Velocity polygons (or velocity triangles) is an important concept that is often used in mechanical engineering to relate the motion of a particle with respect to two different reference frames. While one of the reference frames is fixed, the other has a constant velocity at a given instant. One common application of the velocity triangles is in the turbo machinery. When the blades attached to the turbo machinery rotate quickly, the air particles behavior around the blades is far more complex than you might think. It would be nice to observe the flow of the air from the point of a moving blade, however since we cannot put a person on a quickly rotating blade of a turbo machinery; we have to create a method to visualize the event. For the flowing of air particles there are two velocities: the one observed by a stationary man on the ground, is absolute velocity and will refer to with the character V and the color blue. The second is the flow velocity seen from the moving blade called relative velocity and will refer to with the character W and the color red. The velocity of the moving reference frame with respect to the fixed reference frame is called rotation velocity and represented by the character U and the color green. The velocity polygons is drawn to represent vector addition of:
V = W + U ( this is a vector sum )
This formula would tell us a lot if we knew vector algebra, however most of us do not have these skills. Now we have a dilemma, we can't put a person on the fast rotating blade to see how the air is flowing. And since he/she does not posses the knowledge of vector algebra we canít expect the person to understand the formula above. To overcome this difficulty I will tell you an experience that I had, which may have also happened to you.
While I was driving my car on the highway one night, I spotted a rabbit running TOWARD my car from left side at a high speed. Luckily I did not hit the rabbit, but I was thinking to myself "Why didn't this stupid rabbit just run ACROSS the highway, instead it rushed TOWARD my car? Than it occurred to me what I was teaching all along to my students about velocity polygons. The poor rabbit was darting right across the highway while my car was rushing toward it at a high speed. I saw the rabbit as rushing toward me, because I was sitting on a moving reference frame which was rushing toward the rabbit.
What does the rabbit have to do with turbo machinery or a wind turbine you might ask? A lot if you try to make an analogy here. Let us assume for a moment that the rabbit represents an air particle in the wind blowing across the highway with velocity V, and my car is a turbine blade mowing along the highway with velocity U and I am trying to observe from my car which direction the wind is blowing. Even though the wind is blowing perpendicular (right across) to the highway, I will think that the wind is blowing toward my car at an angle, this is because my car has a high velocity in a direction perpendicular to the wind.
The following flash animation demonstrates this concept. My car location is represented with the "www.flapturbine.com" blue rectangle shown in the bottom right corner. The dashed line shows the path of the rabbit as its darting across the highway. In the bottom left corner the button labeled "CAR FIXED" will show you how I would see the rabbit moving if my car was not moving. Notice that absolute and relative velocities have the same magnitude and direction as shown on the animation when the car not moving. This should be so, because both observer me in the car and the person outside the car is not moving. The other button labeled "CAR MOVING" will show me how I would see the rabbit moving when my car is mowing with velocity U. In both cases the rabbit was running along the fixed dashed line across the highway, however when my car is approaching this dashed line, I would perceive that the line is coming toward me, as such rabbit also rushing toward me with an angle. This angle is very important to know and plays very important role in the wind turbines. This is because HAWT blades is designed such that blades are tangent ( or parallel ) to this angle. ( Do you know why HAWT needs a pitch mechanisms? )
The flash animation displayed below shows the velocity polygons for the flap-turbine. In this turbine there are three sails. There are many flaps on each sail, however in the animation one flap shown on each sails. To understand how the velocity polygons are drawn, we show each flap in different location on the sail. We assume the flap on the center circle is working in optimum condition. Note that the flaps are shown ( with yellow color) and they are opening angle restricted by 90 degrees. In this design the wind blows from top to down direction. The wind velocity shown with blue line for each flap are fixed magnitude. The flap rotation speed magnitude shown by green color is half of the wind speed for the center flap. The outer and inner flap rotation speeds are larger and smaller in magnitude than center flap rotation speed respectively. This is due to their distance from the center of rotation. The relative wind velocities are shown by the red lines and they are constantly changing. Notice that the relative velocities are smaller when the sail moves in the wind direction (positive movement), while they are larger when they are moving up direction (negative motion). (Do you know why the wind velocity is much higher in on arm of the fast mowing hurricane than the other?) Notice also that flaps are closing or opening when ever relative velocity angle is parallel to flap angle. If we did not know velocity polygons, we would think that the flaps would open when sail switches from its positive motion toward negative motion. I saw this kind animation in a couple of places. As it can be seen in this animation, flaps are opened well before the sail reach that point. If you watch some of my YouTube videos, this is really the case.