## Appendix C Analytical Wake Deflection Models  
The following section is adopted from the project assignment of the authors [29].  
[PERSON] et al. [25] derived Equation (11) for the skew angle where deflection is determined by integrating with respect to \(x\). It assumes a uniformly distributed velocity profile inside the velocity deficit region, known as the top-hat assumption.  
\[\theta=\frac{C_{T}cos^{2}\gamma sin\gamma}{2(1+2k_{w}x/D)} \tag{12}\]Here \(\gamma\) is the yaw angle and \(k_{w}\) is the wake expansion factor. However, this model over-predicts wake deflection due to the top-hat assumption not being accurate [46]. A new formula was proposed by [PERSON] and [PERSON] [26] using a Gaussian distribution for the velocity deficit.  
\[\begin{split}\frac{y_{d}}{D}=&\theta_{0}\frac{x_{0}} {D}+\frac{\theta_{0}}{14.7}\sqrt{\frac{cos\gamma}{k_{y}k_{z}C_{T}}}(2.9+1.3 \sqrt{1-C_{T}}-C_{T})\\ &\times\ln\left(\frac{(1.6+\sqrt{C_{T}})(1.6\sqrt{\frac{8\sigma_ {y}v_{z}}{D^{2}cos\gamma}})-\sqrt{C_{T}}}{(1.6-\sqrt{C_{T}})(1.6\sqrt{\frac{8 \sigma_{y}v_{z}}{D^{2}cos\gamma}})+\sqrt{C_{T}}}\right)\end{split} \tag{10}\]  
where \(y_{d}\) is the deflection normalized by the turbine diameter \(D\). Equations for initial skew angle \(\theta_{0}\), potential core length \(x_{0}\), and wake width \(\sigma_{y,z}\) can be found in [PERSON] and [PERSON] [27]. To better specify equations for \(k_{y}\) and \(k_{z}\), [PERSON] and [PERSON] [27] proposed a new model, Equation (11). It is also based on a Gaussian velocity deficit profile together with momentum conservation in the lateral direction.  
\[\begin{split}\frac{y_{d}}{D}=\frac{y_{d0}}{D}+\frac{\sqrt{C_{T} cos\gamma}sin\gamma}{18.24k^{*}}\\ \ln&\left|\frac{\left(\frac{\varphi_{0}}{D}+0.24 \sqrt{C_{T}cos^{3}\gamma}\right)\left(\frac{\varphi}{D}-0.24\sqrt{C_{T}cos^{3 }\gamma}\right)}{\left(\frac{\varphi_{0}}{D}-0.24\sqrt{C_{T}cos^{3}\gamma} \right)\left(\frac{\varphi}{D}+0.24\sqrt{C_{T}cos^{3}\gamma}\right)}\right| \end{split} \tag{11}\]  
A final wake deflection model has been developed by [PERSON] et al. [28] based on [PERSON]'s lifting line theory, classical momentum theory, and [PERSON]'s equation. This gives the wake deflection downstream of the yawed turbine \(\delta v\) as a function of the free stream velocity along the axis orthogonal to the yawed turbine, denoted by \(x^{\prime}\).  
\[y_{d}(x)=\int_{-\infty}^{x}\frac{-\delta v(x^{\prime})}{U_{\infty}}dx^{\prime} \tag{12}\]  
The transverse velocity is then defined in the following manner  
\[\delta v(x)=\frac{\delta v_{0}}{2d_{w}^{2}(x)}\left[1+erf\left(\frac{x}{\sqrt{ 2}\Delta_{w}}\right)\right] \tag{13}\]  
where \(\delta v_{0}\) is the transverse velocity magnitude, \(d_{w}(x)\) is the effective diameter of the wake normalized by the turbine diameter, defined as \(d_{w}(x)=1+k_{w}ln(1+\exp\left[(x-2\Delta_{w})/R\right]),erf\) is an error function, and \(\Delta_{w}\) is a characteristic width, usually set equal \(R\). These equations are completed of with the following expression for \(\delta v_{0}\).  
\[\delta v_{0}=\frac{1}{4}C_{T}U_{\infty}\cos^{2}\gamma\sin\gamma \tag{14}\]  
Selected results for wake deflection of a 5 MW NREL turbine produced in a RANS simulation during the project assignment by [PERSON] and [PERSON] [29] are shown in Figure 19 together with the analytical wake deflection models. The model from [PERSON] overpredicts wake deflection, while the others follow the wake trajectory much closer, with what could be described as a constant offset between the predicted trajectories.