TWO-DIMENSIONAL PROBLEM OF TURBULENT NATURAL CONVECTION IN A SEMI-OPEN CAVITY WITH RADIANT HEATING OF INTERNAL BOUNDARIES

Mathematical modelling of turbulent natural convection in a semi-open cavity with a heat-conducting walls of finite thickness with radiant heating of internal boundaries was performed. Two-dimensional problem of the conjugate heat transfer was solved by means of the finite difference method. Scale influence of open boundaries and radiant heating of the gas – wall interfaces on the formation of differential and integral heat transfer characteristics was established. An increase in the dimensionless time ( ) led to displacement of extremum temperatures in the typical cross section (Y = 0.5) to an open vertical boundary. The average Nusselt number monotonically increased at the gas a wall interfaces in a range of 40


Introduction
It is known [1] that the convective heating of air masses is appropriate only for heating of small premises. In the case of large industrial buildings, where only a small part of the useful area (local working areas) is used prospectively to apply radiant heating systems based on gas infrared emitters (GIE) [2,3]. However, the use of radiant heating systems is currently not widespread due to the lack of a basic theory of the processes occurring when operating the GIE. For this reason, it is advisable to investigate the basic laws of heat transfer under conditions of radiant energy supply to the heating facilities. To solve these complex problems of heat transfer is prospectively to use the methods of mathematical modelling based on the solution of mass, momentum and energy transfer equations in the conjugate formulation.
It should be noted that the main specificity of the heat transfer process in the large-scale areas is turbulent flow regime. At the same time, the working zones of industrial facilities are locally distributed along the useful building area. In these cases, thermal regime modelling of locally based working area is of interest.
The purpose of this study is mathematical modelling of turbulent natural convection in a semi-open cavity under conditions of conjugate heat exchange and radiant heating of the internal boundaries.

Problem formulation and solution method
Two-dimensional problem of turbulent natural convection was considered. Solution domain was presented as a cavity of rectangular cross section with an open vertical boundary. It was assumed that the radiant flux was uniform distributed along the bottom horizontal and vertical gas -wall interfaces. Turbulent flow was calculated by means of the large eddy simulation method [4]. In order to calculate the subgrid scale viscosity, I used a classical Smagorinsky model [5]. It was assumed that the thermal characteristics of the air and the walls were temperature-independent. Conditions of temperature and heat flux equalities were set at the gas -wall interfaces. Thermal insulation conditions were accepted at the external boundaries. I considered the model of a viscous gas in the Boussinesq approximation. "Mild" condition was set for the open boundary. The turbulent natural convection and conjugate heat transfer process under study is described by unsteady two-dimensional Navier -Stokes and energy equations. These equations in vorticity -stream function -temperature dimensionless variables are as follows [4]: Subgrid scale eddy viscosity was calculated as in [5]: The initial conditions for the equations (1) -(4) are as follows: 4 Boundary conditions for equations (1) -(4) are as follows: at the external boundary of the solution domain: w at the gas -wall interfaces parallel to the X -axis: at the gas -wall interfaces parallel to the Y -axis: at the open boundary: Formulated boundary value problem was solved by means of the finite difference method analogically [6,7]. Implicit two-layer scheme [8] was applied for equations (1) -(4) approximation. One-dimensional finite-difference analogues of differential equations were solved by the sweep method [9].
In order to validate mathematical model, solution method and algorithm used were tested on a model problem of turbulent natural convection [10,11] in closed square cavity. Comparison of obtained temperature fields and average Nusselt numbers with [10,11] showed their satisfactory agreement.

Results and discussion
Investigations of conjugate natural convection regimes were conducted for the following dimensionless criteria corresponding to the turbulent flow:  W did not lead to significant modification of the differential heat transfer characteristics in geometrical and physical conditions under consideration. Fig. 3 shows the temperature distribution in the gas cavity in the Y = 0.5 section. It was clearly seen that an increase in dimensionless time led to a rise in the temperature in the typical cross-section of Y=0.5. Extremum of temperature was shifted to the open boundary, which was obviously due to the air flow pattern in this area ( fig. 2 b, d, f).
In order to estimate the heat transfer rate, integral heat exchange characteristics were calculated. Dimensionless heat exchange coefficients at the top (  Figure 4 shows the dependence of the average integral Nusselt numbers versus dimensionless time. Nu ) gas -wall interfaces had a nonlinear dependence before 300 W , which was connected with the radiant energy supply to these interfaces. The air layer was formed with an increase in temperature in the sections of X = 0.1 and Y = 0.1 ( fig. 2 a,  , the average dimensionless heat exchange coefficients at the bottom horizontal and vertical gas -a wall interfaces increased, which was connected with the formation of convective plumes as the result of natural convection. The average integral criteria av Nu monotonically increased in a range of 400 1200 W , which was obviously due to a rise in the circulation gas flow rate. At the same time, the thermophysical process under study gradually entered the quasi-stationary mode. The average temperature increased in the solution domain, but the shape of the isotherms did not substantially modify.

Conclusion
An approach for the modelling of turbulent natural convection in the semi -open cavity under conditions of the radiant heating of gas -wall interfaces was suggested. According to the numerical simulation results, scale influence of open vertical boundary and radiant heating of the gas -wall interfaces on the formation of differential and integral heat transfer characteristics was established. The formulated boundary value problem allows to decrease the spent of computing resources by reducing the modelling area size when the analyzing the thermal regimes of systems heated by infrared emitters.