# Evaluation of Tip Loss Corrections to AD/NS Simulations of Wind Turbine Aerodynamic Performance
## 3 Tip Loss Corrections  
### Glauert Correction  
The tip loss correction of [PERSON] [26] was developed from the study of [PERSON] [36]. A function \(F\), which was later recognized as the first tip loss factor, was derived by [PERSON] making [PERSON]'s optimal circulation [37] go to zero at the blade tip,  
\[F=\frac{2}{\pi}\text{arccos}\Big{(}\exp\left[-\frac{B}{2}\left(1-\frac{r}{R} \right)\sqrt{1+\lambda^{2}}\right]\Big{)}, \tag{25}\]  
where \(B\) is the number of blades, \(r\) is the local radial location, \(R\) is the rotor radius, and \(\lambda\) is the tip speed ratio. The original derivation of [PERSON] was written very briefly and was elaborated more in [4; 27; 38] as for a more detailed derivation.  
Later on, [PERSON] made further contributions. First, he interpreted the physical meaning of factor \(F\) as the ratio between the azimuthally averaged induced velocity and the induced velocity at the blade position, leading to  
\[F=\frac{\overline{a}}{a_{B}}=\frac{\overline{a^{\prime}}}{a^{\prime}_{B}}, \tag{26}\]  
where \(\overline{a}=\frac{1}{2\pi}\int_{0}^{2\pi}a\mathrm{d}\theta\) and \(\overline{a}^{\prime}=\frac{1}{2\pi}\int_{0}^{2\pi}a^{\prime}\mathrm{d}\theta\) are the azimuthally averaged axial and tangential interference factors, and \(a_{B}\) and \(a^{\prime}_{B}\) are the interference factors at the blade position. Second, the local inflow angle \(\phi\) was introduced into the function in order to make it consistent with the local treatment of the BEMT, leading to the following new formula of factor \(F\),  
\[F=\frac{2}{\pi}\text{arccos}\left\{\exp\left[-\frac{B(R-r)}{2r\sin\phi}\right] \right\}. \tag{27}\]  
The factor \(F\) was then applied into the momentum theory through a straightforward way of multiplying the axial velocity at the rotor plane with factor \(F\), resulting in the following thrust and torque for an annular element,  
\[\mathrm{d}T=4\pi r\rho V_{0}^{2}a(1-a)F\mathrm{d}r, \tag{28}\]  
\[\mathrm{d}M=4\pi r^{3}\rho V_{0}\Omega t^{\prime}(1-a)F\mathrm{d}r. \tag{29}\]  
Using another two equations of \(\mathrm{d}T\) and \(\mathrm{d}M\) derived from the blade-element theory,  
\[\mathrm{d}T=\frac{1}{2}\rho V_{rel}^{2}cC_{n}B\mathrm{d}r=\frac{1}{2}\rho V_{ rel}^{2}c(C_{l}\cos\phi+C_{d}\sin\phi)B\mathrm{d}r, \tag{30}\]  
\[\mathrm{d}M=\frac{1}{2}\rho V_{rel}^{2}cC_{t}B\mathrm{d}r=\frac{1}{2}\rho V_{ rel}^{2}c(C_{l}\sin\phi-C_{d}\cos\phi)B\mathrm{d}r, \tag{31}\]  
and the velocity triangle at the rotor plane, the equations for the interference factors in the BEMT approach was derived to be:  
\[\frac{a}{1-a}=\frac{\sigma(C_{l}\cos\phi+C_{d}\sin\phi)}{4F\sin^{2}\phi}, \tag{32}\]  
\[\frac{a^{\prime}}{1+a^{\prime}}=\frac{\sigma(C_{l}\sin\phi-C_{d}\cos\phi)}{4F \sin\phi\cos\phi}. \tag{33}\]The above two equations lead to the final iterative formulae of the BEMT approach with the Glauert tip loss correction. Their difference from the baseline BEMT approach is only the appearance of factor \(F\) in the equations. Obviously, the application of the Glauert correction in BEMT is very simple, which is also a great advantage. There are several variations of the Glauert correction in which the way of applying factor \(F\) is different [39; 40], but the original Glauert tip loss correction is the most commonly used form till today.