AERSP 583
Wind Turbine Aerodynamics

2a.2 The Actuator Disk Model

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Bernoulli Equation

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

 p0+12ρV02=p+12ρu2This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.7)
 

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 pp. We can then write the ‘downstream’ Bernoulli equation as:

 (pΔp) +12ρu2=p0+12ρu12This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.8)
 

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

 Δp=12ρ(V02u12)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.9)
 

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

T=ΔpA=m*(V0u1)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.10)
 

...to obtain u:

 u=12(V0+u1)This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(2a.11)
 

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.

Calculating the axial induction factor to find thrust and power. Refer to text for details.
Figure 2a-4. Calculating the axial induction factor to find thrust and power.
Source: Aerodynamics of Wind Turbines. Hansen (2008). © 2015 Penn State. Reproduced with permission.

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:

Click for video transcript.

Transcript: Thrust and Power

We note a couple of very important things. Number one is thrust is proportional to the square of the wind speed. And that is also reflected in a quadratic dependence on the axial induction factor a. While the the power P is proportional the cube of the wind speed. We had mentioned it earlier. And that is also reflected in a third order dependence on the axial induction factor a.