Welcome to this session about flow and forces around a wind turbine blade. The learning objectives of this session is that you will be able to draw a velocity triangle for the flow around a blade section. You will be able to explain how the aerodynamic forces drives the rotor around. And you will be able to calculate the lift on a blade section based on airfoil data. The role of the wind turbine is to extract velocity from the wind and to generate power and we know that it cannot actually break the wind, because the wind loses velocity. But there's something more going on in the flow, and I'd like to show you that on a real turbine. So here is the turbine. We have the wind coming in in this direction here and we could imagine that we were some fluid passing, which moves say from outside and then into the rotor. Now the turbine rotates. So this means that as this fluid kind of meets the rotor and forces it to go around, because the fluid forces the rotor around, the rosin will also force the fluid around and the fluid pass-through will also get an induced velocity in the an opposite direction as the rotation of the rotor. So, it kind of comes in, leaved the rotating rotor, drives it and gets a velocity in the other direction. That is sketched in this drawing here. So we can see the expansion, and this C theta, that is the induced rotation in the wake. And when we go down in the final position here, there will also be an induced rotational velocity, that is called C theta 1. We are able to describe the flow that goes into the area here around the blade in terms of a velocity triangle. So, if we look at this line here, which is called the rotor plane, then we would have the incoming wind speed coming from below and we know that there will be an induction because it's kind of stopped a little bit. So the turbine will not see, say the full incoming wind speed but just a part of it. Because the turbine moves around the rotor, the rotor being a component which is Omega times r. Omega is the rotational speed, r is the local radius. So that would be a component, and due to this induction, it will not only be omega times r, it will be an axial induction factor, which is called a' times omega r cross omega r itself. So this thing here is called the velocity triangle flow. We can take this reduced action velocity and the induced rotational velocity into account in our analysis. The simplest analysis is the 1D momentum theory. Which tells us that there is a power coefficient, which is called C_p. It has an optimal value of 59% of the incoming flow. But this doesn't take into account that some of the energy is actually used to drive this rotating flow behind the turbine. So we can go one step further. And that is, with this analysis here. Where we actually define also an induction coefficient for the rotational flow, a'. And we can express that the power is an integral from the middle of the rote and then out to the largest ridges along the blade. So the power is 4 pi r cubed rho induction factor prime, 1- a, V_o, omega squared, dr . And this comes out of a turbine theory. From that one here we can also derive a C_p. So now C_p is not just the function of the actual induction, it's an integral which involves both the actual induction and the rotational induction, and we have some numbers involved. Lambda, that's called the tip speed, so the ratio between rotational speed at the tip and incoming wind speed. And x, which is kind of the local ratio between the rotational speed and the incoming speed, not at the tip but somewhere else on the blade. Now as for 1D, we would like to say optimize our Cp and optimize our values for a and a'. So we use our velocity triangle for that. And there is an angle here between the rotor plane and the incoming wind speed. That angle is called phi. It is a flow angle. And by looking at this big triangle here we can express the tangent to that angle is actually 1-a, V_0, this distance here. Divided by 1 + a' omega, which is this distance here. The small triangle up here actually shows us the induced velocities. So this is the actual induction which removes incoming wind velocity. This one is rotational induction, which adds to the rotational speed. Phi can also be found up here, and that is under the assumption that the lift force dominates. And by that assumption, we can express tangent to phi inside the small triangle. Phi a' omega r divided by a, v_0. And that can then lead to this equation here, which kind of puts x, a' and a together in one condition that needs to be satisfied. So now our remaining job is then to optimize the C_p under this restriction here. And this is the resulting equation that comes out of it, which tells you that if you know x, you're also able to solve this equation for a. That's how the optimal a is found. The benefit of all that is that we are then able to compute the forces on the blade. So let's imagine at a certain precision, out along the plate, I know x. I know the rotational speed. I'm able to compute a and a' from the previous formulas. Then I'm actually able to draw my velocity triangle and express what my relative wind speed is an its angle phi. Then from airfoil theory, I could identify an angle of attack. That's the angle between the incoming speed and then the court line going through the front edge and the back edge of airfoil. And the lift is then perpendicular to that line. The drag goes in the same direction as the inflow. The interesting thing now is that lift and drag, they can be projected in the direction of the rotor plane, some contribution from lift, some contribution from drag, which detracts. We have have a final small component. That is actually the component that drives the plate around in the rotor plane. There's another component that is perpendicular that's the normal component. That's the one which contributes to the trust on the rotor Sometimes this takes a little while to get into, so let's have a look at a real turbine and see how these forces they look. So, here is the turbine again, we have the in flow of direction here and now we kind of line it off, so it becomes clear. So this is a copy of the sketch we had before. So we have the inflow of velocity in this direction here. Because of the rotation of the rotor, there will be this induced velocity omega times r and with induction. Then forms the resulting inflow velocity, and we have the lift coming out here as the green line. The drag is perpendicular to lift, and together we can project them into this upward force which then drives the rotor around and there will also be a force in the normal direction. That's the force which is kind of responsible for the thrust on the turbine. So it would force the turbine in this direction here. So are we able to compute these forces? The answer is yes. We have from standard mechanics that the lift force is one half rho times the chord length here, times the lift coefficient, times the V_rel squared. So that's kind of the speed coming in here and it's expressed here in terms of the inflow speed with induction and the rotational speed with induction. And there's a similar formula for the drag. Now since we know the flow angle phi, we're also able to project lift and drag onto the normal and tangential direction and that gives us the expression for the forces that drives the rotate around. How would we know about the seal and sd the we would know that from air foil data, so here are some examples. This is lift as function of angle of attack, so here we see that the bigger the angle of attack the lift goes up until we get to storer, and that's also drag and lift coefficient plotted together here for this same air foil here. So data like that are available and can be used for such computations. So, in summary, here we have learned that say, a rotor uses both an axial and a rotational component. We have learned to draw a velocity triangle for the flow that comes into the blade section, we have been able to explain how the lift force drives and contributes to the torque on the rotor. And we have learned how to calculate lift based on airfoil data.
