User:Smilesgiles89/sandbox

A vertical axis wind turbine rotates about a vertical axis; consequently, as illustrated in the figure below, the angle of attack for a blade segment changes with azimuthal angle. In addition, the relative apparent wind speed changes with azimuthal angle. This means that the lift and drag forces, and subsequently the torque, are strongly cyclic. This has been one of the strong criticisms of vertical axis wind turbines.

However, vertical axis wind turbines do offer some advantages: first, all the heavy drivetrain equipment can be located at the base of the wind turbine. This can reduce the costs by removing the necessity for a strong tower. In addition, in the event of maintenance, the heavy drivetrain equipment can be easily accessed for a vertical axis wind turbine. With an offshore wind farm, if a gearbox on a large horizontal axis wind turbine needs replacing, heavy vessel equipment must be used. In addition, the sea state can strongly influence the downtime and thus the lost revenue. In light of these concerns facing horizontal axis wind turbines far offshore, there has been a revival of interest in vertical axis wind turbines.

Although a horizontal axis wind turbine also experiences cyclic variations in torque (due to wind shear and gravity), for small wind turbines these are not large. That being said, with the upscaling of wind turbine size, blades are becoming heavier, which increases the torque variations produced by a horizontal axis wind turbine.

There are several means of predicting the performance of a VAWT: BEM modelling, vortex modelling, and full CFD modelling. CFD modelling is the most accurate; however, it is a computationally intensive activity. Vortex modelling can either be prescribed-wake form, or free vortex form. For a VAWT, it is required that the wake be non-prescribed since the blades of a VAWT cut through the wake in the downstream half of the cycle. This method of modelling can also be computationally intensive. BEM modelling is the least accurate, at least in theory, of the methods described. BEM modelling when applied to a VAWT is known as DMST modelling (Double multiple streamtube). This involves the division of the swept area into streamtubes.

Blade Element Momentum for a VAWT
The following equations are flexible enough to be applied to either H-rotors or to V-rotors.

Consider a blade segment at a height, $$h$$, above the ground. Its position relative to the direction of the undisturbed wind speed is denoted by the angle, $$\theta$$. This is as shown in figure .... It follows from this that the apparent wind speed, $$W$$, as seen by the blade segment is given by the following expression:

$$ W^2 = (v_\infty(1 - a)\cos\theta)^2 + (v_\infty(1 - a)\sin\theta + \omega r)^2 $$

where $$v_\infty$$ is the undisturbed wind speed, $$a$$ is the induction factor, $$\omega$$ is the rotor speed, and $$r$$ is the distance from the axis of rotation to the blade segment (note for a V-rotor or a Darrieus machine this is a function of height). The associated flow angle, the angle between the apparent wind speed and the tangent to rotation at the point of interest is defined as follows:

$$ \phi = atan2 $$

With the angle convention illustrated in the first figure, the angle of attack, $$\alpha$$, is given as

$$ \alpha = \phi - \beta $$

where $$\beta$$ is the pitch angle (which for general purposes will also be allowed to be a function of azimuth). Coupling this with the Reynolds number at the point under investigation, it is possible to determine the the lift and drag coefficients. It is required that Reynolds number be known also since a lift curve can be strongly influenced by the Reynolds number and the Reynolds number can vary significantly with azimuth for a VAWT. For completeness, Reynolds Number is calculated using

$$ Re = Wc/\mu $$

where $$c$$ is the chord of the aerofoil, and $$\mu$$ is the dynamic viscosity of air. The lift, $$L$$, and drag $$D$$ forces are respectively defined as

$$ L = \frac{1}{2}\rho NcW^2c_L(\alpha) $$ $$ D = \frac{1}{2}\rho NcW^2c_D(\alpha) $$

where $$c_L(\alpha)$$ and $$c_D(\alpha)$$ are the lift and drag coefficients respectively, $$\rho$$ is the density of air, and $$N$$ is the number of blades.

By definition, lift is defined as normal to the apparent wind speed whilst drag is defined as parallel to the apparent wind speed. Of interest are the forces tangential to the motion of the blade and normal to it. Resolving lift and drag into components tangential and normal to the motion of the blade gives the following expressions:

These forces must in turn be resolved to give forces parallel to the undisturbed wind speed and normal to it:

The value of induction factor, $$$$ is determined iteratively. It is known when the axial forces from blade element predictions match those predicted by momentum theory i.e

$$ 4a(1 - a) = $$

With the iteration procedure complete, all the terms such as apparent wind speed, angle of attack etc. are known. The torque produced at that azimuthal position can then be determined.

