# Numerical Study of Mixed Convection and Heat Transfer in Arc-Shaped Cavity with Inner Heat Sources
## 2 Physical Model and Numerical Method
### Numerical Method  
In this study, the governing Equations (1)-(4) are discretized by the finite volume method using ANSYS FLUENT (17.2). The convection term of the equation is discretized by the second-order upwind scheme, and the diffusion term is discretized by the second-order central difference scheme. The time term is discretized using the second-order implicit scheme. The separated solver was selected for solving, and the SIMPLE algorithm is used to solve the coupling problem of pressure and velocity [26]. Due to the large scale of the model studied in this paper and the high Re involved (up to \(10^{5}\)-\(10^{7}\)), the improved turbulence model (RNG \(k\)-\(\varepsilon\) model) is used for numerical solution, and the standard wall function is used to calculate the region near the boundary layer.  
For large-scale models, it is difficult to verify the numerical with experimental results. However, according to the study on the numerical simulation method of natural convection with a high Rayleigh number in a closed cavity mentioned in the literature [27], when the Rayleigh number is high (especially \(\text{Ra}\geq 10^{8}\)), the flow transitions into turbulence and is particularly sensitive to various disturbances. Therefore, the turbulence model should be selected for numerical simulation. In addition, some scholars [28] have used the RNG \(k\)-\(\varepsilon\) model to conduct numerical simulation research on cavities containing independent heat sources. Therefore, the numerical method proposed in this paper is reasonable and suitable.  
Symmetric block-structured grids are used to partition the computational domain. Figure 2 shows a grid division diagram. When dividing the grid, the height of the first layer is set to 0.16 m and the growth factor is set to 1.008. The quality of the divided mesh is tested, and the test results show that the mesh quality is good and that the aspect ratio is less than 20.  
Before the calculation, grid and time step independence studies are carried out to test four groups of grids with mesh quantities: 40,000, 81,000, 97,000 and 160,000 as well as four different time steps: 0.2, 0.4, 0.6 and 0.8. It is worth noting that all the independent validations mentioned were conducted at Re = 3.29 \(\times\)\(10^{6}\). The accuracy of this study was verified by comparing the time-mean temperature of sym  
Figure 2: Diagram of mesh distribution.  
P6(\(-\)0.37, 0.34) and P6*(0.37, 0.34) in the flow-sensitive region. The convergence criteria are less than 10\({}^{-5}\).  
The results of grid and time step sensitivity tests are shown in Tables 1 and 2, respectively. The results in Table 1 are compared with 210 \(\times\) 495 as the benchmark. The results show that when the number of grids is greater than 97,000, the change rate of monitoring variables is less than 1%. Similarly, the comparison of the remaining results in Table 2 with 0.2 as the basis shows that when the time step is less than 0.4, the relative error is almost less than 2%. Therefore, based on the calculation accuracy and calculation time, the grid with a grid quantity of 97,000 is selected as the calculation grid, and the time step is selected as 0.2.