*2.4. Mesh and Time-Step Independence*

Steady and unsteady calculations were performed with a constant guide vane opening (2◦) with different mesh numbers and time steps. The number of nodes ranged from 7.5 million to 12 million, and the time step ranged from 1 × <sup>10</sup>−<sup>4</sup> to 5 × <sup>10</sup>−<sup>3</sup> s. The amplitude of the HT coefficient of the GVs was used as the criterion for this independence study. Figure 4a indicates that when the mesh number is greater than 11 million, the amplitude of the hydraulic torque only has a slight change; thus, 11 million mesh elements were adopted in this paper.

**Figure 4.** Independence study results: (**a**) the time step and (**b**) the mesh size.

A mesh with 11 million elements was used in the calculation of the time-step independence. Figure 4b shows that the amplitude of the hydraulic torque coefficient in the GVs is stable when the time step is below 5 × <sup>10</sup>−<sup>4</sup> s; thus, the time step was set to 5 × <sup>10</sup>−<sup>4</sup> s.

#### *2.5. Numerical Model Validation*

Mesh settings with openings of 3◦ and 6◦ were chosen for the numerical model validation. These meshes were both deformed from the initial mesh. The comparison between the results and the test data from the site shows that they are consistent with each other (the results for the 3◦ opening mesh are shown in Figure 5a; the results of the 6◦ opening mesh are shown in Figure 5b).

#### *2.6. Calculation Approach*

This work was carried out in the following steps, and Figure 6 shows a flowchart of our work:


speed (500 rpm) remained constant. The initialization was based on the result of step 1, and the boundary condition was kept the same as that in step 1. Finally, we set the transient simulation results of 10 complete runner revolutions as the initial flow field.


**Figure 5.** Numerical model validation results for different GVOs of (**a**) 3◦ and (**b**) 6◦.

**Figure 6.** Workflow of the calculations.
