## 3 Results and Discussions
### Wall Effect on Velocity Distribution  
To analyze in greater detail the flow characteristics of helium inside the pebble bed, separate calculations were made for the cross-sectional average velocities of helium at various locations along the flow direction of the bed. These calculations are illustrated in Figure 13a. The results indicate that the velocity of helium flow prior to entering the pebble bed closely matches the designed inlet flow velocity of 0.1 m/s. As helium enters the pebble bed, the flow velocities of the helium gas increase rapidly. The maximum helium flow velocity is reached at approximately 0.5 d into the bed. In the inner region, the velocity is varied from about 0.2 m/s to 0.25 m/s. The cross-sectional averaged flow velocity increases about 2.5 times. Again, the velocity of helium flow reaches its maximum point at a position of 0.5 d before exiting the pebble bed. Subsequently, the flow velocity drops to around 0.1 m/s when leaving the bed. This variation can be mainly attributed the gradual decrease in the effective cross-sectional porosity of the flow channel, from 1 before entering and after leaving the pebble bed to approximately 0.3822-0.4791 inside the pebble bed. The decrease in the effective porosity in the flow channel results in a rapid increase in gas flow rate. The position at approximately 0.5 d to the outlet and inlet of the pebble bed indicates the lowest porosity value, maximizing the helium flow velocity (refer to Figure 13b for details). It has been observed that the wall effect causes a change in the porosity of the pebble bed, resulting in the pebble bed with four periodic boundaries having the smallest porosity. Therefore, its flow velocity is larger than that of the other two pebble beds, as shown by the average flow velocity in the central region of the pebble bed in Figure 13a.  
In addition, the localized velocity distribution of helium flow within the pebble bed was also analyzed. Figure 14a-c illustrate the streamline inside the three pebble beds with different boundaries. It is observable that the helium flow quickens upon entering the pebble bed and decelerates upon leaving, ultimately returning to its initial state. Furthermore, the cut-plane method was utilized to calculate the average flow velocity and corresponding porosity of the pebble bed in the \(x\) and \(y\) directions parallel to the flow. The resulting data are presented in Figure 15, illustrating the velocity distribution of helium flow in a pebble bed with two fixed walls and two periodic boundaries in the \(x\)- and \(y\)-axis directions. Due to the fixed wall on the \(x\)-axis, the porosity along the \(x\)-axis varies significantly, leading to a perturbation of the average flow velocity within the pebble bed. As the pebble bed's fixed wall is non-slip, the velocity near it is 0. As the distance from the wall increases, the helium velocity increases rapidly, reaching a maximum value within a range of less than 0.2d from the wall. As the distance from the fixed wall further increases, the average flow velocity of helium fluid varies in correspondence with the porosity variation. This is explained by the fact that larger porosities parallel to the direction of flow offer less resistance to helium, resulting in a similar flow velocity perturbation law to that of the porosity perpendicular to the wall direction, which allows a portion of the bypassed helium flow to exist in the area near the fixed wall.  
Figure 14: Streamline of purge gas in pebble beds with different boundary walls: (**a**) 4 fixed walls, (**b**) 2 fixed walls and 2 periodic boundaries, and (**c**) 4 periodic boundaries.  
Figure 13: Velocity (**a**) and porosity distribution (**b**) along the flow direction in pebble beds with different boundary walls.