Basic aerodynamic theory is rooted in the actuator- or rotor disk model. In this lesson, we are going to begin with a power curve of a modern pitch-controlled wind turbine, then continue with the basics of the actuator- and rotor disk models.

Learning Outcomes

By the end of this lesson, you should be able to:

• Understand the power curve of pitch-controlled HAWTs
• Understand the basics of Momentum Theory
• Derive the ‘Betz Limit’
• Include Rotational Effects
• Plot the Power Coefficient vs. Tip-Speed Ratio

What is due for Lesson 2a?

This lesson will take us one week to complete.

Lesson 2a Assignments

Quiz:	Lesson 2a quiz

Questions?

If you have any questions about the course, please post them to our Class Discussion Forum, located in Canvas under Modules in Orientation and Resources. I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you are able to help out a classmate.

If you have a question that you would prefer not to share with the rest of the class, you are welcome to contact me by e-mail, located in Canvas in the Inbox tab.

Power Curve

Figure 2a-1 below illustrates a typical power curve of a modern, pitch-controlled, horizontal-axis wind turbine. It is plotted as power versus wind speed. The solid blue line (P_wind) denotes the power available in the wind. Wind power is proportional to the cube of the wind speed, as shown by the P_wind line. What does this mean? If you double the wind speed, the result is an eight fold increase in the power you can capture from the wind.

The following video explains the 3 wind (or power) regions from Fig. 2a-1:

Actuator Disk Model

The flow field around a wind turbine rotor is very complex. The blades of today’s utility-scale wind turbines are equipped with various airfoils that are optimized to perform at their respective position along the blade span. Depending on the wind speed, blades are pitched collectively to either maximize the power extraction at a given wind speed or to control the power production for wind speeds larger than rated. Furthermore, the blades experience elastic deformations and are exposed to wind gusts, turbulent eddies, and an incoming sheared wind profile of the surrounding Atmospheric Boundary Layer (ABL).

The apparent problem complexity includes time-varying, three-dimensional, turbulent flow with vortical structures. It seems impractical to gain some basic understanding in power production and blade loads without significantly simplifying the problem. For this purpose, let us think about the wind turbine as a device that extracts momentum and energy from the wind, and does so ‘uniformly’ over the rotor swept area (as indicated in Fig. 2a-2). We can hence consider the rotor as an infinitesimally thick actuator disk (surface) that reduces the momentum of the axial wind.

A one-dimensional and axi-symmetric model in fluid dynamics is that of the streamtube, see again Fig. 2a-2. As the wind turbine extracts momentum and energy from the axial free stream, the wind inside the streamtube slows down. Consequently, the streamtube has to expand in order to satisfy mass conservation in the former.

We add a further assumption to the problem that concerns neglecting shear effects within the streamtube. This means, in particular, that there is a uniform wind profile entering and exiting the streamtube. Hence, viscous losses are neglected (inviscid) that prohibit the generation of vorticity and associated vortical structures such as root- and tip vortices (irrotational). We further assume that the actuator disk be stationary (no rotation) and that the uniform wind entering the streamtube does not change in time (steady).

Let us summarize the assumptions the Actuator Disk Model: steady, 1-D, inviscid, irrotational, no rotation.

Basic Streamtube Analysis

Let us refer to Fig. 2a-3 (top plot) for a cross-sectional view through the streamtube. We can see that a uniform wind profile V₀ enters the streamtube. The wind speed inside the streamtube continuously decreases (lower plot) as the wind turbine extracts momentum and energy from the wind profile that enters the streamtube. At the actuator disk, the uniform wind profile is of magnitude u; at the exit plane of the streamtube, the velocity magnitude has reduced to u₁. We note that the wind speed V₀ outside the streamtube remains unchanged.

Before proceeding, let us have a closer look at the exit plane where two uniform velocity profiles come together. An abrupt velocity gradient is present, however our assumption of inviscid and hence irrotational flow allows for ‘free’ slip along the streamtube. Therefore, no friction losses are present in the analysis. – While the velocity distribution along the streamtube is continuous, the situation is different for the static pressure. It is obvious that the local static pressure has to equal the ambient pressure p₀ at the streamtube entrance and exit. However, the continuous flow deceleration inside the streamtube demands an increase in pressure according to Bernoulli’s law (which we will refer to a little later). Therefore, the only way to satisfy that the static pressure at streamtube entrance and exit equals p₀ (ambient pressure) is to have a pressure discontinuity Δp at the actuator disk. It is this pressure jump that acts as an external force to the fluid and extracts the axial momentum, respectively energy and power.

