# Numerical Study of Mixed Convection and Heat Transfer in Arc-Shaped Cavity with Inner Heat Sources
## 3 Results and Discussion
### Heat Transfer  
In order to study the dependence of heat transfer on Re in the arc cavity and near the heat source, the convective heat flux \(q_{cv}\), conductive heat flux \(q_{cd}\) and total thermal heat flux \(q_{t}\) in the cavity and the Nusselt number on heat source wall surface are monitored [29], in view of the phenomenon according to which the temperature in the middle region of the two heat sources changes regularly under the influence of Re. We monitor the average temperature \(T_{m}\) in the middle region of the two heat sources at the full development stage and introduce the effective heat coefficient \(m\) to evaluate the effect of convective heat dissipation in the arc-shaped cavity under different Re. Monitoring variables are defined as follows:  
Figure 6: Fully developed stage of isotherms and streamlines in the arc-shaped cavities for different Re. Heat flux:  
\[q_{t}=\frac{1}{s}{\iint_{s}}\bigg{(}vT-\frac{\partial T}{\partial y}\bigg{)}ds \tag{12}\]  
\[q_{cv}=\frac{1}{s}{\iint_{s}}(vT)ds \tag{13}\]  
\[q_{cd}=\frac{1}{s}{\iint_{s}}\bigg{(}-\frac{\partial T}{\partial y}\bigg{)}ds \tag{14}\]  
Effective heat coefficient:  
\[m=\frac{T_{n}-T_{c}}{T_{p}-T_{c}} \tag{15}\]  
Figure 7 shows the variation of the average heat flux versus Re in the curved cavity with two heat sources. The total heat flux \(q_{t}\) and convection heat flux \(q_{cv}\) are essentially similar, and the magnitude of conduction heat flux \(q_{cd}\) is relatively small. The heat transfer in the arc cavity is controlled by convection heat transfer. When Re is small, \(q_{cv}\) and \(q_{cd}\) are of the same order of magnitude. When Re is greater than 2.0 \(\times\) 10\({}^{6}\), there is a linear relationship, \(q_{t}\)\(\sim\)4.9 Re, \(q_{cv}\)\(\sim\)4.89 Re, \(q_{cd}\)\(\sim\)0.001 Re. It can be seen that the increase in Re can increase the convective heat transfer in the curved cavity structure, while the heat conduction only increases slightly. The gap between heat convection and heat conduction increases with the increase in Re, which is similar to the phenomenon observed in the cavity with a single heat source. In actual industrial systems, heat transfer near the heat source deserves more attention. Therefore, Figure 8 shows the variation of Nu on the heat source wall with Re. As Re is less than 1.75 \(\times\) 10\({}^{6}\), Nu gradually increases to reach the peak. When Re ranges from 1.75 \(\times\) 10\({}^{6}\) to 6.14 \(\times\) 10\({}^{6}\), Nu\(\sim\)\(-\)0.0059 Re. As Re ranges from 6.15 \(\times\) 10\({}^{6}\) to 1.1 \(\times\) 10\({}^{7}\), Nu\(\sim\)\(-\)0.017 Re. As Re continued to increase, with Re in the range of 1.31 \(\times\) 107 to 4.38 \(\times\) 107, Nu\(\sim\)\(-\)0.001 Re. Figure 7: The values of various heat fluxes in curved cavity under different Re numbers and their fitting curves. In the range of Re from 6.14 \(\times\) 10\({}^{6}\) to 6.15 \(\times\) 10\({}^{6}\), the decreasing rate of Nu increases with Re, which corresponds to the flow state transformation described in Section 3.1. The flow changes from weak turbulence to periodic flow, the flow disturbance decreases, the convection is not so intense as before and the heat transfer effect of the heat source decreases. In the range of Re from 1.14 \(\times\) 10\({}^{7}\) to 1.18 \(\times\) 10\({}^{7}\), a small temporary rise of Nu appears. After this stage, Nu still maintains a linear decline, but the rate of declination is lower than before. This corresponds to the phenomenon that the temperature in the middle region of the two heat sources experiences a sudden rise with Re between 1.14 \(\times\) 10\({}^{7}\) and 1.18 \(\times\) 10\({}^{7}\), as reflected in Figure 6.  
Figure 9 shows the variation trend of the average temperature \(T_{m}\) in the middle area of the two heat sources with Re. It can be divided into three ranges, as shown in Figure 9. When Re is less than 6.14 \(\times\) 10\({}^{6}\), \(T_{m}\) presents a quasi-power function decline with a gradual decline rate: \(T_{m}\)\(\sim\)0.38\(\rm{Re}^{-0.64}\).