An Explanation of Aspect Ratio and how it affects sailingbyIvor G. Slater, P. Eng. |
NOTE: | The suggestion in recent side messages that I would refine the CIRCULATION part of the aero-hydrodynamic story in an addendum, to be labelled AR-2a, will not apply. The additional explanation and development appears below, after the short reviews of AR-1 and AR-2. This way, you needn't wonder if you're missing something. If you get the integral numbers, AR-1, AR-2, AR-3, AR-4... etc. you have it all. |
REVIEW OF PART 1 In part 1, we saw how dynamic sail forces of Lift and Drag (and also hull-rig parasitic drag) could be be related to the Apparent Wind. We then combined those forces into a Resultant. We resolved that resultant into Thrust in the track, and side force, the latter to be overcome by the underwater hull and its appendages (if any) at a cost of hydrodynamic drag opposed to motion that has to be overcome by Thrust.
REVIEW OF PART 2 In part 2, after a somewhat bumbling start on my part (warning people not to misuse data that they may not even have known about, and in the process hinting at subjects that should have been reserved till later - for which I do apologize - we got down to the subjects of Vorticity and Circulation. We discussed the Karmann Vortex Street, which is the alternate shedding of vortices from opposite edges of bluff bodies. The potentially strong, or even violent, action of the KVS is a common experience on highways around large fast transports, and in heavy downwind rolling of vessels whose sails are stalled. We then looked at vortices caused by a simple body like an oar in water. We conducted a mental experiment (which can be done in practice any time) showing Lift. Finally we saw that a rotating cylinder in a stream (a spinning mop handle was the experimental tool) can generate lift. From that we were warming up to the idea that the function of an aero-hydro foil is to generate circulation. Then, the circulation combining with free fluid flow is the direct cause of Lift. That brought us to the threshold of the "lifting line" theory of wings, in which the wing is replaced by a line.
Then Bill Heinlen exploded onto the list, with what sounded like frustration of some years' duration. He suggested, and I concur, that he may be speaking for many inquisitive but mute sailors, who have been effectively shamed into silence by past treatments of circulation. Those interested in a well-grounded understanding are indebted to Bill for speaking up, because he has forced me to sharpen a too-casual approach. I am grateful for the opportunity to try do better. It will delay our arrival at Aspect Ratio a little. But the more solidly we build our understanding, the more approachable and easier the rest will be.
We might summarize Bill's statements and questions this way, where
the numbers correspond to my later answers and comments:
() | I've seen those circulation diagrams, and sidestepped, feeling skunked. Is circulation real(1), or is it something imaginary like the long-disproved aether that was once supposed to fill the universe as a medium for light transmission? Does it have a valid theoretical base(2), at least equal to the theories of internal combustion engines? What the !@#$%^ is circulating anyway?(3) They tell me there's circulation around my sails, but my tell-tales all point aft (4). Is this circulation something like a giant slinky wrapped around the wing, wasting energy as it spews off into space (5)? Is my confusion just because I don't have mathematics(6)? |
Ultimately, the mathematics of vorticity do reach a rarefied stratum, but we don't need them. Properly presented (as I'll try to do) it's not hard to understand circulation at an intuitive level.
Circulation is real (answer to 1). It can actually be seen and photographed in certain conditions. Here I'm not speaking of evidence of cast-off vortices, but the equivalent of windward-side tell-tales blowing forward. To see the circulation alone, you must suddenly stop the free stream (or the motion of the foil, which dynamically is the same thing.) About a sailboat of course, that is scarcely possible. But it can be done experimentally.* When the circulation is unhampered by the free stream, it causes the FLUID (answer to 3) to circulate,** and it continues to do so until viscosity wears it out.
Ordinarily, you don't see the circulation alone, because its velocities add vectorially (we'll review and extend this idea soon) to the free stream velocities. Usually, free stream velocities are higher than circulation velocities. To begin, let's put it very crudely. If you have a free stream velocity of 4, and a circulation velocity at a given distance from the foil of 1, then the altered pattern of flow produces 4 + 1 = 5 velocity units (aft) above the foil (or lee side of the sail) and 4 - 1 = 3 velocity units (aft) below the foil (weather side.) All tell tales point aft. Yet there is (masked) circulation in the picture.
