# Experimental Study of the Performance of a Novel Vertical-Axis Wind Turbine
## 2 Basic Equations of Fluid Dynamics  
Stating without proof, for baroclinic flows, the integral of N-S equations [19; 20; 21] results in the Bernoulli's equation given by  
\[\frac{1}{2}\abs{\vec{V}}^{2}+\int\frac{\partial p}{\rho}+gz=\Re \tag{1}\]  
where \(\abs{\vec{V}}\) is the magnitude of the fluid velocity, \(p\) is the fluid pressure, \(\rho\) is the density of the fluid, \(g\) is the gravitational acceleration, \(z\) is the elevation from a fixed point and \(\Re\) is a constant. For a one-dimensional (1-D) baroclinic flow, \(\abs{\vec{V}}=U\), where \(U\) is the free-stream speed of the fluid, and \(\int\frac{\partial p}{\rho}=\frac{p}{\rho}\), yielding  
\[\frac{1}{2}U^{2}+\frac{p}{\rho}+gz=\Re \tag{2}\]  
Applying Equation (2) to stations 0 and 1 in Figure 1 yields \(\frac{1}{2}U_{0}^{2}+\frac{p_{0}}{\rho}+gz=\frac{1}{2}U_{1}^{2}+\frac{p_{1}}{ \rho}+gz\), which, on further simplification, yields \(\frac{1}{2}U_{0}^{2}+\frac{p_{0}}{\rho}=\frac{1}{2}U_{1}^{2}+\frac{p_{1}}{\rho}\) and  
\[\frac{p_{1}}{\rho}-\frac{p_{0}}{\rho}=\frac{1}{2}U_{0}^{2}-\frac{1}{2}U_{1}^{2} \tag{3}\]  
Applying Equation (2) to stations 2 and 3 in Figure 1 yields \(\frac{1}{2}U_{2}^{2}+\frac{p_{2}}{\rho}+gz=\frac{1}{2}U_{3}^{2}+\frac{p_{2}}{ \rho}+gz\), which, on further simplification, yields \(\frac{1}{2}U_{2}^{2}+\frac{p_{2}}{\rho}=\frac{1}{2}U_{3}^{2}+\frac{p_{3}}{\rho}\) and  
\[\frac{p_{0}}{\rho}-\frac{p_{2}}{\rho}=\frac{1}{2}U_{2}^{2}-\frac{1}{2}U_{3}^{2} \tag{4}\]  
Adding Equations (3) and (4) yields  
\[\frac{p_{1}}{\rho}+\frac{p_{2}}{\rho}=\frac{1}{2}U_{0}^{2}+\frac{1}{2}U_{1}^{ 2}+\frac{1}{2}U_{2}^{2}-\frac{1}{2}U_{3}^{2} \tag{5}\]  
Applying Equation (2) to stations 1 and 2 in Figure 1 yields \(\frac{1}{2}U_{1}^{2}+\frac{p_{1}}{\rho}+gz=\frac{1}{2}U_{2}^{2}+\frac{p_{2}}{ \rho}+gz\), which, on further simplification, yields \(\frac{1}{2}U_{1}^{2}+\frac{p_{1}}{\rho}=\frac{1}{2}U_{2}^{2}+\frac{p_{2}}{\rho}\) and  
Figure 1: Control volume of the AVAWT.  
\[\frac{p_{1}}{\rho}-\frac{p_{2}}{\rho}=\frac{1}{2}U_{2}^{2}-\frac{1}{2}U_{1}^{2} \tag{6}\]  
Adding Equations (5) and (6) yields  
\[2\bigg{(}\frac{p_{1}}{\rho}-\frac{p_{2}}{\rho}\bigg{)}=\frac{1}{2}U_{0}^{2}-U_{ 1}^{2}+U_{2}^{2}-\frac{1}{2}U_{3}^{2} \tag{7}\]  
Momentum is recovered far downwind the VAWT due to the turbulence of the atmospheric boundary layer; thus, \(U_{3}=U_{0}\). Equation (7) then becomes \(\frac{p_{1}}{\rho}-\frac{p_{2}}{\rho}=\frac{1}{2}U_{2}^{2}-\frac{1}{2}U_{1}^{2}\), which, on further simplification, yields the pressure drop across the VAWT:  
\[p_{2}-p_{1}=\frac{1}{2}\rho\big{(}U_{1}^{2}-U_{2}^{2}\big{)} \tag{8}\]  
The wind power captured is calculated from the rate of change of the kinetic energy of the wind:  
\[Power=\frac{1}{2}\dot{m}\big{(}U_{0}^{2}-U_{2}^{2}\big{)} \tag{9}\]  
where \(\dot{m}=\rho AU_{1}\) is the mass flow rate, in which \(A\) is the area of the AVAWT seen by the wind. Equation (9) then becomes  
\[Power=\frac{1}{2}\rho AU_{1}\big{(}U_{0}^{2}-U_{2}^{2}\big{)} \tag{10}\]  
The total energy borne by the wind seen by the VAWT is given by  
\[P_{\rm max}=\frac{1}{2}\rho AU_{0}^{3} \tag{11}\]  
The power coefficient \(C_{p}\) is given by  
\[C_{p}=P/P_{\rm max} \tag{12}\]