IE010520A

Chapter 9
The Verticals

The two-dimensional blade aerodynamics vector diagrams for the vertically oriented wind turbines have not seen widespread publication, if they have seen any publication at all. So what is presented in this chapter both fills a longstanding need and has a not-to-be-denied heuristic quality that opens the door to new wind generator insights. Everyone sees and knows how wind moves across the face of the land and approaches wind machines. But what happens then must be looked at from the standpoint of the moving blades. The world looks different when one bends down and looks at it upside down between one's two legs. The same is true for vector diagrams, within which the blades are fixed in place and everything else is moving relative to them.

By now the reader should be familiar with these diagrams and how they work. By way of review the diagram that applies for the horizontal wind machines that was discussed in Chapter 7 previously is reprinted below. What continues to be important about it is the clarity with which it shows that the blades work not only quite well at a zero degree pitch angle but that the driving force reaches a maximum at this angle and it even continues into the negative pitch angle region.

Some qualifications to this observation apply due to the occurrence of flow separation, as was discussed in the previous chapter. Since in most cases flow separation limits the power obtainable by the machine its occurrence presumably is controlled by means of engineering factors in design and, if so, then the flow at some distance from the surface of the blades is subject to deflection as well. All flow deflection, even that at some distance from the blade surfaces, causes forces in the blades. By keeping the blade pitch angles at values close to zero degrees and even into the negative region and limiting the flow separation, it is seen from these diagrams that the driving forces present are substantial. The way it is represented here is by means of the colored areas to be seen to the right of the y-axis. The wind is deflected and provides driving force vectors within the entire regions colored yellow and orange. A measure of the amount of total driving force ("integrated" over these areas, in terms of the vocabulary of calculus) is therefore the size of these areas.

In the below diagram, the "T" stands for "tip", as in "blade tip", and the "R" stands for "root", as in "blade root". Bear in mind that, even though the driving force is greater at the blade root, the energy conversion is also proportional to the tangential velocity of the blade at the location in question. Certainly, the tangential velocity of the blade tips is much greater than that at the blade roots and so the one factor makes up for the other and horizontal machine blades are known to be quite efficient for their entire lengths.

The vector diagrams for the verticals are presented in what is to follow in a series. All these diagrams have certain common features about them and they are shown in the first of the series, the diagram immediately below. What is to be noticed first is that the blade circuit is shown as a circle to the left of the y-axis. The center of the circle is displaced to the left by an amount proportional to the tangential velocity of the blades as they go around their paths, the faster the blades are rotating, the farther left the displacement. The radius of the circle, meanwhile, is proportional to the wind velocity, the greater the wind speed, the larger the circle. The wind is shown as a vector pointing from the periphery of this circle in toward its center and, as the blades turn, the tail of this wind vector moves around the circle in a counterclockwise direction, now pointing from one direction, now another. The direction of rotation is arbitrary but it is always the opposite of that seen in the actual movement of the blades in the machine from the standpoint of an observer looking from above. The small circle near the bottom of the diagram and to the right is intended to so depict the actual machine and its rotor orbit and it is seen that the blades in this view rotate in a clockwise direction. Naturally, the blades may rotate in the opposite direction as well.

The resultant wind velocity actually seen by the blades is not shown but follows the same rule as applies to previous diagrams, that is, it is a vector named "A" that is obtained by adding the V and the W vectors. In this case, this vector is simply that obtained by drawing a new vector from the tail of the W vector (as it goes around the circle in its path) to the head of the V vector at the origin of the graph where the x and y axes cross. Clearly some features exist here that are not present in the case of the horizontals. The A vector moves and is sometimes above the x-axis and sometimes below.

In this next diagram, the A vector is shown in this "snapshot" taken of one blade at the point P' on its circuit of rotation. This is a point about halfway between the point where the blade is moving parallel to the wind and in the same direction (point B) and the point where the blade is moving across the wind on the downwind side of its orbit (point C). The vector A is shorter than the V vector, something different from the case of the horizontals, but even so can be seen to be generating power due to the large deflection angle as it is converted into the vector B by the blade. In fact, the blades never see a reversal of the direction of the wind and generate power all around the entire circular path except for the brief moments at points B and D when no airflow deflection occurs.

The activity below the x-axis is similar to that above and similarly results in a positive driving force on the blade.

In fact, the next diagram shows the vectors for the cases of blades both above and below the x-axis and at the points of maximum driving force, points A and C. If the machine has an even number of blades (2, 4, etc.), then two of them would be at these locations in this snapshot, resulting in the driving forces shown. This is the General Case and most of the Darrieus machines, especially the two-bladers can be represented by this diagram. Note that due to the curvature of the Darrieus blades, however, the tangential velocities vary along the blade lengths, and so, while the diameter of the circle may stay the same, the distance of its center from the origin of the diagram varies and so some portion of the circle may cross over into the area to the right of the y-axis at locations near the blade roots. Since not much power is expected to be produced in this region, the flow reversals are not of any significance but can be expected to occur.

