## 3 Results and Discussion
### Buoyancy Effect on the Heat Transfer  
Although there has been extensive research on the convective heat transfer of supercritical fluids in different channels under both heating and cooling conditions, a clear understanding of the special convective heat transfer of supercritical fluids at low mass fluxes is still missing in the literature. Further investigation on the mixed convective heat transfer in a supercritical fluid at low mass fluxes is undoubtedly necessary, and could improve the understanding of this heat transfer mechanism. Figure 11 shows the qualitative analysis of forces in the near-wall fluid for the vertical flow. For fluids with constant thermo-physical properties, the fluid in the vicinity of wall achieves a balance under the combined effect of gravity, shear stress, and driving force (F\({}_{1}\) = mg + \(\tau_{1}\), F\({}_{2}\) = mg + \(\tau_{12}\)). However, for strongly-varying density fluids, the buoyancy induced by the density difference between the central domain and the near-wall region may enhance and promote the mixing of fluid, and thus, intensify the mass transfer in the heated pipe. As can be seen from Figure 11, the direction of the resultant force was not perpendicular to the wall, and exhibited a certain angle between the flow direction and the wall. This was because the high-density fluid was always located in the central domain. The deflection angle was related to the magnitude of the heat fluxes. For low heat fluxes, there was not much density difference between the fluid in the near-wall region and the fluid in the central domain, while the direction of the resultant force almost became perpendicular to the wall. Additionally, the effect of buoyancy on the heat transfer induced by a small density gradient can be neglected. However, for high heat fluxes, the direction of buoyancy was similar to the flow direction as  
Figure 11: Force analysis of near-wall fluid in the vertical tube. the large density gradient fluid changed position. Since the buoyancy was very strong at high heat flux (as seen in Figure 6b), natural convection was far higher than that of forced convection. Thus, the fluid in the near-wall region was hard to detach from the wall, and entered the mainstream area in the small channel. At the same time, the fluid close to the solid wall accelerated along the flow direction due to the drop in density, and the shear stress in the near-wall region reduced. Under the effects of buoyancy and thermal acceleration, the fluid with low density and low conductivity adhered to the near-wall region; thus, the heat transfer behavior in small heating channels at low mass flux was similar to \"film boiling\", where the sublayer was occupied by a \"vapor\", and the low mass flux did not cause enough mixing/instability to bring it to the bulk flow. The \"vapor film\" did get disturbed along the channel, leading to some decrease in HTC. The abovementioned modeling analysis can be verified using the results presented in Figure 12. Figure 12 shows the variations in buoyancy (\(Bo^{\bullet}\)) with the increase in bulk enthalpy at different heat fluxes for the pressure and mass flux of 7.5 MPa and 200 kg/m\({}^{2}\)s.  
At a rather low heat flux (\(q<14.1\) kW/m\({}^{2}\)), which corresponded to a small temperature difference (\(t_{w}-t_{b}\)), the effect of thermal acceleration and buoyancy forces can be neglected (\(Bo^{\ast}<1.2\times 10^{-6}\)) [39]. However, for a moderate or high heat flux, the manifestation of buoyancy forces and thermal acceleration was coupled with the density change that became more intense with the increase in heat flux [40]. However, the intensive buoyancy only took effect in the low-enthalpy region (\(H<300\) kJ/kg), which was accompanied by a strong variation in physical properties. As the bulk enthalpy exceeded the value of 300 kJ/kg, the buoyancy quickly decreased and reached the minimum value.