AERODYNAMICS OF WIND TURBINES

Classification of wind turbines according to principle of transformation of kinetic energy of wind into work

According to the orientation of the axis of rotation, wind turbines are classified as horizontal axis wind turbine (HAWT) and vertical axis wind turbine (VAWT). Horizontal axis rotating turbines operate on the principle of the turbomachine in which the air does the work as it flows through the rotor blade passages, see Figure 1283a. Vertical axis turbines operate on the principle of variable blade drag during a single rotation, so that, given the same wind direction, they provide different drag at different phases of the rotor rotation, see Figure 1283b.

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Basic types of wind turbine rotors

(a) frontal view of HAWT and cylindrical section A-A of its blade cascade; (b) frontal view of VAWT and section A-A of its rotor (in picture is so-called Darrieus wind turbine, but there are other VAWT types). (c), (d) detail of blade orientation changes and their aerodynamic drag as rotor rotates - in position (c), blade has more drag than in position (d), thus creating moment of force on rotor that causes turbine to rotate. A_S [m²] rotor swept area; r [m] rotor radius under investigation; r_t [m] tip radius; U [m·s^-1] blade speed at radius under investigation; V [m·s^-1] absolute airflow velocity; V_θ [m·s^-1] tangential component of absolute velocity; W [m·s^-1] relative airflow velocity. Index ₁ indicates the condition in front of the rotor, index ₂ behind the rotor.

HAWT vs. VAWT

Turbines with the vertical axis of rotation have some advantages (mainly they are not dependent on the wind direction and are able to process the energy of wind gusts), but the dominant type used are horizontal axis turbines, which can have significantly more power and are more efficient - that is what this article is about.

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Definition of Power and Thrust coefficient

Part of the kinetic energy of the air flowing through the rotor is transformed into work, respectively power in the form of rotor torque. In addition, the air flow acts in the direction of the flow on the turbine with a thrust in the same way that wind acts on trees, houses and other objects. Quantification of the power and thrust of the wind turbine is done using the power C_P and rotor thrust coefficient C_T. The power coefficient is defined as the power transmitted by the wind to the turbine to the kinetic power of the wind flowing through the same area as the rotor swept area, see Equation 260a. The thrust coefficient is defined as the ratio of the force acting on the rotor in the direction of the axis of rotation (thrust) from the wind flow to the force that would be caused by the dynamic wind pressure on the rotor swept area, see Equation 260a. It is necessary to distinguish between the power and thrust coefficients for the whole rotor and the local one at the radius under investigation, see Equation 260b.

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(a) parameters for whole rotor; (b) local value of coefficient. C_P [1] power coefficient; C_T [1] thrust coefficient; P_i [W] wind power transferred to wind turbine rotor; P_wind [W] kinetic wind power; P_d [Pa] dynamic wind pressure; T [N] thrust on rotor. ρ [kg·m^-3] density of air. The equation for the wind kinetic power P_wind is derived in Appendix 260.

Comparison of ideal and actual wind turbine rotor

The parameters of an ideal wind turbine rotor are referred to as the Betz limit. Actual rotors cannot achieve these limit due to losses that necessarily occur during energy transformations. In addition, in actual rotors, the stream tube collapse, which increases the value of the thrust coefficient compared to the ideal state.

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Power coefficient of wind rotor and losses according to Betz

L_CP [1] power coefficient losses; U_t [m·s^-1] blade speed at blade tip; λ [1] tip speed ratio; λ_start [1] tip speed ratio at which wind turbine is able to run independently. The index _opt indicates the optimum parameters. Data from [Hau, 2006, p. 98].

Optimal values of Tip speed ratio with regard to losses

The value of the maximum or optimum rotor power coefficient C_{P, opt} depends mainly on the number of blades. It only reaches this value at a particular value of the tip speed ratio, referred to as λ_opt. The value of λ_opt depends not only on the number of blades but also on the shapes of the blade profiles, see Problem 900, p. 10.14. Table 615 shows the usual parameters for wind turbine rotors - full C_P-λ curves for different blade numbers are given in [Hau, 2006, p. 98].

Helical vortex loss behind rotor

The dominant loss is the kinetic energy loss of the helical vortex behind the turbine. The absolute value of V_2θ, according to the Euler equation for the work of a turbomachine, for the same Euler work decreases with blade speed, therefore the loss in kinetic energy of the vortex increases with decreasing rotational speed. The vortex loss behind the turbine is manifested by the value of the angular induction coefficient a' not being equal to zero as in an ideal rotor.

Rotor profile losses

Profile losses are caused by the air flow around the blade and are reflected in the profile drag, more about profile losses is in the article Aerodynamics of airfoils [Škorpík, 2022]. A typical problem with wind turbine blades is the small Reynolds number at the blade root and hence the low turbulence of the flow. This also implies an increased sensitivity to flow separation. This is avoided, for example, by installing vortex generators, see Figure 599, p. 10.8. Vortex generators stabilise the boundary layer even at lower than optimum wind velocities and are therefore most commonly used in onshore turbines where wind velocities are lower.