Note, again, that the W vectors always point to the center of the circle. Note also that the Darrieus machines are normally constructed in such a way that the blade pitch angles are nominally always about zero degrees and with no provision for dynamic pitch changes during rotation. Why this is so can probably be ascertained from the history of this invention in that it has always been found to work well. The diagram shows why. Good airflow deflection is obtained both on the upwind and downwind sides and power is produced quite efficiently. No variation of the pitch angles is provided around the circuit and none is necessary. The blade shapes also are seen to be symmetrical on the one side as opposed to the other and this allows for some simplicity of manufacture, at least in that this avoids complexities resulting from unusual blade shapes. The Darrieus seems to, in other words, be a simpler machine and seems also to have resisted most efforts to introduce complexities within it, like a recipe that is known to satisfy the palate without anyone quite knowing why.

Having said this, this material now takes it upon itself to do just that. It has always seemed to be something of a question as to whether other blade pitch angles, never anything other than fixed and unvarying but different from zero degrees, would be more efficient. These diagrams provide a way of determining this. In this next diagram, the effects of changing the blade pitch angle to a negative 10 degrees are shown. The B vectors now have an upward slope to them due to the necessity that they be parallel to the blade chord as can be seen by carefully reviewing the diagram. What happens is that the size of the yellow area representing the driving force from the upwind side of the machine has been reduced and is now much smaller. Meanwhile, the size of the corresponding yellow area on the downwind side (above the x-axis) has seen a considerable increase. The practical result, then, is that the wind passes through the blades rotating on the upwind side with little interference or deflection and interact strongly with them on the downwind side. Thus the power production is shifted from a sort of balance between the upwind and downwind sides to an unbalance, with the greatest part being obtained on the downwind side.

At this point, some further aerodynamics not shown by the vector diagrams comes into play. For power to be produced on the upwind side, some deflection and other interference is caused in the airflow which limits the power that the downwind side has available to it. The concave or "cup" shape of the downwind side is more highly beneficial to efficiency as well. Between the two sides, it is seen that, given a choice, the downwind side would be preferred and here is a method in which such can be provided. Again, the blade pitch angle is fixed yet this fine nuance can be added to wind machine operation as a result predicted by making use of these vector diagrams.

The last three diagrams below show what happens as the blade rotation rate changes for a fixed windspeed. When the machine is stopped, the circle is centered at the origin of the diagram, crossing over into the area to the right of the y-axis. The A and B vectors show good deflection (not shown) if drawn from points A and C on the circle and hence startup torque can be quite good, despite the reputation these machines have of not being good starters. The windflow is reversed at point B and on the rest of the circle to the right but is not seen as causing undue interference. Naturally, at least one blade must not be in the neutral positions (B and D).

As the machine gathers rotational speed, the blade tangential velocity reaches the velocity of the wind. This is shown in the second of the three diagrams. The circle is now entirely to the left of the diagram but point B passes through the origin. This means that the blades now no longer see flow reversal and at only this one point the blades are moving downwind with no apparent airflow over their surfaces. This seems to be the situation most lay observers see as the usual operating mode of the verticals. Point D is seen as a point at which the blades are travelling upwind and therefore a source of loss of efficiency.

But more is to follow. The third diagram shows something resembling fully developed operation, with the tangential blade speed at some multiple of the wind speed. It is, after all, the advantages of the lift principle, as a great improvement over the drag principle, that have allowed wind energy to find its place as a source of energy. In the lift principle, recall that the blade is driven forward while traveling faster, even much faster, than the windspeed. Here we see this principle in action. The blades at both points B and D see significant airflow over their surfaces, several times the windspeed, and so there is very little difference between them. This, again, is another case in which these diagrams are useful in depicting the realities of how the aerodynamics plays out.

Formulae For The Blade Driving And Lateral Forces

In the case of the horizontals wind machine configuration, the subject of the last chapter, a series of mathematical expressions was presented that, derived as they were readily from the geometry of the vector diagrams depicted there, provide a way of computing what everyone always wants to know - the actual forces developed by the blades in these machines under various conditions. If the reader takes away nothing else from this material it would be sufficient only to have placed on file a reference to these. Herewith below now, in exactly the same fashion, are provided the set of formulae of which use may be made to find the exact same unknowns in the case of the verticals wind machine configurations. Another angle must be included, the angle of the blade on its way around its rotational circuit, and so some additional complexity is involved. However, again as before, simplification is obtained by defining and making use of the apparent wind acting on the blade, which is a combination of the actual wind and the "wind" the blade sees due to its forward motion at its design velocities.
Here are, then, the full, detailed expressions for the blade driving force and blade lateral force:

The simplified versions of the above formulae, obtained by substituting into them an expression for the apparent wind (which, again, is normally several times greater than the actual wind) are now provided as well:

It is hoped that the reader will agree that the ready explanations and clean cut nature of this chapter in particular are attractive and a source of satisfaction. As such, this particular material may have merit as a contribution to the technology of renewables.