Numerical Study of Natural Convection-Surface Radiation Coupling in Air-Filled Square Cavity with Curved Wall Using the Gauss’s Theorem on Triangular Meshes

Abstract

A numerical code is developed for the solution of the two-dimensional, time-dependent coupling natural convection- surface radiation equations in square cavity with curved wall. The cavity consists of one curved wall and three straight walls while the two vertical walls are kept isothermal, the cold one is curved (right) and the two horizontal walls are considered adiabatic. The computations are performed for an air- filled cavity, whose four walls have the same emissivity. The governing equations are formulated in Helmholtz variables (ψ, ω). The current method was developed for being used on grid made up of triangles. The basic principle of this method is the Gauss’s theorem (integrals over a closed line around an area). The results are presented in the form of main convective and radiative Nusselt numbers distributions for various Rayleigh number, emissivity and amplitude value of curve, and for Prandtl number equal to 0.71 and finally discussed. Streamlines and isothermal lines are also presented. The effect of emissivity and amplitude value of curve on the temperature, net radiative flux and velocity profiles has been analyzed. The influences of emissivity in a square cavity with outwards curve are the same in a square cavity; with the increase of the emissivity, the net radiative flux will increase at the top horizontal wall and will decrease at the bottom, this means that it cools down the top horizontal wall and heats up the bottom. Hence the convective Nusselt numbers are decreased at the active walls and the horizontal velocity is increased near the adiabatic walls. On the contrary with the increase of the inwards curve, the net radiative flux will increase on the curved wall and will decreases on all the other walls; this leads to heat up both the horizontal walls and to increase their temperature respect to the emissivity that results smaller. For consequence, at the hot wall, the convective Nusselt number is decreased with the increasing of emissivity while at the cold wall the Nusselt Number will be increased; further the horizontal velocity is decreased near the top horizontal wall and it is increased near the bottom.