# A Simplified Free Vortex Wake Model of Wind Turbines for Axial Steady Conditions
## 3 Length of Near Wake  
To validate the VSRW model, the National Renewable Energy Laboratory (NREL) Phase VI wind turbine is used in this paper as a test case. The NREL Phase VI is a stall-regulated turbine. This turbine was designed by the NREL. The experiments were performed in the National Aeronautics and Space Administration (NASA) Ames wind tunnel (24.4 \(\times\) 36.6 m) [35], which is considered a benchmark for the evaluation of wind turbine aerodynamic methods.  
In the TVW model, the near wake is truncated after a short azimuthal distance, which is typically an azimuth angle of about 60\({}^{\circ}\)[36] in the helicopter field. Beyond this point, the far wake extends and is comprised of a single tip vortex filament. In the proposed VSRW model, one should ensure that the vortex sheet extends sufficiently far downstream. The aerodynamic performance of the rotor should be independent of the length of the vortex sheet. The length of the vortex sheet (or length of the near wake) can be represented by the wake age angle of the near wake. The spatial step equals 10\({}^{\circ}\) as mentioned above.  
The LSST has been calculated used different lengths of the near wake ranging from 10\({}^{\circ}\) to 360\({}^{\circ}\). Figure 7 shows the LSST along the length of the near wake at different wind speeds. As the length of the near wake increases from 10\({}^{\circ}\), the LSST increases at the beginning and reaches the first maximum value at a certain near wake length. When the length of the near wake is longer than this certain length, the LSST is nearly constant at this maximum value. This certain length of the near wake is defined as the dividing length in this study. It is evident that the dividing length of the near wake is about 120\({}^{\circ}\) at wind speeds of 13 m/s, 20 m/s and 25 m/s, about 150\({}^{\circ}\) at wind speeds of 7 m/s and 10 m/s and about 140\({}^{\circ}\) at a wind speed of 15 m/s. At wind speeds of 7 m/s, 10 m/s and 15 m/s, the changes of the LSST between the near wake length of 120\({}^{\circ}\) and the dividing length are about 0.73%, 0.08% and 0.04% of the first maximum value. This illustrates that when the length of the near wake is about 120\({}^{\circ}\), the LSSTs achieve nearly constant values at all wind speeds.  
Figure 7: LSST along the length of the near wake at different wind speeds (The red points are the LSST values with a near wake age angle of 120\({}^{\circ}\)).  
The calculation time of the VSRW model along the length of the near wake was also examined. The central processing unit (CPU) of the computer is an Intel Core i5-4200U and the memory is 8 GB. Figure 8 shows the wake iteration time along the length of the near wake at 7 m/s and 15 m/s. It is apparent that the number of discrete nodes used to approximate the near wake increases as the length of the near wake increases. However, the computational cost for the induced-velocity calculation changes as the square of the number of discrete nodes [25].  
The vortex sheet of the near wake is cut off at a 120\({}^{\circ}\) wake age angle in the VSRW model after considering the accuracy and calculation time. The red points in Figure 7 are the LSST values with a near wake age angle of 120\({}^{\circ}\) at every wind speed. It is observed that all LSST values reach the stationary values when the length of the near wake is 120\({}^{\circ}\). It should be noted that this cut-off wake age angle is just an empirical angle rather than a real roll-up angle. In the following section, the aerodynamic analysis is conducted using the VSRW model with the near wake age angle of 120\({}^{\circ}\).