Dynamic considerations
By virtue of the fact that the blades take a curved trajectory, there is what is known as the virtual camber effect. This is equivalent to a uniform pitching of a blade. A modification of thin aerofoil theory reveals that the

In addition, due to the fact that the angle of attack varies around the azimuth, it also follows that the blades will, in some cases, be affected by the phenomenon known as dynamic stall.

The onset of static stall is when the angle of attack is such that the flow around an aerofoil becomes separated from the object (boundary layer separation). The result is that eddies and vortices form behind the aerofoil, which leads to both a reduction in the lift force generated and an increase in the drag, specifically the parasitic drag arising from the pressure difference fore and aft the aerofoil.

When the angle of attack is dynamic and undergoes significant swings, static force coefficients are not representative of the process. This is because following a severe change in angle of attack, a vortex can be shed from the leading edge of the aerofoil. This vortex helps maintain flow attachment up until the point the leading-edge vortex has propagated down and off the aerofoil and into the wake.

Such an effect has to be represented in VAWT models. Two methods for representing the hysteresis loop associated with dynamic stall are commonly employed. The first is the Gormont model; the second, the Beddoes-Leishman model.

Streamtube expansion
In the previous section, it was shown that a blade in the upstream section of the cycle would take energy out of the wind, thus slowing it down. By the conservation of mass flow rate, the streamtubes must expand. A means of representing the streamtube expansion in a relatively simple manner was developed by Sharpe.

Tip losses
A blade is not an infinitely long item.

By Kelvin's theorem, where there is a change in vorticity, there is a change in the lift produced on the aerofoil. Thus, as one moves towards the outer sections of a blade, the lift produced varies (even when ignoring effects such as wind shear).

Negative induction
If the rotor features a cambered aerofoil, then the possibility of flow acceleration must be considered also. This is also true for pitched symmetric aerofoils. In both situations, there are scenarios where the rotor accelerates the flow and thus produces negative torque at different azimuthal positions.

In this case, rather than the streamtubes expanding, they could in fact contract.

Multiple solutions
The variation of blade element forces with induction factor is non-linear. It is possible to get multiple crossing events as illustrated below:

Sample output
The following is the prediction of torque variation for a H-rotor with NACA0015 aerofoils around one revolution neglecting dynamic stall.

The tip-speed ratio has been chosen to be 3. Typically, H-rotors achieve maximum $$c_P$$ at a tip-speed ratio of between 3 and 4. V-rotors and Darrieus rotors operate optimally at higher tip-speed ratios as demonstrated in the following figure.

In contrast, rotors with cambered aerofoils typically operate at much lower tip-speed ratios.

All of the above simulations are with zero pitch.

J
The loads on both horizontal axis wind turbines (HAWTs) and vertical axis wind turbines (VAWTs) are cyclic; that is, the thrust and torque acting on the blades is dependent on where the blade is. In a horizontal axis wind turbine, both the apparent wind speed seen by the blade and the angle of attack depend on the position of the blade. This phenomenon is described as rotational sampling. This article will provide an insight into the cyclic nature of the loads that arise because of rotational sampling for a horizontal axis wind turbine.

Background
Analysis of the loads on a wind turbine are usually carried out through use of power spectra. A power spectrum is defined as the power spectral density function of a signal plotted against frequency. The power spectral density function of a plot is defined as the Fourier transform of the covariance function. Regarding analysis of loads, the analysis involves time series, in which case the covariance function becomes the autocovariance function.

In statistics, given a real stochastic process Z(t), the autocovariance is the covariance of the variable against a time-shifted version of itself. If the process has the mean $$E[Z_t] = \mu_t$$, then the autocovariance is given by


 * $$C_{ZZ}(t,s) = E[(Z_t - \mu_t)(Z_s - \mu_s)] = E[Z_t Z_s] - \mu_t \mu_s.\,$$

where E is the expectation operator.