Since submitting AR-2, I have realized that a combination of haste, carelessness, and a little un-oiled rust in my brain may have caused me to mislead you somewhat. The trailing vortex is NOT the circulation somehow unwinding itself off the wing. I apologize for giving grounds that might have led to that wrong impression. The starting vortex, which you can see very clearly in the oar experiment described in AR-2, is the equal and opposite partner of the circulation around the wing. But if you look closely, you'll see that the trailing vortex (joining bottom of oar to bottom of starting vortex) looks thin and spindly by comparison with the starting vortex. In some conditions of interest, which we'll examine, circulation and the starting vortex can exist without trailing vortices. The trailing vortex is a separate effect, on a wing of finite size, of the differing pressures that cause Lift. [More shape and pressure-effect detail in a later part.]
The trailing vortex represents a price you pay in a finite system, for what you want, namely Lift. It isn't necessarily associated with waste, meaning it isn't always bad. Think of money you keep on your dashboard to pay bridge tolls. If a thief steals your money, it does you no good. That's like the action of "parasitic" drag (AR-1) in all conditions except downwind. But if you want to cross the bridge (gain Lift), you pay the toll. In upwind conditions, you might find a cheaper bridge, in the form of a higher Aspect Ratio. But in offwind conditions, what looks like the enormous "Lift-cost" of a low Aspect Ratio suddenly turns into free "manna" that scuds you along with a power unavailable to high aspect ratios. I believe that this is the answer to the part (5) of the query, dealing with waste.
In what follows in the series, we're going to see and understand in detail exactly how all this rises, and hang numbers on it.
Just above, I "added some velocities" representing free fluid flow and circulation, to make the point that circulation is usually masked by the stream flow. The addition procedure carried a sloppiness of thought that is not good enough to deal with vectors properly, and we're going to repair it.
The following discussion of vectors will help us to understand circulation better, and reinforce our tools for dealing with the dynamics of the yacht system. But it has more than academic value, for the examples deal with navigation, useful to every sailor. It will delay our arrival at Aspect Ratio somewhat, but I believe that it will put our understanding on a stronger base.
Suppose you are sailing South. Physically, you are sailing in the "north-south direction." Direction is taken as the orientation of a line, in this case your line of travel. The "sense" of your motion is South. Say your speed through the water is 6 knots. This is the "magnitude" of the total entity known as your "velocity vector."
A "vector" is a way of representing and handling physical entities that require definition of magnitude, direction and sense for their full description. Things like force, acceleration, velocity, and even spin, are vector quantities. The most puzzling of these at first glance might be the notion of spin as a vector quantity. Look at it this way. Think of a child's toy top. The axis of the spinning top is clearly related to the "direction" of its spin. We might decide arbitarily to call counter-clockwise spin positive, and relate that to the upward "sense" on the line. If we reverse the spin (going clockwise), we show that by pointing the "spin sense" down. Intuitively, we can grasp the idea that a big top may have a greater "magnitude" of spin than a small one, or that the magnitude of spin is increased when the same top is spun faster. If we were able to measure or describe the magnitude of the spin, we could choose a scale and draw an image of the spin vector, showing all three elements of a vector quantity.
I used the term "drawing an image of the vector" to drive home the idea that vector quantities (force, physical displacement, velocity, acceleration (i.e. rate of change of velocity), and spin, remain vector quantities whether we think of them that way or not. If you and I bump one another slightly in passing, our speeds may not change, but we may be slightly misdirecterd by the carom. So our velocities have been altered and a reactive force was required to do that.
The magnitudes of vectors are called "scalar" quantities. We are quite accustomed to scalars in notions like units of money, potatoes, and buckets of water. We add and subtract them as numbers, as 2 + 2 = 4, and 12 - 7 = 5. But we cannot add or subtract the scalar magnitudes of vectors to get anything meaningful. Twelve knots East MINUS 7 knots South doesn't equal 5 of anything. Neither does 12 knots East PLUS 5 knots South equal 17 of anything. Vectors can only be added and subtracted by vector methods.
To ADD vector B to vector A, place the tail of B on the head of A. The Sum, or Resultant, is the shown in magnitude, direction and sense by the line from the tail of A to the head of B. |
You will see that the ADDITION of vectors may well lead to scalar magnitudes lower than those of the original vectors. If you sail East 12 knots against a 4 knot Westbound current, vector addition shows you that your speed over the bottom is 8 knots East.
The "sloppiness" that I referred to above was in apparently adding and subtracting circulation vectors to the free flow as though they were scalars. Since we were assuming fully parallel flows at the points of combination, I wrote 4 + 1 = 5, and 4 - 1 = 3 as the combined velocities of flow above and below the foil. You will see that even in this short-form approach, it would better to write, 4 + (+1) = 5 and 4 + (-1) = 3 to indicate that we are dealing with vector addition.
Generally speaking, in work of this type, we assume that one set of senses is positive, and the opposites are negative.
