 10^{9}, which is determined by the following formula:where is the acceleration due to gravity, is the thermal expansion coefficient, is the characteristic dimension, is the thermal conductivity coefficient, is the kinematic viscosity, and is the temperature difference. Water is used as a coolant, the properties of which are expressed in the form of dependencies [27]:

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 + no higher than 5 and also allows small-scale turbulence to be resolved within the LES calculation.

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 [9, 21].

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 1 shows calculated and experimental fields of instantaneous and averaged velocity in the central section of the experimental setup.