Stationarity
If Z(t) is stationary process, then the following are true:


 * $$\mu_t = \mu_s = \mu \,$$ for all t, s



[ -(r_1 + r_2 + l_1s + l_2s + w^4(c^2l_1^2r_2 + c^2l_1^2l_2s) + w^2(2l_2c^2l_1^2s^3 + 2r_2c^2l_1^2s^2 + 2l_2c^2l_1r_1s^2 + 2r_2c^2l_1r_1s + l_2c^2r_1^2s + r_2c^2r_1^2 + cl_1^2s + 2*l2*c*l1*s + 2*r2*c*l1) + c*r1^2*s + c*l1^2*s^3 + 2*c*r1*r2*s + c^2*l1^2*l2*s^5 + c^2*l2*r1^2*s^3 + c^2*l1^2*r2*s^4 + c^2*r1^2*r2*s^2 + 2*c*l1*l2*s^3 + 2*c*l1*r1*s^2 + 2*c*l1*r2*s^2 + 2*c*l2*r1*s^2 + 2*c^2*l1*l2*r1*s^4 + 2*c^2*l1*r1*r2*s^3)/(w^4*(3*c^2*l1^2*l2^2*s^2 + 2*c^2*l1^2*l2*r2*s + c^2*l1^2*r2^2 + 2*c^2*l1*l2^2*r1*s + c^2*l2^2*r1^2 + 2*c*l1^2*l2 + 2*c*l1*l2^2) + 2*r1*r2 + w^2*(3*c^2*l1^2*l2^2*s^4 + 4*c^2*l1^2*l2*r2*s^3 + 2*c^2*l1^2*r2^2*s^2 + 4*c^2*l1*l2^2*r1*s^3 + 4*c^2*l1*l2*r1*r2*s^2 + 2*c^2*l1*r1*r2^2*s + 2*c^2*l2^2*r1^2*s^2 + 2*c^2*l2*r1^2*r2*s + c^2*r1^2*r2^2 + 4*c*l1^2*l2*s^2 + 2*c*l1^2*r2*s + 4*c*l1*l2^2*s^2 + 4*c*l1*l2*r1*s + 4*c*l1*l2*r2*s + 2*c*l1*r2^2 + 2*c*l2^2*r1*s + 2*c*l2*r1^2 + l1^2 + 2*l1*l2 + l2^2) + r1^2 + r2^2 + l1^2*s^2 + l2^2*s^2 + 2*l1*l2*s^2 + 2*l1*r1*s + 2*l1*r2*s + 2*l2*r1*s + 2*l2*r2*s + 2*c*r1*r2^2*s + 2*c*r1^2*r2*s + c^2*l1^2*l2^2*s^6 + c^2*l1^2*l2^2*w^6 + c^2*l1^2*r2^2*s^4 + c^2*l2^2*r1^2*s^4 + c^2*r1^2*r2^2*s^2 + 2*c*l1*l2^2*s^4 + 2*c*l1^2*l2*s^4 + 2*c*l1*r2^2*s^2 + 2*c*l2*r1^2*s^2 + 2*c*l1^2*r2*s^3 + 2*c*l2^2*r1*s^3 + 2*c^2*l1*l2^2*r1*s^5 + 2*c^2*l1^2*l2*r2*s^5 + 2*c^2*l1*r1*r2^2*s^3 + 2*c^2*l2*r1^2*r2*s^3 + 4*c*l1*l2*r1*s^3 + 4*c*l1*l2*r2*s^3 + 4*c*l1*r1*r2*s^2 + 4*c*l2*r1*r2*s^2 + 4*c^2*l1*l2*r1*r2*s^4), -(w^3*(2*l2*c^2*l1^2*s^2 + 2*l2*c^2*l1*r1*s + l2*c^2*r1^2 + c*l1^2 + 2*l2*c*l1) + w*(l2*c^2*l1^2*s^4 + 2*l2*c^2*l1*r1*s^3 + l2*c^2*r1^2*s^2 + c*l1^2*s^2 + 2*c*l1*r1*s + 2*l2*c*l1*s^2 + c*r1^2 + 2*l2*c*r1*s + l1 + l2) + c^2*l1^2*l2*w^5)/(w^4*(3*c^2*l1^2*l2^2*s^2 + 2*c^2*l1^2*l2*r2*s + c^2*l1^2*r2^2 + 2*c^2*l1*l2^2*r1*s + c^2*l2^2*r1^2 + 2*c*l1^2*l2 + 2*c*l1*l2^2) + 2*r1*r2 + w^2*(3*c^2*l1^2*l2^2*s^4 + 4*c^2*l1^2*l2*r2*s^3 + 2*c^2*l1^2*r2^2*s^2 + 4*c^2*l1*l2^2*r1*s^3 + 4*c^2*l1*l2*r1*r2*s^2 + 2*c^2*l1*r1*r2^2*s + c^2*l2^2*r1^2*s^4 + c^2*r1^2*r2^2*s^2 + 2*c*l1*l2^2*s^4 + 2*c*l1^2*l2*s^4 + 2*c*l1*r2^2*s^2 + 2*c*l2*r1^2*s^2 + 2*c*l1^2*r2*s^3 + 2*c*l2^2*r1*s^3 + 2*c^2*l1*l2^2*r1*s^5 + 2*c^2*l1^2*l2*r2*s^5 + 2*c^2*l1*r1*r2^2*s^3 + 2*c^2*l2*r1^2*r2*s^3 + 4*c*l1*l2*r1*s^3 + 4*c*l1*l2*r2*s^3 + 4*c*l1*r1*r2*s^2 + 4*c*l2*r1*r2*s^2 + 4*c^2*l1*l2*r1*r2*s^4), -(r1 + r2 + l1*s + l2*s + w^2*(c*l1*r2 + c*l2*r1 + c*l1*l2*s) + c*r1*r2*s + c*l1*l2*s^3 + c*l1*r2*s^2 + c*l2*r1*s^2)/(w^4*(3*c^2*l1^2*l2^2*s^2 + 2*c^2*l1^2*l2*r2*s + c^2*l1^2*r2^2 + 2*c^2*l1*l2^2*r1*s + c^2*l2^2*r1^2 + 2*c*l1^2*l2 + 2*c*l1*l2^2) + 2*r1*r2 + w^2*(3*c^2*l1^2*l2^2*s^4 + 4*c^2*l1^2*l2*r2*s^3 + 2*c^2*l1^2*r2^2*s^2 + 4*c^2*l1*l2^2*r1*s^3 + 4*c^2*l1*l2^2*s^2 + 4*c*l1*l2*r1*s + 4*c*l1*l2*r2*s + 2*c*l1*r2^2 + 2*c*l2^2*r1*s + 2*c*l2*r1^2 + l1^2 + 2*l1*l2 + l2^2) + r1^2 + r2^2 + l1^2*s^2 + l2^2*s^2 + 2*l1*l2*s^2 + 2*l1*r1*s + 2*l1*r2*s + 2*l2*r1*s + 2*l2*r2*s + 2*c*r1*r2^2*s + 2*c*r1^2*r2*s + c^2*l1^2*l2^2*s^6 + c^2*l1^2*l2^2*w^6 + c^2*l1^2*r2^2*s^4 + c^2*l2^2*r1^2*s^4 + c^2*r1^2*r2^2*s^2 + 2*c*l1*l2^2*s^4 + 2*c*l1^2*l2*s^4 + 2*c*l1*r2^2*s^2 + 2*c*l2*r1^2*s^2 + 2*c*l1^2*r2*s^3 + 2*c*l2^2*r1*s^3 + 2*c^2*l1*l2^2*r1*s^5 + 2*c^2*l1^2*l2*r2*s^5 + 2*c^2*l1*r1*r2^2*s^3 + 2*c^2*l2*r1^2*r2*s^3 + 4*c*l1*l2*r1*s^3 + 4*c*l1*l2*r2*s^3 + 4*c*l1*r1*r2*s^2 + 4*c*l2*r1*r2*s^2 + 4*c^2*l1*l2*r1*r2*s^4), -(c*l1*l2*w^3 + (c*l1*l2*s^2 + l1 + l2 - c*r1*r2)*w)/(w^4*(3*c^2*l1^2*l2^2*s^2 + 2*c^2*l1^2*l2*r2*s + c^2*l1^2*r2^2 + 2*c^2*l1*l2^2*r1*s + c^2*l2^2*r1^s + 2*c*r1^2*r2*s + c^2*l1^2*l2^2*s^6 + c^2*l1^2*l2^2*w^6 + c^2*l1^2*r2^2*s^4 + c^2*l2^2*r1^2*s^4 + c^2*r1^2*r2^2*s^2 + 2*c*l1*l2^2*s^4 + 2*c*l1^2*l2*s^4 + 2*c*l1*r2^2*s^2 + 2*c*l2*r1^2*s^2 + 2*c*l1^2*r2*s^3 + 2*c*l2^2*r1*s^3 + 2*c^2*l1*l2^2*r1*s^5 + 2*c^2*l1^2*l2*r2*s^5 + 2*c^2*l1*r1*r2^2*s^3 + 2*c^2*l2*r1^2*r2*s^3 + 4*c*l1*l2*r1*s^3 + 4*c*l1*l2*r2*s^3 + 4*c*l1*r1*r2*s^2 + 4*c*l2*r1*r2*s^2 + 4*c^2*l1*l2*r1*r2*s^4)] $$ and


 * $$C_{ZZ}(t,s) = C_{ZZ}(s-t) = C_{ZZ}(\tau)\,$$

where


 * $$\tau = s - t\,$$

is the lag time, or the amount of time by which the signal has been shifted.

