# Numerical Study of Mixed Convection and Heat Transfer in Arc-Shaped Cavity with Inner Heat Sources
## 3 Results and Discussion
### Transient Characteristics  
As it is discussed in Figure 3, with the increase in Re, the flow in the arc cavity changes from weak turbulence to periodic and then to steady normal. When Re = 7.67 \(\times\) 10\({}^{6}\), a periodic oscillation of the flow can be observed. In order to quantitatively observe this transformation and obtain the critical Reynold numbers, this study involves a large range of Re from 2.19 \(\times\) 10\({}^{5}\) to 4.38 \(\times\) 10\({}^{7}\). Figure 4 shows the time evolution and frequency spectrum analysis results at monitoring points under different Re. It can be seen that with the increase in Re, the temperature at the monitoring point does have a rule of transition from turbulent flow to periodic flow and then to constant transition, which is consistent with the results of Figure 3. As shown in Figure 4a,b, the temperature time series oscillates visibly, and the power spectrum has no clear main harmonic frequency. When Re = 6.14 \(\times\) 10\({}^{6}\), the flow remains aperiodic. Although the frequency spectrum cannot clearly show the main frequency and harmonious frequency, there is a trend of transition to periodicity. In Figure 4c (Re = 6.15 \(\times\) 10\({}^{6}\)), the temperature development sequence at the monitoring point is visibly periodic, and a clear master frequency appears in the spectrum diagram, with the master frequency of \(f\) = 0.00656. It can be seen that the critical Re for the transition from weak turbulence to periodic flow is between 6.14 \(\times\) 10\({}^{6}\) and 6.15 \(\times\) 10\({}^{6}\) in the case of mixed convection in a curved cavity with air as the medium. In Figure 4d-f (Re = 7.67 \(\times\) 10\({}^{6}\), 1.1 \(\times\) 10\({}^{7}\) and 1.38 \(\times\) 10\({}^{7}\), respectively), the time series of temperature at the monitoring points remain periodic, but the amplitude decreases with the increase in Re. The main frequency \(f\) in the corresponding frequency spectrum is 0.008, 0.01343 and 0.03052, respectively. The double harmonic frequency becomes clearer, and the number of harmonic frequencies decreases, and finally even disappears. As Re increases to Re = 1.39 \(\times\) 10\({}^{7}\), the temperatures at the monitoring point no longer change with time, and the temperature time series at the symmetric point coincides with a steady state. Therefore, the critical Re for the transition from periodic flow to steady flow is between 1.38 \(\times\) 10\({}^{7}\) and 1.39 \(\times\) 10\({}^{7}\). In conclusion, with a fixed Ra (Ra = 2.36 \(\times\) 10\({}^{12}\)), with the increase in Re (10\({}^{5}\)-10\({}^{7}\)), the fluid in the arc cavity undergoes the evolution from turbulent flow to periodic flow and then to steady flow. The flow period and amplitude become smaller with the increase in Re, the main harmonic frequency of the corresponding spectrum becomes clearer, the main frequency increases with the increase in Re and the number of harmonic frequencies decreases to disappear. Finally, the flow tends to be steady. Figure 3: The time development series of averaged Nu on the heated wall in the fully developed stage. Figure 4: _Cont_. Figure 4: _Cont._  
In Figures 3 and 4, periodic characteristics of flows are observed from the transient perspective. In order to display the changes of flow structure in the curved cavity more intuitively from spatial dimension, we select the temperature isotherm and streamline distribution in one cycle after the flow enters the full development stage when Re = 7.67 \(\times\) 10\({}^{6}\). Figure 5 shows the temperature isotherm and streamline in a period (\(\tau\) = 6192\(\sim\)6314). The cold plume invades from the two corners of the cavity along both inclined sides of the heat source and moves upward near the top of the heat source due to the entrainment of the rising hot plume in the core region. The cold and hot plume front generate rapid energy exchange, and the speed decreases and tilts toward the middle area.