In the following, let us write down some familiar engineering relations.

We can write the Rotor Disk Area as a function of rotor diameter D as:

As for the Mass Flow Rate (m*) through the streamtube, we can evaluate it at the actuator disk:

and at the exit plane (noting that the exit area A₁ at the streamtube is thus far unknown to us):

The pressure jump Δp across the actuator disk acts on the rotor disk area A to produce the thrust force T of the wind turbine:

We note that the thrust force acts downwind in the streamwise direction. This is a consequence of the fact that a wind turbine extracts axial momentum and energy, which is opposite to the behavior of a propeller that generates an upwind thrust force by accelerating the fluid at the expense of engine power. From Newton’s second law, we know that a force equals a change in momentum. Considering the streamtube as a control volume we find that the thrust force T is the only external force acting on the control volume and that this force is balanced by the momentum fluxes that enter and leave the control volume:

As for the power P (which is what we really want in Wind Energy), we remember that power equals energy per unit time. The wind turbine extracts ‘kinetic’  energy from the wind, hence the amount of power being extracted equals the difference in kinetic energy fluxes entering and exiting the streamtube control volume:

In order to compute rotor thrust T and power P we need to find the velocities u and u₁ at the actuator disk and at the exit plane. In addition to mass conservation and Newton’s 2nd law we can apply the classical incompressible Bernoulli equation to the flow inside the streamtube. We note, however, from Figure 2.2 that the pressure is discontinuous across the actuator disk. Hence the Bernoulli equation can only be applied upstream and downstream of the pressure discontinuity.

Video Question: What are the conditions in order to apply Bernouli's Equation?

Bernoulli Equation

We begin by writing the Bernoulli equation ‘upstream’ of the actuator disk as:

In this equation, p is the pressure slightly upstream of the rotor disk. If we define the pressure drop across the actuator disk as Δp, then the pressure just downstream of the actuator disk becomes p-Δp. We can then write the ‘downstream’ Bernoulli equation as:

Subtracting the upstream and downstream Bernoulli equations from one another relates the pressure drop Δp to the unknown velocities u and u₁:

We can use the relation for the pressure drop Δp in the equation for the thrust:

...to obtain u:

It is interesting to observe that the velocity u at the actuator disk is the arithmetic average of the velocities entering and exiting the streamtube.

It is practical in many engineering problems to use dimensionless coefficients (figure 2a-4). Let us define an ‘axial induction factor a through u=(1-a)⋅V₀ which describes the velocity u at the actuator disk as a fraction of the free stream wind speed V₀.

Using u=½⋅(V₀+u₁) we can write the velocity u₁ at the exit plane as u₁=(1-2a)⋅V₀. It becomes evident that in the special case of a=0, the unknown velocities u and u₁ simply equal the free stream wind speed V₀ and both thrust T and power P are zero.

A couple things about thrust and power:

Rotor Thrust and Power

Next, we use the derived equations for u and u₁ in those for the thrust T and power P and obtain:

We hence obtained thrust T and power P as a function of air density ρ, wind speed V₀, actuator disk area A, and the ‘axial induction factor’ a. It is convenient to write both physical quantities in an appropriate dimensionless form.

As for the thrust T, we use the dynamic pressure acting on the actuator disk as the normalization factor:

For the power P, we use the available ‘power in the wind’ P_avail passing through the rotor disk area A as the normalization factor:

We can thus define a thrust coefficient C_T and power coefficient C_P as:

Both are dimensionless quantities and are sole functions of the axial induction factor a.

This means in particular that thrust T and power P can be written as:

The previous equation illustrates the key dependencies for thrust and power:

• The thrust T is proportional to the wind speed squared, the rotor diameter squared, and the thrust coefficient.

In general, one aims at minimizing the rotor thrust for a given wind speed and rotor diameter, i.e. as small values of C_T as possible are desired.

• The power P is proportional to the wind speed cubed, the rotor diameter squared, and the power coefficient.

