### 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

*A*and increasing exit area

_{0}*A*. Note that there is always the wind speed

_{1}*V*entering the streamtube, while we saw that the exit velocity

_{0}*u*is a linear function of the axial induction factor

_{1}*a*. As a consequence, there is a discontinuity between the exit velocity

*u*and the ambient wind speed

_{1}*V*.

_{0}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 *u _{1}/V_{0}<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
*u*_{1}/V_{0} - Rotor Thrust
*C*_{T}

**Wake Shear**

- Dominant for
*u*_{1}/V_{0}< 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*curve for momentum theory to the limiting value of

_{T}*C*that represents the drag coefficient of a flat plate at 90 degrees angle-of-attack.

_{T}=2.0The 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*at

_{P,max}*a=1/3.*

*Click for video transcript.*

#### Transcript: Momentum Theory

Wow, there's a lot of information on this diagram, so let's put some order in this. What we're actually showing here is the thrust coefficient on the vertical axis, again plotted verses the axial induction factor a. So here we have the thrust coefficient and here we have the axial induction factor a. And we're doing that for all kinds of rotary wings, aerodynamic flow states.

What we know so far is, well, we know the parabola, we recognize the parabola for the thrust coefficient, which is the green line here. And yes, the green line turns from solid into a dashed line for axial induction factors a>1/2 because we saw that in this case the exit velocity at the end of the streamtube reverses and because of that, momentum theory is no longer valid. What happens now is actually at some point this reverse flow is reaching the rotor disk area itself. It moves progressively towards the rotor disk area as the axial induction factor a increase. Ultimately, if the reverse flow region hits the rotor disk area, we enter the so-called vortex ring state. That's something that's very dangerous if you're flying in a helicopter, and the only was a helicopter pilot can get out of this situation is by moving the helicopter to the side or moving forward. If the axial induction factor is yet further increased, the vortex ring states turns into the propeller brake state. Everything here is highly non- linear and absolutely not something momentum theory can help us with.

So what do we do if we get into the turbulent wake state? What happens here? Well, there is a number of experiments, you see these symbols here that were conducted a NACA and BRC, and for propellers of different solidity and advanced ratios or tip speed ratios you name it, it's kind of a shot gun approach. And what the solid blue line is showing here is sort of a curve fit, and we know the curve fit has to be tangent at some point to the solid green line, the momentum theory is valid. And in the end, for an axial induction factor of a=1, the thrust coefficient becomes 2, which is related to a flat plate drag coefficient. Later on in the course, when we talk about blade element solution algorithm, this empirical relation that attributed to Glauert, will be further explained and be worked with.

Wait a minute, what happens actually if the axial induction factor is zero and become negative? Well we saw the reason why the wind turbine takes energy is because it slows down the main flow. Meaning, the velocity at the rotor disk itself is equal to V knot, the wind speed, times 1-a. So it reduces it. So now if a become negative, 1-a become positive, so we're actually accelerating the flow at the rotor disk. In that case, the windmill becomes a propeller. And still in propeller theory, we also have a region where momentum theory is indeed valid.

One more thing I'd like to point out here, in the wind mill state, the steamtube is expanding, we know that, and the thrust is directed in the flow direction. So it's really a drag, not a thrust. This changes in the propeller state. The propeller accelerates the flow thus the streamtube contracts and the propeller generates a jet in its wake and the result of that is that the thrust force is in opposite direction towards the incoming air speed and the thrust is really a thrust. The propeller is propelling us forward, while the wind turbine in that case, the thrust is more of a drag and it's pulling the rotor backwards and generating a large bending moment on the tower and foundation.

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) / V*_{0} - Rotor Power:
*P = P(V*_{0}^{3}, D^{2}, C_{P}) - Wind Turbine Power Production driven by …
- Wind Resource,
*V*_{0} - Rotor Size,
*D* - Blade/Rotor Design,
*C*&_{P}*C*_{T}

- Wind Resource,
- “Betz Limit”
*C*_{P,max}≈ 0.59- We can only capture a Max. of 59% of the energy in the wind!