**4.3 Rms of velocity fields**

**Figure 9** shows the root mean square of the velocity field of the curved blade turbine. The root mean velocity presents the fluctuation of the periodic velocity. According to these results, the highest value region is localized at the same direction of the trailing vortices. In fact, for the convex configuration, the fluctuated velocity is localized close to the blade tip. For the flat blade turbine, the turbulent fluctuation is propagated to the vessel wall. For the concave blade, the fluctuation is localized at the upper and the downer edges of the blade. The maximum variability of the flow occurs due to the trailing vortices following the recirculation loops.

For the staged system (**Figure 10**), the greatest fluctuation is localized at the blade that is placed at the bottom of the tank. These fluctuations can be explained by the suction of the flow from the bottom of the tank. In addition, the maximum values of the turbulent fluctuations are created between the two blades due to the interaction between the blades. The fluctuations generated due to the association of the convex impeller with the flat impeller are narrowed when they are compared to the association of the concave and the flat impeller. This effect reveals that the

**Figure 9.** *Rms velocity for the one-staged system.*

narrowed one is localized besides the wall of the tank at the bottom. The development of the maximum value of the axial velocity component is larger with the convex blade, which confirms that the recirculation loops associated to the convex blade are more important according to the other configurations. According to the velocity field, it can be seen that the recirculation loops associated to the convex blade are larger than the other configurations that are in conjunction with the amelioration of the axial velocity component with the convex blade. The highest maximum axial velocity component is generated by the flat turbine by vmax = 0.22 Utip followed by the concave blade by vmax = 0.16 Utip and by the convex blade by

For the staged system (**Figure 8**), the maximum value of the axial velocity component is localized at the lowest blade that represents the suction of the flow. The highest maximum value is produced by the second configuration by vmax = 0.4 Utip. For the first configuration, the highest value is equal to vmax = 0.29 Utip. In addition, the maximum value of the axial velocity is almost similar while we use the

flat blade turbine at the bottom (vmax = 0.33 Utip).

vmax = 0.15 Utip.

**170**

**Figure 6.**

**Figure 7.**

*Axial velocity for one-staged system.*

*Radial velocity for staged system.*

*Vortex Dynamics Theories and Applications*

**Figure 10.** *Rms velocity for the staged system.*

convex blade is not able to create a large fluctuation on the turbulent flow and local vortices are created.