It is obvious that one would aim at maximizing the power coefficient C_P given a wind speed and rotor diameter. However, we note that there is just a linear proportionality of rotor power to the power coefficient. The biggest effect on the total power P is due to the wind resource V₀³ and rotor size D².

Let us consider again the relations for the thrust coefficient C_T and power coefficient C_P as sole functions of the axial induction factor a.

We note that C_T is a quadratic function in a, while C_P is a cubic function in a. Next, we plot the dimensionless quantities C_T and C_P versus their dependent a in order to find some basic limitations on rotor thrust and power.

As for the thrust coefficient C_T, we find it to be of parabolic shape symmetric about a=0.5. Its maximum value is C_T=1, which makes the rotor thrust being equal to the dynamic pressure force on a solid actuator disk. (Figure 2a-5.) See definition of C_T in equation (2.16).

As for the power coefficient C_P, we find that its maximum value is approximately 0.59 for an axial induction factor of a=1/3. (Figure 2a-5.)

With reference to the definition of the power coefficient C_P in equation (2a.17), it becomes clear that we can at the most harvest approximately 59% of all the kinetic energy in the wind that passes through the streamtube.

The upper limit for the power coefficient C_P,_max=0.59 is called the “Betz Limit”.

Figure 2a-5 also illustrates a diagonal dashed line representing the ratio of the streamtube exit velocity u₁ and the wind speed V₀. From Figure 2a-4, we can see that u₁/V₀ depends linearly on the axial induction factor a. It is not surprising that u₁/V₀=1 for a=0, as in this case of zero induction both rotor thrust and power equal zero. However, the ratio u₁/V₀ becomes negative for a>0.5. This means in particular that there is reverse flow at the exit of the streamtube. This flow state is called the “turbulent wake state” and violates the assumptions of inviscid and irrotational flow of the actuator disk model. We thus conclude that the actuator disk model is valid for axial induction factors a<0.5.

Wake Expansion and Wake Shear

Figure 2a-6 shows that as the thrust coefficient C_T increases, i.e. increasing axial induction factor a, the streamtube expands with decreasing entrance area A₀ and increasing exit area A₁. Note that there is always the wind speed V₀ entering the streamtube, while we saw that the exit velocity u₁ is a linear function of the axial induction factor a. As a consequence, there is a discontinuity between the exit velocity u₁ and the ambient wind speed V₀.

Now remember that the streamtube is a fictitious surface that allowed us to apply mass, momentum, and energy balances within the assumptions of the actuator disk model. In reality, the discontinuity between the streamwise velocities inside and outside the streamtube generates a ‘viscous shear layer’ in the wake. It has been found that wake shear becomes very strong for u₁/V₀<0.2 or a>0.4. This further limits the validity of the actuator disk model to axial induction factors a<0.4.

Wake Expansion

• Velocity Jump u₁/V₀
• Rotor Thrust C_T

Wake Shear

• Dominant for u₁/V₀ < 0.2
• Turbulent eddies

Validity of Momentum Theory

Figure 2a-7 below presents another illustration concerning the validity of the actuator disk (or momentum) theory. Plotted is again the thrust coefficient C_T versus the axial induction factor a. For a<0, we are in the propeller state where the streamtube is contracting, and the thrust force is directed upstream and acts as a propulsion device. Note that one has to provide energy to the fluid for a<0. For 0<a<0.4, we are in the windmill state where momentum theory is indeed valid. The streamtube is expanding, and the thrust force is directed downstream and acts as a drag force. In this case, we are extracting energy from the main flow stream.

The second half of the C_T parabola (a>0.5) is plotted as a dashed line with an indicator that momentum theory is invalid. This is the ‘turbulent wake state’ that begins with a=0.5 and reverse flow at the exit plane that progresses towards the actuator disk with increasing axial induction factor a. For a=1.0, the reverse flow has reached the actuator disk, and the rotor enters the ‘vortex ring state’. This special flow state can be very dangerous for helicopter rotors. At approximately a=0.4, a solid line ‘Glauert Empirical Relation’  connects the C_T curve for momentum theory to the limiting value of C_T=2.0 that represents the drag coefficient of a flat plate at 90 degrees angle-of-attack.

The symbols denote experimental data obtained by Glauert in the 1930s. The steep increase of the thrust coefficient C_T for a>0.4 is attributed to flow separation and stall. In general, one would aim at operating a wind turbine rotor between 0<a<0.4 as close as possible to the C_P,max at a=1/3.

