*3.1. Mesh Independence*

Two levels of the mesh are compared to demonstrate the mesh independence of the simulation results of this study. Table 1 shows comparison of element number and element size between three mesh levels. The element size in the rotor and wake region is designed to be 10 mm in normal mesh and 8 mm in refined mesh. Furthermore, a non-uniform mesh is specially designed to take advantage of the anisotropic kernel. The element size is 10 mm in rotor plane and 6 mm in the main flow direction.

**Table 1.** The comparison of element number and element size between three mesh levels (Unit: number of elements).


Figure 8 shows the torque result calculated by different meshes and different values, which are 16 mm, 20 mm, 24 mm, and 32 mm. Since the Gaussian > 2Δgrid must be guaranteed to avoid numerical oscillation, the torque result calculated with = 16 mm is only achieved on the refined mesh. All results shown are calculated with the standard regularization kernel.

**Figure 8.** The comparison of torque result with different meshes and different values. Curves with the same color represent results using different level of mesh but the same Gaussian width.

The torque result shows that the normal mesh gives a similar prediction of the rotor torque compared with the results of the refined mesh when using the same value. Figure 9 shows the normal velocity (illustrated in Figure 1) of each blade element. It shows that normal mesh gives a similar prediction of the normal velocity compared with the results of the refined mesh. Therefore, it can be concluded that the simulations using a normal mesh are mesh independent.

It should be noticed that = 2Δgrid is not sufficient to guarantee a reliable simulation, because simulations with = 16 mm and refined mesh give a totally different result compared with simulations with = 20 mm and normal mesh. Furthermore, a rotating rotor generating torque is a physical phenomenon and it must not relate to the element size. On the other hand, Figure 8 also shows that the torque result is strongly affected by the value and does not going to converge when the value grows too much. This phenomenon will be discussed in the next section.

**Figure 9.** The normal velocity along the blade calculated by two levels of meshes with different Gaussian width when rotating speed is (**a**) 400 RPM, (**b**) 550 RPM. Curves with the same color represent results using the same Gaussian width but different level of mesh.