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Stream tube collapse

The thrust coefficient equation, derived from the theorem of momentum change (Equation 357a), aligns with experimental results up to a≈1/3, behind which stream tube collapse increases the thrust coefficient (Figure 357c). While for most turbines the maximum value of the axial coefficient a does not exceed 0,6 [Sharpe, 1990]. According to [Anderson, 1980], the collapse occurs at a=0,3262 and he proposed to approximate the measured data by the straight line defined by Equation 357b, but there are other approximations, see [Wilson et al., 1976, p. 49], [Liew et al., 2024].

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Thrust coefficient of wind turbine rotor

(a) equation derived for ideal rotor; (b) approximation of measured values of rotor thrust coefficient starting at a=0,3262; Axial induction coefficient a corresponds to mean value along the length of blades. The derivation of equation (a) is shown in Appendix 357.

Calculation of wind turbine blade shape using aerodynamic coefficients

The design of the blade is carried out by analytical 2D calculation, i.e. dividing the blade into elementary stages – Blade element method (BEM), see Figure 642, p. 10.10. The design begins with the selection of a suitable blade profile and the estimation of the axial and angular induction coefficient values at the investigated radius. From these predictions, the velocities in the velocity triangles, the stagger angle, the chord length and the forces acting on the investigated blade element can be calculated using airfoil theory, see the article Aerodynamics of airfoils [Škorpík, 2022]. From there, the local values of the power and drag coefficients can be determined. Verification of the accuracy of the estimation of the axial and angular induction coefficient values at the radius under investigation can be done using the energy and force balance of the blade element being calculated. The total power of the turbine is then the sum of the powers of all blade elements.

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Calculation of stagger angle of profile blade

Using the estimated values of the axial and angular induction coefficients and the optimum value of the tip speed ratio, the velocity angles, in particular the angle of mean aerodynamic velocity W_m, can be calculated (according to Formula 1065c). The angle of attack of the mean aerodynamic velocity corresponds to the optimal angle of attack of the blade airfoil and for the proposed Reynolds number. From these data, the stagger angle of the profile γ in the cascade at the investigated radius can also be calculated, see Figure 1065 and the solution procedure of Problem 166, p. 10.12.

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W_m [m·s^-1] mean aerodynamic velocity; β_m [°] angle of mean aerodynamic velocity; Δβ [°] camber of flow; i_m [°] angle of attack of mean aerodynamic velocity (angle between mean aerodynamic velocity and chord). The index _m indicates the mean aerodynamic velocity. The derivation of this formula is shown in Appendix 1065.

Blades with constant stagger angle

Some wind turbine rotors have straight blades (c=const., γ=const.). In this case, the optimum profile parameters are achieved at only one radius, usually the mean square radius. The reasons for building such blades are technological. For example, for blades made of metal or wood, respectively, material that cannot be cold-formed into the necessary complex shapes or only very hard at higher cost. For example, the first 1 MW wind turbine in 1941 (see Figure 172, p. 10.12) had steel straight blades because the idea was to minimize manufacturing costs [Hau, 2006]. On the other hand, such a simple blade shape obviously leads to higher aerodynamic losses and noise.

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Energy balance equation of blade element under investigation

The accuracy of the estimation of the axial and angular induction coefficients can be verified from the energy balance of the local values of the power coefficient at the investigated radius, which can be calculated in two ways. Firstly, from the tangential force F_θ, respectively the torque from this force (Eq. 358a), and secondly, from the Euler turbine equation (Eq. 358b). The two values must be equal (Eq. 358c), see Problem 900, p. 10.14.

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(a) power coefficient according to Euler work definition; (b) power coefficient according to airfoil theory; (c) energy balance condition. m [kg·s^-1] mass flow; w_E [kg·s^-1] Euler work on investigated radius. The derivation of the equations is shown in Appendix 358.

Correction of energy balance equations using Prandtl coefficient

The equality defined by Eq. 358c is based on the assumption that the Euler work w_E is the same over the entire radius under investigation and corresponds to the velocity triangle, respectively the camber of flow in the vicinity of the blades. In reality, the camber of flow varies slightly around the circumference, being largest around the blades and smallest in the axis of the blade passage. The difference increases as the number of blades decreases and near the blade tips, where the effect of tip clearence losses and the radial component of the flow velocity are significant. For this reason, Prandtl proposed to correct the value of Euler work using the C_F factor, see Eq. 1089. Fig. 448, p. 10.14 shows the results of the wind turbine blade calculation from Problems 166, 153 and 900 with the Prandtl correction.

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(a) condition for correct design of axial and angular induction coefficient values with Pradtl correction; (b) Prandtl coefficient. C_F [1] Prandtl coefficient. The product w_E·C_F represents the mean value of Euler work on the investigated radius. The Prandtl coefficient C_F and was originally derived for propellers, but is still used in wind turbine calculations today [Wilson et al., 1976], [Hansen, 2008, p. 52].