As a result, the autocovariance becomes


 * $$C_{ZZ}(\tau) = E[(Z(t) - \mu)(Z(t+\tau) - \mu)]\,$$


 * $$ = E[Z(t) Z(t+\tau)] - \mu^2\,$$


 * $$ = R_{ZZ}(\tau) - \mu^2,\,$$

where RZZ represents the autocorrelation in the signal processing sense.

Sources of deterministic processes
Upon completing a single revolution, a blade has produced an ever changing torque, and so power. Some of these changes are due to deterministic processes i.e. processes that can be determined and do not require statistical methods. Examples of deterministic processes are listed below:


 * Gravitational loading
 * Tower shadow
 * Wind shear

Gravitational loading
As a blade sweeps through each cycle, gravity is acting on the blade. Depending on the part of the cycle, gravity might be acting to accelerate the blade, or decelerate it. The additional torque that arises on a blade due to gravity is given by

$$ T_{grav} = rmg\cos(\Omega_0t) $$

where r is the length of the blade, m is the mass of the blade, g is the gravitational field strength, t is the time, and $$\Omega_0$$ is the angular velocity of the blade.

Tower shadow
In fluid dynamics, the flow of a fluid is dependent upon boundary conditions. Boundary conditions are influenced by the presence of solid bodies. In a wind turbine, the presence of the tower results in a reduction of the wind speed directly in front of it; that is, the blades experience a reduced wind speed when they pass in front of the tower.

Wind shear
In fluid dynamics, there exists the no slip boundary condition. This states that the velocity of a fluid at the surface of a solid body, such as the Earth, is zero. A consequence of that is that the wind speed varies with height above ground. This effect is known as wind shear. As a result, a blade at the highest part of its cycle will experience a greater wind speed than that of one at the lowest part of its cycle.

Power spectral density functions
Consider an $$n$$ bladed wind turbine. Each blade is separated angularly from a neighbouring blade by $$360/n$$ degrees. That is, for a 3-bladed wind turbine, the blades are 120 degrees apart.

The torque acting on the blade is defined as the z-component of $$\textbf{r}\times\mathbf{F}$$, where r is the radius from the axis of rotation (in this case the hub), and F is the force acting on the blade. If the torque is defined as the z-component of this cross product, then the torque is simply rFperp where Fperp is the force perpendicular to the radius vector, or tangential to the instantaneous velocity of the blade (See figure below)

From the figure above, it can be seen that the torque, T, due to gravitational forces acting on a single blade is given by the following expression:

$$ T = \sum_{k}rmg\cos(k\Omega_0t) $$

where m is the mass of the blade, g is the gravitational field strength, k is a multiplicative integer, $$\Omega_0$$ is the angular velocity of the blade, and t is the time.

For an n-bladed rotor, the instantaneous torque at the hub from all n blades by gravity is determined by summing the effects of all the blades at any one instant. Remembering that the blades are offset from each other by 360/n, the instantaneous torque at the hub from gravity is given by the following expression:

$$ T = \sum_k \left(rmg\cos(k\Omega_0t) + rmg\cos(k\Omega_0t - 360/n) + rmg\cos(k\Omega_0t - 2(360/n)) + \dots\right) $$

Simple trigonometry reveals that only non-zero terms arise when k is a multiple of n. Thus, the overall effect of gravity on the torque is

$$ T = n\sum_{k = 1}\cos(3k\Omega_0t) $$

The covariance function of a sum of sinusoids is itself a sum of sinusoidal functions. Thus, the power spectral density function is a set of Dirac delta functions. The locations of these are at multiples of n. Thus, on a power spectrum, deterministic processes such as gravitational loading manifest themselves as spikes.

Stochastic processes
Stochastic processes are those belonging to turbulence. These require a purely probabilistic approach. A stochastic process such as turbulence is stationary; that is, ...