Actuator Disk Model Summary

Let us summarize once more the assumptions and main results of the actuator disk model :

• 1-D, Inviscid, Irrotational, NO Rotation, Steady
• Includes effect of tip speed ratio λ = (Ω · R) / V₀
• Rotor Power: P = P(V₀³, D², C_P)
• Wind Turbine Power Production driven by …
  • Wind Resource, V₀
  • Rotor Size, D
  • Blade/Rotor Design, C_P & C_T
• “Betz Limit” C_P,max ≈ 0.59
  • We can only capture a Max. of 59% of the energy in the wind!

Introduction

Our next step is to develop “The Rotor Disk Model” that allows for rotation of the actuator disk and the downstream wake. The following assumptions, though, remain:

• One-Dimensional
• Inviscid & Irrotational
• Steady

In the simplified model of a rotor disk, we still consider a streamtube, however we proceed one step towards an actual wind turbine rotor by adding a ‘swirl’ or ‘rotation’ to the wake. Without further analysis, we surmise that the addition of wake rotation to the flow inside the streamtube is associated with a loss in power produced by the rotor disk as additional kinetic energy is needed to sustain the rotation of the wake.

The actuator disk theory revealed that the maximum power coefficient C_P is governed by the Betz limit of C_P,max=0.59. We will find that adding wake rotation will decrease the theoretical maximum. – The actuator disk theory generated the rotor power via the rotor thrust and the axial induction factor a. In rotor disk theory, on the contrary, we will discover that rotor power is generated through rotor torque and the angular momentum in the wake via a combination of axial- and angular induction factors.

Rotor Torque and Power

At first, let us recall from classical mechanics that rotor torque is associated with a change in wake angular momentum:

Rotor Torque = Change in Wake Angular Momentum

We denote the rotor speed or the angular velocity of the rotor disk by Ω (Omega). The wake speed, respectively, is the sum of the rotor speed Ω and the swirl speed ω (omega). We thus have:

Rotor Speed: Ω

Wake Speed: Ω + ω

Next, we consider an annulus of width dr of the rotor disk and write the incremental torque dQ acting on the annulus as...

$$dQ=(d{m}^{*}\cdot r^2)\cdot \omega$$

...where the term in brackets represents a mass moment of inertia flowing through the considered annulus of the rotor disk.

We make use of the definition of the mass flow rate in eqn. (2a.2) dm* = ρ · u · dA and write the incremental torque as:

We also remember from previous analysis that the velocity at the rotor disk is u = (1-a)·V₀ , and note that an incremental area dA of a rotor disk annulus is dA = 2 πr · dr

We can also define an incremental power and an incremental power coefficient in the following way:

Rotor Torque and Power

Our next task is to find an expression for the incremental power coefficient d C_P. Before proceeding, though, let us define some additional dimensionless parameters that will he helpful in the analysis and useful to a thorough understanding of the effects of wake rotation.

Angular Induction Factor:  a' = ω / (2 Ω)

Tip Speed Ratio:  λ = (Ω · R) / V₀

Local Tip Speed Ratio:  λ_r = λ · r/R

The ‘angular’ induction factor a’ is defined as being proportional to ω. It is apparent that in the special case of ω = 0 the angular induction factor becomes zero, and the rotor disk model reduces to the original actuator disk model. The next parameter, i.e. Tip Speed Ratio λ, involves the rotor speed Ω and describes the ratio of the blade tip speed Ω · R to the wind speed V₀. We will see later in this course that typical values for λ range between 4 and 7. The third parameter shown is the Local Tip Speed Ratio λ_r, which is simply a fraction of λ based on the local blade position r/R.

Using these newly defined parameters and performing some algebra on the incremental power dP we obtain

...which gives us the incremental power dP as a function of wind speed V₀, blade radius R, the rotational parameters λ and λ_r, and the dimensionless axial- and angular induction factors a and a’.

Hence, the differential power coefficient dC_P becomes

The power coefficient C_P of the wind turbine is obtained by integrating the previous equation along the entire rotor disk, specifically for λ_r ranging between 0 and the tip speed ratio λ.