To SUBTRACT Vector B from Vector A, place the head of B against the head of A. The vector difference is shown in magnitude, direction and sense by the line from the tail of B to the tail of A. |
Conversely, vector ADDITION gives us the True Wind vector from our knowledge of Boat velocity and Apparent wind velocity that we observe while sailing. Prediction of True wind helps us know what conditions to expect, and the requirements for sails and trim after we change course. To determine the True wind, place the tail of the Apparent Wind vector (say SE 14.1 knots) on the head of the boat velocity vector (say East 10 knots.) The SUM is a vector North 10 knots, meaning a 10-knot South wind. (Remember currents are defined by the way they are heading, winds by the direction from which they are blowing.) Practical sailors learn to make these assessments with acceptable accuracy by "waving their arms." When it comes to America's Cup racing or efforts of similar intensity, there is a need for more precision. In such fields, each astronomically expensive sail is at its best in a limited range of Apparent winds (direction and speed.) Preparing the wrong one for the next leg means losing the race. So careful prediction of True wind, new boat Velocity, and new Apparent wind is done by computer (and some sweating by the skipper.)
I will mention only for completeness that determining what course to steer to OVERCOME the effects of a current involves the subtraction of vectors.
In AR-1 we touched on the RESOLUTION of a given vector into "components" in other directions that had more meaning and value for us. You may recall that we combined sail Lift and Drag by vector addition to get a single Resultant sail force. That still didn't have any direct value in assessing the performance of the boat. So, we "resolved" the Resultant into two directions that ARE of interest, namely in-track and cross-track forces. Even in the generalities of the discussion, we were able to see how this would help us relate sail and hull forces.
This resolution of vectors into components in other directions of interest to us is the basis of Velocity Made Good (VMG.) If our destination lies to the north, but we are constrained from sailing directly towards it by wind or obstructions, we may want to know how quickly we are making ground towards it. To do that we draw a line through the tail of our boat velocity vector (Vb) in the direction from us to the destination. We "drop a perpendicular" from the head of the Vb vector onto the line representing the track to our destination. The length of the intercept is the Vmg.
One more thought about resolving vectors, and we're done with our review of them. Imagine that you have to sail around an irregular closed loop. Suppose you cannot follow it exactly, so you end up meandering back and forth across it as you sail the circuit. Still, at every point and instant in your circuit, should you choose to do so, you could resolve your Vb into a component parallel to the loop, and a component at right angles. This is an idea that will help us to understand the idea of the "Strength" of a circulation.
Just one more bit of general background before we go on.
It is a general rule of mathematics, physics, and engineering, that we may make whatever assumptions we like, and use whatever symbols we like, as long as we state them clearly, and remain consistent. However, in some fields and applications, custom has established terminology and standard views that it would be wasteful and discourteous to others to ignore. Typical of these are the measurement of angles positive clockwise from North in navigation, and positive counter-clockwise from "paper East" in mathematics and physics. Some symbols, (like F or f for force, V or v for velocity, M or m for mass, a for acceleration, lowercase Greek letter "rho" for density, upper case "Gamma" (like an upside down L, for circulation), "Pi" for the ratio of circumference to diameter of a circle - this latter appearing a million times a day in mathematical physics in the most unlikely seeming places - are so common that it is scarcely necessary to define them when they are used in context. |
Therefore, in absolute and unquestionable terms, the force between a rig and and the wind is equal to the rate of change of momentum of the wind. That appears mostly as its change of direction, not speed. Unfortunately, we cannot assess how great a mass of air is being accelerated to what extent. All early attempts to assess transverse (i.e. Lift) fluid forces on bodies in a flow, based on "mass reaction" ideas failed. They underestimated how much fluid was being redirected by how much.
Only circulation theory can tell us what the Lift will be.
Now we're getting ready to think seriously about circulation. We'll be helped in that by the notion of a "velocity field." Let's start with the free flow, because it's easier. Imagine you are looking across a uniform flow coming in from your right at a speed of 10 units. At every point in that "flow field" you could represent what was going on with a velocity vector of magnitude 10, heading level from right to left. This is the velocity field of the flow.
Now let's stop the free flow, and introduce a circulation. Is it related to the spinning of the top? Not quite. The top spins as a solid. The velocity of every point in or on it is directly proportional to its distance from the centre of spin. But in fluid circulation, the velocities closest to the source of the "spin" are highest. They reduce with distance from the centre, tending towards zero at some indefinite distance from the centre. You can see this by watching a starting vortex caused by an oar. Spiral galaxies like our own Milky Way also reveal this kind of vorticity.