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Force balance equation of blade element under investigation

Simultaneously with the energy balance (Eq. 358c) the force balance for the thrust coefficient must be valid. The local values of thrust coefficient can be calculated in two ways. First, using the momentum change theorem (Eq. 1088a), and second, using the airfoil theory, respectively from the force F_a (Eq. 1088b). The two values must be equal (Eq. 1088c), see Problem 900.

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Wind turbine power changes with wind speed

A wind power plant is only able to generate power within a certain interval of wind velocities. Thus, there is a starting wind speed (the minimum wind speed at which the power plant operates) and a shutdown wind speed (the maximum wind speed allowed). When a wind turbine starts, its output increases with increasing wind speed up to its nominal power or installed power, which is given by the maximum power output of the generator. From the nominal wind speed the power of the wind turbine is maintained at a constant value even when the wind speed increases, see Figure 1098, p. 10.16.

Operation of wind power plants

The control of the wind power plant responds to changes in wind speed with respect to the power characteristics of the wind power plant and its safety. Wind turbine control starts with the start of the turbine and can be done in four basic ways, namely stall control, turning of the blades, rotational speed change and yaw control.

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Stall control

Stall control is a simple way to keep the wind turbine rotational speed within the required range. It is used in the case where the wind turbine drives an asynchronous generator, which increases its rotational speed with increasing power, see Figure 988a, p. 10.16. This is because as the rotational speed increases, the forces acting on the blade also increase. These forces are used to extend the aerodynamic brakes at the blade tips (Figure 988b, p. 10.16) or along the blades at nominal rotational speeds - one possible mechanism for the aerodynamic brake is shown in [Miller et al., 1972, p. 1255]. At the rotational speed of the maximum power output P_shutdown, the turbine will stop because the wind speed has already reached a level where the aerodynamic brakes are no longer able to effectively control the turbine rotational speed.

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Wind turbine power control by changing rotational speed

The combination of turning blade control with turbine speed variation enables the maximum annual installed capacity of the wind turbine (Capacity coefficient) to be achieved. If the turbine speed can be varied, this means that different generator outputs can be achieved at the same torque depending on the characteristics of the generator. The turbine speed can be varied in steps using a multi-stage gearbox, or continuously using a special gearbox or a multi-frequency generator without gearbox. Multi-frequency generators, or generators with gearboxes, are used where there are large fluctuations in wind velocities, respectively in areas where Wind speed is often below the nominal value. These are onshore power plants.

Changing rotational speed of wind turbines with multi-frequency generators

Figure 1095a shows the T-N characteristic of the multi-frequency generator. Note that over a wide range of rotational speeds, the torque of the generator and therefore the turbine remains constant. This allows, to keep the tip Speed Ratio in the optimal region and the turbine operates with a high power coefficient value over a wide range of wind velocities.

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Circuit diagram of multi-frequency generator of wind power plant

(a) characteristic of multi-frequency generator; (b) power plants with multi-frequency generator must also include power electronics to change frequency of current. 1-multi-frequency generator; 2-AC to DC converter; 3-DC converter; 4-connection of power plant to transmission grid. AC- alternating current; DC-direct current.

Changing rotational speed of wind turbines with asynchronous generators

The aerodynamic brake is not the only control element in the connection of a wind turbine to an asynchronous generator. As the rotational speed and velocities change, the angle of attack i_m also changes, increasing the profile losses and reducing the power output.

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Yaw control of wind turbine

Wind turbine yaw control is performed by turning the turbine around the vertical axis, see Figure 526, p. 10.18. It is used to turn the rotor into the wind, or to turn it to a position that prevents the rotor from destroying when the wind exceeds the operational capability of the turbine. Some small turbines also allow turning along the horizontal axis, which acts as an aerodynamic brake.

Blade testing

Certification

The blades are tested not only for high-cycle fatigue but also for other weathering resistance, see Figure 218. Finally, the blade must be certified according to IECRE OD-501 (System for Certification to Standards Relating to Equipment for Use in Renewable Energy Applications – Type and Component Certification Scheme - Wind Turbines). This certification can currently only be obtained from about ten certification authorities.

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Testing of blade resistance to lightning strikes

Testing of blade resistance to lightning strikes at LM Wind Power factory (Netherlands) [Thomsen, 2004].

Wind turbine blade equipment

In addition to deformation sensors and vibration sensors, the blades are equipped with lightning arrestors, temperature sensors and other accessories not directly related to their primary function - for example, heating to prevent humidity condensation and frost formation in the blade hollow.

Bending of long blades

The operational problem with long blades is their bending in the axial direction and the danger of the bent blade hitting the monopile. For this reason, wind turbines are not designed for high wind speeds and the rotors have a certain inclination to the horizontal plane so that the blades are further away from the monopile at the bottom.

Service life of wind rotor blades

The blades are designed for a service life of approximately 20 years. During this time they must be fully functional without the need for repainting, but even then their surface can be refurbished on site (they do not have to be removed from the wind turbine), see Figure 419, p. 10.20.