As the integral for the power coefficient C_P is performed over the local tip speed ratio λ_r, our next task is to find two relations between λ_r and the axial- and angular induction factors a and a’, which will enable us to compute the integral in (2a.27). We will do so by considering the following:

First, let us remember equation (2a.9) from actuator disk theory that related the pressure jump Δp_a across the actuator disk to the axial induction factor a.

Actuator Disk:   Δp_a = ½ ρ (V₀² - u₁²) = 2 ρ V₀² a (1 - a)

What approach would we take for the Rotor Disk?

We will use a similar approach here, however considering only the angular (or rotational) velocity. We write the Bernoulli equation as:

And substitute ω using the definition of the angular induction factor, i.e. a’ = ω / (2 Ω). Thus, we find for the pressure jump Δp_a’:

Equating both formulations for the pressure drop across the rotor disk, i.e. Δp_a = Δp_a’ , we obtain the following after some algebra:

$$△p_a=△p_{a^{\prime }}\Rightarrow 2\rho V_0{}^{2}a\left(1-a\right)=2\rho {\Omega }^2{r}^2{a}^{\prime }(1+a^{\prime })$$

$$\Rightarrow a\left(1-a\right)={\left(\Omega \cdot \frac{r}{V_0}\right)}^2{a}^{\prime }(1+a^{\prime })$$

The above equation constitutes a first relation between a, a’, and λ_r . In order to evaluate the integral for the power coefficient C_P in (2a.27), we must find a second relation.

We mentioned earlier that the addition of wake rotation is likely to reduce the maximum power coefficient C_P,max=0.59 due to Betz in actuator disk theory. One of our objectives in rotor disk theory is to understand to what extent do the newly introduced parameters a’, λ, and λ_r  reduce the Betz limit. By inspecting the integrand in (2a.27) we realize that the maximum power coefficient occurs for the integrand factor a’(1-a) attaining a maximum for each λ_r. We therefore want to maximize the function f(a, a’) = a’(1-a). As a first step, let us compute the first and second derivatives of f(a,a’) using the product rule:

Next, we must perform the necessary algebra find the second relation between a and a'.

We've just shown that the 2^nd relationship between a and a' may be an extreme, but we don't know yet if this relation will result in maximized or minimized power. In order to better understand this, we have to look at the second derivative of  d²f / da².

Check the second derivative...

...from the first derivative...

...and the first derivative of eqn. (2a.30) ...

Remember the first equation (2a.30) equated the pressure differences. We now also have its first derivative, and we will take its second derivative as well and obtain:

Now we use all of the above relations in the second deriviative:

Assuming that the 1^st derivative is equal to 0, we know that:

Thus, we can indeed find that …

So far we have eqns. (2a.30) and (2a.32), for a and a'. Equation (2a.30) was a result of equating the pressure drop across the disk:

And equation (2a.32) from maximizing the integral for the pressure coefficient:

Now we have 2 equations and 2 unknowns. We can take a' from eqn. (2a.32), which is a sole function of a and plug it into eqn. (2a.30) to find an explicit relation for a.

$$a(1-a)={\lambda }_r^2\frac{(3a-1)}{(1-4a)}\cdot \left(1+\frac{(3a-1)}{(1-4a)}\right)$$

$$a(1-a)={\lambda }_r^2\frac{(3a-1)}{(1-4a)}\cdot \frac{(1-4a+3a-1)}{(1-4a)}$$

$$1-a={\lambda }_r^2\frac{(1-3a)}{{(1-4a)}^2}$$

That gives us now the axial inductor induction factor a as a function of λ_r , which is scaled tip speed ratio, along the blade for maximum power.

Now, what is the actual C_P,Max?

We take the derivitive d/da of boxed equation (2a.42) above and obtain:

Now take eqns. (2a.32), (2a.42), and (2a.44) in the relation for C_P,max, and we get the following:

At the root where  λ_r = 0, this will define the lower bound of the integral of C_P,max with respect to a.

For the upper bound at the tip where λ_r becomes equal to the actual tip speed ratio λ, it defines the upper bound for a.

 

The table below provides different values for the tip speed ratio, λ, and solves eqn. (2a.42) for the appropriate upper integration bound, a₂. It then provides the associated maximum power coefficient. Note that the quicker the rotor spins (λ increases), the closer it gets to the Betz limit.

