**3. Double, multiple streamtube model (DMS), CARDAAV code**

In the Double, Multiple Streamtube models (DMS) developed initially been by Paraschivoiu [11, 29], the variation in the upwind and downwind induced velocities as a function of the azimuthal angle has been included. In this case, the wind turbine blade is divided into several elements assuming no interaction between the components. This method's main principle is to determine the axial and angular induction factors using an iteration method. Then, the forces and power generated are defined in a similar manner to the VAWT. **Figure 1** shows the double streamtube model principle where the machine is represented by a pair of actuator disks in tandem in the upwind and downwind zone, CARDAAV Code. The induced velocity decreases in the axial direction so that the downwind induced velocity is smaller than the upwind zone. With *V*∞*<sup>i</sup>* representing the local ambient wind velocity, *Ve* the equilibriuminduced velocity as shown in **Figure 1**, we can write the induced velocities as a function of the induced factor in the upwind zone named *a* and downwind zone named *a*<sup>0</sup> *:*

$$V = a \; V\_{\approx i} \tag{1}$$

$$V\_{\epsilon} = (2\mathfrak{a} - \mathfrak{1})V\_{\lnot \mathfrak{i}} \tag{2}$$

$$V' = a'V\_{\varepsilon} = a'(2a - 1)V\_{\infty i} \tag{3}$$

$$W^{\prime\prime} = (2a^{\prime} - \mathbf{1})(2a - \mathbf{1}) \text{ V}\_{\approx i} \tag{4}$$

**Figure 1.** *Principle of the double multiple streamtube model.*

The atmospheric wind shear in Eq. (5) is based on the Frost *et al.* model [30] where *ZEQ* represents the height at the equator, *V*∞*<sup>i</sup>* is the local ambient wind velocity in the vertical direction, *V*<sup>∞</sup> is the ambient wind velocity at the equator, and *α*∞*i*is the exponent of the power-law, which in our simulation is taken to be 1*=*7.

$$\frac{V\_{\infty i}}{V\_{\infty}} = \left(\frac{Z\_{\infty i}}{Z\_{EQ}}\right)^{a\_{\infty i}}\tag{5}$$

The relative velocity and the local angle of attack in the upstream zone where the azimuth angle varies between �*π=*2≤ *θ* ≤*π=*2, are given by:

$$\mathcal{W}^2 = V^2 \left[ \left( X - \sin \theta \right)^2 + \cos^2 \theta \cos^2 \delta \right] \tag{6}$$

$$a = \arcsin\left[\frac{\cos\theta\cos\delta}{\sqrt{\left(X - \sin\theta\right)^2 + \cos^2\theta\cos^2\delta}}\right] \tag{7}$$

where *X* ¼ *ωr*is the tip-speed ratio, and *ω* is the turbine rotational speed. For the downwind zone where the azimuth angle varies between *π=*2≤*θ* ≤3*π=*2 similar equations can be derived using *W*<sup>0</sup> and *α*<sup>0</sup> . The nondimensional normal and tangential forces as a function of the azimuth angle *θ* are given by

$$F\_N(\theta) = \frac{c}{S} \int\_{Z\_R}^{Z\_R + 2\mathcal{H}} \left(\frac{W}{V\_{\infty}}\right)^2 \mathcal{C}\_N dZ \tag{8}$$

$$F\_T(\theta) = \frac{c}{S} \int\_{Z\_R}^{Z\_R + 2H} \left(\frac{W}{V\_{\infty}}\right)^2 \frac{C\_T}{\cos \delta} dZ \tag{9}$$

Using the same modeling above for the upwind region, the downwind area can also be calculated with the azimuth angle between *π=*2 <*θ* < 3*π=*2*:* The normal and tangential force coefficients of the blade section are given by

$$\mathbf{C}\_{N} = \mathbf{C}\_{L}\cos\alpha + \mathbf{C}\_{D}\sin\alpha \tag{10}$$

$$\mathbf{C}\_{T} = \mathbf{C}\_{L} \sin a - \mathbf{C}\_{D} \cos a \tag{11}$$

where the lift and drag coefficients CL and CD are computed by interpolating the available test data using both the local Reynolds number Re <sup>¼</sup> Wc μ∞ � � and the local angle of attack. These coefficients are used up to an angle close to the static stall angle, from which point a dynamic-stall model is considered to estimate the dynamic lift and drag coefficients. The equations for the interference factor *a*ð Þ*θ* are given by [11].

$$K[\mathbf{1} - a(\theta)] \cos \theta = a(\theta) f(\theta) \tag{12}$$

$$f(\theta) = \left(\frac{W}{V}\right)^2 \left[\mathbf{C}\_N \cos \theta + \mathbf{C}\_T \left(\frac{\sin \theta}{\cos \delta}\right)\right] \tag{13}$$

where K <sup>¼</sup> <sup>8</sup>π<sup>r</sup> Nc and N is the number of blades. The upwind half-cycle of the rotor is divided into several angular tubes Δ*θ*, assuming a constant induced velocity for each of these tubes, **Figure 2**. The interference factor *a*ð Þ*θ* can be written as

$$a(\theta) = \frac{K\mathcal{K}\_0}{K\mathcal{K}\_0 + \int\_{\theta - \frac{\theta^\theta}{2}}^{\theta + \frac{\theta^\theta}{2}} f(\theta)d\theta} \tag{14}$$

*Aerodynamic Analysis and Performance Prediction of VAWT and HAWT Using CARDAAV… DOI: http://dx.doi.org/10.5772/intechopen.96343*

**Figure 2.** *VAWT model: (a) integration points, (b) DMS model.*

where K0 <sup>¼</sup> sin *<sup>θ</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>θ</sup>* 2 � � � sin ð Þ *<sup>θ</sup>* � <sup>Δ</sup>*θ=*<sup>2</sup> and *<sup>a</sup>*ð Þ*<sup>θ</sup>* is computed numerically. A similar technique is then used for the downwind zone to determine the interference factor *a*<sup>0</sup> ð Þ*θ* . The torque produced by the blade element as a function of the azimuth angle is calculated based on the lift and drag contributions *CQL* and *CQD*

$$\mathbf{C}\_{\mathrm{Q}\_{L}} = \frac{\mathrm{Nc}}{2\pi \mathrm{SR}} \int\_{-\pi/2}^{3\pi/2} \int\_{Z\_{R}}^{Z\_{R} + 2H} \mathbf{C}\_{L} \mathrm{s} \mathrm{s} \mathrm{a} \left(\frac{W}{V\_{\infty}}\right)^{2} \times \left(\frac{r}{\cos \delta}\right) d\mathbf{Z} d\theta \tag{15}$$

$$\mathbf{C}\_{\rm Q\_D} = \frac{\rm Nc}{2\pi \rm SR} \int\_{-\pi/2}^{3\pi/2} \int\_{Z\_R}^{Z\_R + 2H} \mathbf{C}\_D cos \alpha \left(\frac{W}{V\_{\infty}}\right)^2 \times \left(\frac{r}{cos \delta}\right) d\mathbf{Z} d\theta \tag{16}$$

Finally, the power generated due to the lift and the losses due to the drag is given by

$$P\_L = \frac{1}{2} \rho\_{\text{ss}} V\_{\text{ss}}^3 \text{S}(\text{C}\_Q)\_L \frac{\text{o}R}{V\_{\text{ss}}} \text{ and } P\_D = \frac{1}{2} \rho\_{\text{ss}} V\_{\text{ss}}^3 \text{S}(\text{C}\_Q)\_D \frac{\text{o}R}{V\_{\text{ss}}} \tag{17}$$

The impact of atmospheric wind turbulence on a wind turbine's aerodynamic loads can be included in the DMS model using a stochastic atmospheric wind. The induced velocities necessary for predicting the forces were computed for each Streamtube in both the upwind and downwind zones of the rotor, including the longitudinal and lateral fluctuation velocities resulting from the fluctuating atmospheric wind. A wind time series in the rotor's upwind zone is created to produce the turbulent wind velocity. Using a time delay in the time series based on a linear variation, the turbulent wind speed in the downwind zone is generated [11]. The total velocity of the fluctuating atmospheric wind is a superposition of a mean part and a stochastic fluctuating part. The values of the fluctuations velocities are performed by using the Fast Fourier Transform. The time series are assumed to propagate downstream of the rotor with the speed of the airflow disturbed by the presence of the wind turbine. The simulation uses Double-Multiple Streamtube model (DMS) with up to 1512 stream tubes for the whole Darrieus rotor (21 vertical stream tubes and 72 lateral stream tubes at every five degrees [11]. In the presence of atmospheric turbulence, the relative velocity and the local angle of attack are given by [18]:

$$\mathbf{W}^2 = \left[ (\mathbf{V} + \mathbf{u}\_\mathbf{f}) \sin \theta + \mathbf{v}\_\mathbf{f} \cos \theta - \alpha \mathbf{r} \right]^2 + \left[ \mathbf{v}\_\mathbf{f} \sin \theta - (\mathbf{V} + \mathbf{u}\_\mathbf{f}) \cos \theta \right]^2 \cos^2 \theta \tag{18}$$

$$\mathbf{a} = \arcsin\left[\frac{\mathbf{v}\_{\text{f}}\sin\theta - (\mathbf{V} + \mathbf{u}\_{\text{f}})\cos\theta}{\mathbf{W}}\right] \cos\delta\tag{19}$$

where *V* is the upwind velocity and *u <sup>f</sup>* and *v <sup>f</sup>* the fluctuation velocity components. The overall model developed "CARDAAS" performs the computation of steady-state, CARDAAV as well as the stochastic response of aerodynamic loads when turbulence is included [11, 18, 22].
