**2. Theory**

The first requirement of any simulation is the initial estimation of quantities of interest. This calculation will give an idea about the expected results and will act as a prerequisite for the last step: verification and validation. Without this, the simulation might produce some other results that need not always be correct.

### **2.1 One dimensional momentum**

One-dimensional momentum theory is one of the oldest theories of wind turbines. Much literature is dedicated to the same. It relates the velocities upstream, at the turbine blades, and downstream with a mathematical induction factor "a." The induction factor is essential and has some implications for the wind turbine as a whole. The exact derivation may be seen in [2, 4]. The formulae alone will be listed here.

Let the velocity of wind upstream be *u*; at the turbine be *y*; and downstream be *v*; then:

*Fluid Dynamics Simulation of an NREL-S Series Wind Turbine Blade DOI: http://dx.doi.org/10.5772/intechopen.107013*

$$y = \frac{(u+v)}{2} \tag{1}$$

The maximum power that can be generated from the wind is:

$$P\_{wind} = 0.5 \rho u^3 A \tag{2}$$

The maximum torque that can be extracted from the wind is:

$$T\_{wind} = \mathbf{0}.5 \rho u^2 A \tag{3}$$

It is convenient to use a non-dimensional power coefficient (Cp) and torque coefficient (Ct) as it is a ratio from 0<*C* <1. They are defined as:

$$\mathbf{C}\_p = \frac{P\_{out}}{P\_{wind}} \tag{4}$$

$$\mathbf{C}\_t = \frac{T\_{out}}{T\_{wind}}\tag{5}$$

"a" is the mathematical induction factor defined as:

$$\mathbf{C\_p} = \mathbf{4}\mathbf{a}(\mathbf{1} - \mathbf{a})^2 \tag{6}$$

$$\mathbf{C}\_{\mathbf{t}} = \mathbf{4}\mathbf{a}(\mathbf{1} - \mathbf{a})\tag{7}$$

These are the two primary verification and validation formulae. Through Fluent simulation, the velocity distribution and torque/power will be found separately. Notice how the induction factor links them. One can use Eq. (6) to compute the induction factor as a ballpark figure. Eq. (7) can then be used to provide theoretical Ct. But Ct can be directly found through Eq. (5) using Fluent. By comparing these two values, we can check for the correctness of our results.

The maximum power coefficient (percentage) that can be delivered by a wind turbine is 59.3%, and the induction factor must be less than 0.5. Anything above that is impossible or has no practical significance. This limit in power coefficient is known as Betz Limit, and it arises because some energy needs to be present in the wind to move past the turbine blades to prevent local wind accumulation. The same can be figured out by finding the maximum value of Eq. (6). Verification and validation will be performed in the end using these points. Large deviations can be expected as this theory does not account for turbulence and is a significant approximation of the underlying physics.

#### **2.2 Blade element momentum**

A wind turbine blade is known for its characteristic twist as one moves along the body. This twist is the characteristic angle defined by *β*. An airfoil provides max lift only for a particular angle of attack. That is easy in an airplane , exhibiting translational motion where one adjusts the pitch angle for maximum lift. However, a wind turbine exhibits rotational motion. The fundamental problem is that any two points on the circle's radius never move at the same speed. Since the angle of attack is computed keeping the net velocity wind vector as a reference, the angle with this

must remain constant. If one looks at this velocity vector, it has a k component of wind velocity (incoming wind) and an i component of the rotational velocity. The net velocity vector is:

$$V\_{relativotalde} = \mathfrak{u}\_{wind}\hat{\mathfrak{k}} + r\alpha\hat{\mathfrak{i}}\tag{8}$$

One can see that as we move across the blade, the net velocity vector will change and is, in fact, a function of radii. The only way to make a constant angle with this vector at all points is to compensate with a twist angle of our own. Hence, the characteristic curve.

BEM theory will help us with the exact mathematical formulae to compute this twist angle and hence, will be the starting point of our design. For exact derivation, please refer to [1–3, 5–7].

The local-speed ratio for arbitrary radii r and with incoming wind velocity u from the center is defined as:

$$
\lambda\_r = \frac{r\nu}{u} \tag{9}
$$

Note the local speed ratio changes as one moves along the blade. The angle made by the horizontal and the net velocity vector (as defined earlier) is:

$$\phi = \frac{2}{3} \tan^{-1} \frac{1}{\lambda\_r} \tag{10}$$

Here, the angles can be written as below for zero pitch. Also, *α* was assumed to be equal to 5.25 degrees. Please refer to **Figure 2** for visualizing the exact angles.

$$a = \phi - \beta \tag{11}$$

The chord length of the airfoil is given by:

$$\mathcal{L} = \frac{8\pi r}{B\mathcal{C}\_l} (\mathbf{1} - \cos\phi) \tag{12}$$

where B is the number of blades (3 in our case), Cl is the lift coefficient, and r is the radii from the center. However, one can notice that except for Cl, the net blade length and the rotational speed *ω*, everything else is defined and can easily be computed. Many approaches, such as calculating axial and tangential induction factors have been

**Figure 2.** *Airfoil angles (image referred from [1]).*

*Fluid Dynamics Simulation of an NREL-S Series Wind Turbine Blade DOI: http://dx.doi.org/10.5772/intechopen.107013*

developed [3, 5]. This method suffers from the limitation that the airfoil data must be present, and the airfoil must be uniform. Neither is applicable in our case, as NREL has not released airfoil data separately, and the root, primary, and tip are composed of totally different airfoils. The net blade length can be determined from previous designs for a chosen power level. Furthermore, 35 meters from the hub center to the tip was assumed to be sufficient. NREL has mentioned the maximum obtainable lift coefficient for each of the three airfoils. These values were averaged, and a slightly lower figure of 1.3 was chosen for Cl, and Cd was chosen as 0.1. Finally, the wind turbine was designed for an optimal TSR of 7. The TSR and blade length give an omega of 2.42 rad/sec.

The chord length and twist angle give us everything to design our blade in CAD. As a cross verification, XFoils and QBlade will be used to generate performance curves. **Tables 1** and **2** give the variation of twist angle and chord length as one moves across the blade.

QBlade will take the various airfoil parameters and generate curves, such as Cl vs TSR, Cd vs TSR, and Cl/Cd vs TSR. These are the three curves of interest. Our earlier value of Cl can be cross-checked with this software-generated graph. It is worth noting that QBlade does assume varying airfoils across the blade. The blade itself is designed through a selection tab for different airfoils. After this, the simulation is performed using highly simplified models for turbulence and airfoils. Later, CFD simulations will verify and produce highly accurate results.