Values for the tip speed ratio λ, upper integration bound a₂, and associated maximum power coefficient.

λ	a2	CP,max
0.5	0.2983	0.289
1.0	0.3170	0.416
1.5	0.3245	0.477
2.0	0.3279	0.511
2.5	0.3297	0.533
5.0	0.3324	0.570
7.5	0.3329	0.581
10.0	0.3330	0.585

Let us get back to the integral for C_p,max:

If we use the transformation x = 1- 3a and its derivative dx/da = -3 we obtain:

If we use this to look at the distribution, we find the induction factors a and a' as functions of the local tip speed ratio, λ_r, for maximum power for an example reference case of a tip speed ratio that equals 7.5.

Let us recall these numbers...

The axial induction factor $a=1-\frac{u}{V_0}$ where u is the axial velocity of the actuator or rotor disk, and V₀ is the wind speed.

$a^{\prime }=\frac{\omega }{2\Omega }$ where Ω is the rotor speed.

And for the optimum case $a^{\prime }=\frac{(1-3a)}{4a-1}$

The graph below (Fig. 2a-10) illustrates this distribution. Keep in mind that this is for a specific tip speed ratio that is fairly high at 7.5. The graph shows a and a' versus the radial position along the blade (multipy this number by the tip speed ratio 7.5. to get λ_r.)

The solid line represents the axial induction factor a. At the root where r or λ_r equals zero, the induction factor a is 0.25, and it approaches the ideal value of 1/3 towards the tip.

The dotted line represents the angular induction factor a'. In the relationship $a^{\prime }=\frac{(1-3a)}{4a-1}$ , a approaches 1/3 when a' goes to zero. Out at the tip the effect or the loss due to rotation becomes zero, and a approaches indeed the Betz limit. However, when a becomes close to 1/4, as it does at the root, we're dividing by zero and a' actually has a singularity.

Video question: In Fig. 2a-10, why does a' have a singularity?

The plot below (Fig. 2a-11) illustrates the maximum power we can get including the wake rotation. We have the tip speed ratio as a free parameter λ, which equals the velocity at the tip Ω • R, divided by the wind speed V₀ : $\lambda =\frac{(\Omega \cdot R)}{V_0}$

We're plotting power coefficient versus tip speed ratio. The solid line represents actuator disk theory, or the the Betz limit, which is 16/27 or approximately equal to 0.593. This is the maximum you can acheive. If your tip speed ratio is 0, the power will be zero. As you increase the tip speed ratio, the power coefficient gets closer to the Betz limit.

Video question: Is there an ideal tip speed ratio?

Validity of the Rotor Disk Model

Parameters:

$$a=1-\frac{u}{V_0}$$
$$a^{\prime }=\frac{\omega }{2\Omega }$$
$$a^{\prime }=\frac{(1-3a)}{4a-1}$$

The two relations between a and a':

$$\frac{a(1-a)}{a^{\prime }(1+a^{\prime })}={\lambda }_r^2$$
$${\lambda }_r^2=\frac{(1-a){(4a-1)}^2}{(1-3a)}$$

Eliminate the λ_r² by subtracting the two equations from one another and you'll find:

$$a^{\prime }=\frac{1-3a}{4a-1}$$

As we approach locally the ideal axial induction factor of a = 1/3, a' naturally goes to zero.  It will also do this for high tip speed ratios. An interesting twist is that the effects of wake rotation are smallest for a quickly spinning rotor. This seems counter-intuitive at first as the loss due to wake rotation is related to the 'swirl' that's being added to the downstream flow.

In general, the higher the rotor speed, the closer one approaches the limit of the uniform-flow actuator disk. And the faster the tip speed, the smaller the relation between the ω (the swirl you're adding to the wake) and the actual rotor speed Ω.

Rotor Disk Model Summary

Let us summarize a few lines about the rotor disk model: 

• 1-Dimensional, Inviscid, Irrotational, Steady
• Includes Effect of Tip Speed Ratio  λ = (Ω·R) / V₀
• Rotor Power still remains a function of the 3-2-1 law:  P = P(V₀³, D², C_P(λ) )
• Approaches “Betz Limit” of C_P,max ≈ 0.59 for high λ

This is the essense you should you know as well as how to sketch the relative distributions of a and a' prime versus radius and for different tip speed ratios.