Th Tc Insulated
Insulated g
H u= v = 0
u= v = 0
u= v = 0 u= v = 0
(a) (b)
Figure 6.17: Schematic of the computational domain. (a) Three-dimensional differ- entially heated cavity. (b) Two-dimensional differentially heated cavity.
The temperature variation for pure convection case as seen from Fig. 6.18 (a) de- picts two-dimensional nature with the growth of boundary layers both hydrodynamic and thermal along the vertical isothermal walls with thermal stratification along the horizontal midplane at the center core of the cavity. With the inclusion of thermal radiation, the two-dimensional nature of flow and heat transfer is completely dis- torted and strong three-dimensionality can easily be perceived from the iso-surface of temperature as seen from Fig. 6.18 (b-d). The inclusion of radiation leads to the increased mean temperature inside the enclosure as evident from the clustering of iso-surface of temperature near the cold vertical wall. The increased temperature levels due to radiation reduce the temperature gradients near the walls and hence decreases the convection currents which leads to the formation of thicker boundary layers as compared to a pure convection case. The influence of gas radiation causes an increased heating of the top wall due to the combined influence of buoyancy and the role of radiation in elevating the mean temperatures. The inclusion of surface and gas radiation causes the entire cavity to be thermally active as regions of cold fluid are limited only to the bottom of the cold wall which influences the flow and heat transfer immensely.
6.5 The Importance of Three-Dimensional Non-Boussinesq Analysis for Convective
Radiative Heat Transfer 161
(a) Isosurface of temperature (b)
(c) Isosurface of temperature (d)
Figure 6.18: Isosurface of temperature for natural convection flow with and without the influence of thermal radiation. (a) pure convection, (b) combined convection with surface radiation, (c) combined convection with gas radiation of optical thick- nessτ = 0.2, (d) combined convection with gas radiation of optical thicknessτ = 5.0.
As the optical thickness of the medium increases, a major part of the radiation is absorbed by the participating medium and as a consequence, the temperature lev-
10 10 15
15 20
20 25 30
Z
Y
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Quasi-Incompressible Incompressible
6 6
6
5 5
4 3
3 2
2
3 6
Z
Y
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Quasi-Incompressible Incompressible
Y NuC
0.2 0.4 0.6 0.8
5 10 15 20 25 30 35
Quasi-Incompressible Incompressible
(a) (b) (e)
4747 43
43 39 39 3535 31
Z
Y
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
56 56
52 52 48 44 40 36
Z
Y
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Y NuR
0 0.2 0.4 0.6 0.8 1
0 5 10 15 20 25 30 35 40
Quasi-Incompressible Incompressible
(c) (d) (f)
Figure 6.19: Local variation of convective and radiative Nusselt number at the hot and cold wall for three-dimensional (a, b, c, d) and two-dimensional (e, f) combined convection with gas radiation of optical thickness τ = 0.2 using incompressible and quasi-incompressible low-Mach number model respectively: (a) convective Nusselt number variation at the cold wall, (b) convective Nusselt number variation at the hot wall, (c) radiative Nusselt number variation at the cold wall, (d) radiative Nusselt number variation at the hot wall, (e) convective Nusselt number variation at the hot and cold walls, (f) radiative Nusselt number variation at the hot and cold walls.
els are further elevated as seen in Fig. 6.18 (c). The participating gas of optical thickness τ = 5 attenuates most of the radiation as a consequence the flow and temperature distribution depicts strong two-dimensional behavior with very little three-dimensionality observed near the adiabatic walls. The isotherms in case of combined convection with radiation case depict sharp bending near the adiabatic walls. The curved iso-surface of temperature near the adiabatic top and bottom
6.5 The Importance of Three-Dimensional Non-Boussinesq Analysis for Convective
Radiative Heat Transfer 163
walls follow opposite curvature due to the combined effect of convection and radi- ation. The curved iso-surface of opposite curvature depicts enhanced heating and cooling of the top and bottom wall respectively due to convection. The influence of surface and gas radiation on overall heat transfer is realized from the average convective and radiative Nusselt number variation at the isothermal hot and cold walls from Table 6.7. The Nusselt number reveals that the symmetric nature of heat transfer in case of pure convection turns asymmetric with the inclusion of ra- diation. The asymmetric nature of heat transfer with the inclusion of surface and gas radiation is easily realized from the enhanced convective heat transfer at the cold wall in comparison to the hot wall. Large convective heat transfer at the cold wall is due to the enhanced temperature levels as a consequence of radiation to equalize the temperature. Moreover, the radiative Nusselt number reveal enhanced heat transfer due to radiation at the hot wall. Similar results are also obtained for the local variation of convective and radiative Nusselt number as seen from Fig. 6.19.
Figure. 6.19 (a, b) represents the local variation of the convective Nusselt number at the cold and hot walls respectively. It can be easily seen that large convection heat transfer rates are obtained at the top and base of cold and hot walls due to the interaction of the hot rising and cold descending fluid with the cold and hot walls respectively. Figures 6.19 (c, d) represents the local variation of the radiative Nusselt number at the cold and hot walls respectively, the variations signify larger radiative heat transfer rates at the hot wall in comparison to the cold wall. En- hanced heat transfer at the hot wall due to radiation is due to the attenuation of radiation by the absorbing medium. Similar variations can also be observed from the two-dimensional simulation of the same case as shown in Fig. 6.19 (e, f).
On comparing the average and local Nusselt number variation obtained using the incompressible and quasi-incompressible low-Mach number model in Table 6.7 and Fig. 6.19 it is seen that considerable deviations are obtained. The deviations are remarkably high for convective heat transfer, specifically at the cold wall. The max- imum deviation of 18.5% in the convective heat transfer is obtained at the cold wall for a combined convection with gas radiation of optical thickness τ = 0.2 case. The maximum variation in the radiative Nusselt number of 5.29% is obtained at the cold wall for combined convection with gas radiation of optical thickness τ = 0.2 case.
Furthermore, Table 6.8 represents the deviation in the results obtained between
Table 6.7: Average convective and radiative Nusselt number at the isothermal hot and cold walls using incompressible and LMN model for pure convection and con- vection with surface and gas radiation.
Geometry Hot Wall Cold Wall
Model NuC NuR NuC NuR
pure convection
Incompressible 8.683 NA 8.683 NA
Quasi-Incompressible (LMN) 9.442 NA 9.422 NA
Percentage deviation 8.03% NA 8.03% NA
Convection with surface radiation
Incompressible 4.757 49.218 11.633 42.342 Quasi-Incompressible (LMN) 4.546 50.620 14.096 41.070 Percentage deviation 4.64 % 2.75% 17.4% 3.09%
Convection with gas radiation τ = 0.2 Incompressible 3.943 47.430 12.353 39.020 Quasi-Incompressible (LMN) 3.798 48.953 15.171 37.580 Percentage deviation 3.81% 3.11% 18.5% 3.83%
Convection with gas radiation τ = 5.0 Incompressible 4.031 25.568 13.760 15.840 Quasi-Incompressible (LMN) 4.058 26.650 15.662 15.044 Percentage deviation 0.6% 4.06% 12.14% 5.29%
an incompressible and quasi-incompressible model for a combined convection with surface radiation problem using a two-dimensional and three-dimensional analysis.
The results from Table 6.8 reveal that the deviation between the incompressible and quasi-incompressible model increases in case of three-dimensional simulation in comparison to the two-dimensional approximation. In case of two-dimensional simulations, it is shown that the quasi-incompressibility effects, influence the con-
6.6 Closure 165