Effects of Finite Wall Conductivity on Flow Structures in Natural Convection

Natural convection in a box is driven by the thermal boundary conditions at the active walls and is profoundly affected by the thermal conditions at the passive walls. The effects of these conditions were studied numerically and compared with experiments for natural convection in two cavities. The results indicate the importance of these conditions for proper modelling of three-dimensional flow structures, and also their inevitable influence on the Nusselt number. INTRODUCTION In the past, computational limitations have enforced several simplifications in the numerical modelling of thermally driven flows. One of the common misconceptions usually found is related to the thermal boundary conditions at the so called “side walls” of a box in which natural convection is occurring, i.e. walls which are not active in generating the flow, but simply play the role of a boundary for the flow domain. To simplify the problem, either adiabatic or perfectly conducting walls can be specified, or a specified heat flux can be imposed. However, this approach in many cases leads to solutions which are only approximately similar to the observed physical situations. On the one hand, this creates doubts about the quality of the numerical solutions, and on the other, is a serious obstacle in using experimental results for code validation and improvement. Our previous investigations (Hiller et al. 1989,1990) have shown that gentle natural convection flows can be extraordinarily sensitive to small changes in the thermal boundary conditions. This is reflected in the results of numerical simulations, which show a similar sensitivity, and it is not easy to formulate proper mathematical boundary conditions to mimic those found in the laboratory. Having this in mind, in this paper we describe our numerical and experimental attempts to understand and properly describe changes appearing in the flow structure due to the finite conductivity of both passive [nominally adiabatic] and active [heated or cooled] walls. It is worth noting that these are mainly three-dimensional variations of the flow patterns (a fact which perhaps partly excuses those who use classical two-dimensional programmes). PROBLEM FORMULATION We consider natural convection in two configurations: a horizontal temperature gradient in a side wall heated cavity, and a vertical temperature gradient in a lid cooled cavity. The first configuration comprises low Rayleigh number natural convection in a cubical cavity with differentially heated active side walls. Two opposite vertical walls were isothermal and kept at temperatures Th and Tc; the other four walls were passive: nominally insulators of finite thermal diffusivity. A heat flux, both through and along the walls, was generated due to temperature gradients existing between the fluid inside the cavity and the surrounding environment and also along the front and back walls, the lid and the floor of the box. In the second configuration, the top wall of the cube was isothermal at a low temperature Tc. The other five walls were non-adiabatic, allowing a heat flux to cross from the external fluid surrounding the box. The temperature Th of the external bath was kept constant. Due to forced convection in the bath it could be assumed that the temperature at the external surfaces of the box was close to the bath temperature. The temperature field at the inner surfaces of the walls adjusted itself depending on both the flow inside the box and the heat flux through and along the walls. Both configurations were used here to investigate the convective flow without phase change (and have previously been used to study the freezing of water at the cold wall, Kowalewski & Rebow 1998). Experimental Set-up A typical experimental set-up used to acquire temperature and velocity fields consisted of the convection box, a xenon flash or halogen tube lamp, and a CCD colour camera. Most of the experiments described here were performed using a cube-shaped cavity of 38 mm inner dimension. Either the two active vertical walls, or the active top wall were made of a black anodised metal to maintain isothermal boundary conditions. The remaining walls were made of 6 mm Plexiglas or 2 mm glass. The temperature of the isothermal walls was controlled by thermostats. As flow media, pure glycerine, its aqueous solutions and pure water were used. By varying the liquid composition and the temperature difference ∆T = Th Tc, it was possible to cover a relatively wide range of Rayleigh and Prandtl numbers (Ra = 2.10 3.10, Pr = 7-6900). The flow was observed at the vertical and horizontal cross sections of the cavity using a light sheet technique. The details of the experimental setup have been given elsewhere (Kowalewski et al. 1997, 1998). Thermochromic liquid crystals (TLC) suspended in the working fluid were used both as flow tracers and temperature indicators. The computational analysis of the colour and displacement of the liquid crystal tracers allows us to determine both the temperature and velocity fields of the flow. It combines Digital Particle Image Thermometry (DPIT) and Digital Particle Image Velocimetry (DPIV) (Kowalewski et al. 1998). To obtain a general view of the flow pattern, several images recorded periodically within a given time interval were added in the computer memory, generating particle tracks. Numerical method A numerical simulation of the problem was performed using a three-dimensional finite difference vorticity-vector potential formulation of the Navier-Stokes and energy equations for laminar flow of a viscous, incompressible fluid. Solutions were obtained for Cartesian coordinates with the origin placed at a lower corner of the box. The xaxis is horizontal, the y-axis points upward and the zaxis is perpendicular to the active vertical plane. Modified versions of the FRECON3V (Timchenko et al. 1997) false transient solver have been used for the analysis of steady convection. To study transient convection with phase change a modified version of the code FREEZE3D (Yeoh 1993) was used. The computational models were adapted to simulate as closely as possible the physical experiment. The main problem which arises in the simulation of experimental conditions is the proper definition of thermal boundary conditions (TBC). Either two opposite vertical walls or the horizontal top wall were assumed to be isothermal. Two approaches have been used to apply TBCs to the remaining walls, which are in reality neither adiabatic nor isothermal. In gthe first approach, TBCs were estimated by using heat transfer theory applied to a thick, infinitely wide plane plate of uniform conductivity exposed to an external constant temperature environment. In this one-dimensional approach, an arbitrary fixed temperature, a specific heat flux or a specific heat transfer coefficient on each of the six surfaces of the box wasimposed in the calculations. The general form of non-dimensional conditions used for the temperature θ at the boundary is: