| 53 | | where ''U'',,N,, is the velocity in axial direction and ''U'',,θ,, the velocity in tangential direction (along the circular movement of the rotor blade element, as shown in Figure 2). Ω denotes the rotational speed of the rotor. |
| | 53 | where ''U'',,N,, is the velocity in axial direction and ''U'',,θ,, the velocity in tangential direction (along the circular movement of the rotor blade element, as shown in Figure 2).''Ω'' denotes the rotational speed of the rotor. |
| 55 | | Finally, the forces are smeared and interpolated to the PALM grid points. To optimise the performance of the time-consuming 3d smearing process. the smearing is done with a Polynomial function instead of the standard Gaussian smearing and confined to the region around the rotor (). |
| | 55 | Finally, the forces are smeared and interpolated to the cartesian PALM grid. To optimise the performance of the time-consuming 3d smearing process. the smearing is done with a Polynomial function instead of the standard Gaussian smearing and confined to the region around the rotor: |
| | 56 | |
| | 57 | {{{ |
| | 58 | #!Latex |
| | 59 | \begin{equation*} |
| | 60 | f_{\epsilon}(x,y,z) = \sum_{seg=1}^{N_{seg}}\left( f_{N} + f_{\theta} \right) \eta \left( d_{b,r}(x,y,z) \right) |
| | 61 | \end{equation*} |
| | 62 | }}} |
| | 63 | |
| | 64 | It is being summed over the number of rotor segments ''N'',,seg,,. ''η'' is the regularisation kernel which is a function of distance ''d'',,b,r,,''(x,y,z)'' between the center of the current rotor segment and each PALM grid point: |
| | 65 | |
| | 66 | {{{ |
| | 67 | #!Latex |
| | 68 | \begin{equation*} |
| | 69 | \eta \left( d_{b,r}(x,y,z) \right) = \frac{105}{32 \pi \epsilon_{min}^{3}} \left( \left( \frac{1}{\epsilon_{min}^{4}} d(x,y,z) - \frac{2}{\epsilon_{min}^{2}} \right) d(x,y,z) + 1 \right) |
| | 70 | \end{equation*} |
| | 71 | }}} |
| | 72 | |
| | 73 | ''ϵ'',,min,, denotes the zero of the polynomial function and thus confines the region where the smearing is applied (position of current rotor segment +/- ''ϵ'',,min,, in all directions): |
| | 74 | |
| | 75 | {{{ |
| | 76 | #!Latex |
| | 77 | \begin{equation*} |
| | 78 | \epsilon_{min} = \left( \frac{105}{32}\right)^{\frac{1}{3}} \pi^{\frac{1}{6}} \epsilon |
| | 79 | \end{equation*} |
| | 80 | }}} |
| | 81 | |
| | 82 | The smearing width ''ϵ'' is set to twice the grid spacing the horizontal grid spacing which has turned out to be a reasonable value (see e.g. [#troldborg Troldborg et al. (2013)]). |
| | 83 | |
| | 84 | |