How does anyone ever deal with such circulation? How do you "total" and account for this effect over some great indefinite range? Well, fortunately we won't have to. But with the appropriate mathematics, the computation is surprisingly simple for aerodynamicists. By thinking about a velocity field for the circulation we'll see how their mathematics makes the assessment, and the surprising result.
Let's imagine that we are looking into our fluid field with a counterclockwise circulation centred more or less before our eyes. We can imagine a vector at every point in the fluid showing the magnitude, direction, and counter clockwise sense of the velocity of every particle in the field. Magnitudes are higher close in, lower further out. Now, let's draw an arbitrary closed loop around the source (centre) of the circulation. It can be any shape we like, as long as it doesn't cross itself. It can stretch near or far. Now let's RESOLVE the circulation velocity vectors at every point on that loop into a tangential and radial component. We make the points indefinitely close. Then we sum up, by a mathematical process called "integration" the total of all tangential components around the loop. That sum is called the STRENGTH of the circulation, which aerodynamicists have characterized as "Gamma" (Greek upper case letter G.)
Now, the remarkable thing (to me at least) is that it doesn't matter where or how we draw that loop. For any given circulation, the summation of tangential components on any loop around it is always the same. You might perhaps account for it with the idea that, as you go further out, you pick up more points, but with lower tangential velocities, and so keep a balance. Mathematicians would jump all over us for such a sloppy statement, but if it helps us to accept the idea of the strength of a circulation, I for one will take it.
The "distortability" of circulation that is implied in the last paragraph also helps us to see how it can wrap around an airfoil, increasing in loop size and changing shape to accomodate the size and shape of the airfoil without losing itself, or its strength. Obviously, because of its greater size, even the inner velocities of the foil circulation will be lower than the inner velocities of the starting vortex.
Nature does not support single unbalanced vortices, although sometimes their complement is seen only as a "virtual" or "mirror" image in a solid or fluid interface. The trailing vortex of the oar experiment has its virtual image above the water surface, and the masthead trailing vortex of your rig has its virtual image below the sea. If you have space between rig and hull, you also have a trailing vortex there, with its own mirror image under the sea. You might imagine, correctly, that this is a factor in aspect ratio, as we'll come to see.
Several times, I have suggested that the starting vortex and the circulation are an opposite-equal pair. Let's try to get an intuitive sense of what that means. Let's do a slight revision of our oar experiment from AR-1. Put the oar in the water, flat of blade athwart the flow, as a bluff body. The Karmann Vortex Street will appear, shedding alternating vortices from opposite edges. Note in particular, the form of those vortices. To help you understand, if you're on a moving boat, the port side of the oar is producing counter-clockwise vortices, and the starboard side clockwise vortices. There is no lateral force, except perhaps for a fluttering caused by the shedding of the vortices. (Because of that, if you're going too fast, you may have trouble holding things steady enough to see what's going on.) Now slowly rotate the oar clockwise, so reducing the angle of attack. The shedding of vortices will continue, although they may change in size and frequency.
At some point in the rotation of the oar, the building vortex on the leading-port side disappears, without having being shed. It has wrapped itself around the oar as counter-clockwise circulation. Lift suddenly appears, very much larger than the drag, and the KVS and its fluttering stop. The last vortex shed from the starboard, now "trailing," edge is what we call the starting vortex.
If you do this experiment, use it to show something else as well. Start with a fairly deep immersion of the oar blade, say three or four times its width. Note the fairly large rotation from the bluff body position before the KVS stops and lift starts. Next immerse the blade about its own width. Note that Lift starts and the KVS disappears at a much smaller rotation from the bluff body position, which is to say at a much coarser angle of attack. This is another observation of the effect of Aspect Ratio that we are in the proces of understanding in detail.
Now we come to another view of Circulation and Lift.
Have you ever noticed the action of a spinning top, or gyroscope, when you push on it? Its spin is in one direction: you push one end at right angles to the spin: and the gyro moves off at right angles to both push and spin in a third direction (and left or right) determined by the sense of the spin.
So it is, in a similar way, that when we add (positive) counter- clockwise circulation of strength Gamma, whose direction is away from us, to (positive) free stream velocity V from the right, there is a (positive) upward Lift. According to the science of aerodynamics, its magnitude L (in pounds) is given as,
L = | (rho) * V * (Gamma) or the product of mass density of the fluid, times the free stream Velocity times the strength of the circulation. when all are expressed in appropriate units (that we'll discuss in AR-4) |
You'll notice that we didn't have to guess what masses of air were being accelerated downward how much to produce the upward reaction of Lift. I offer the next as a reassurance that we're not about to go into mathematical orbit. Fortunately, as practical users, we don't have to use this formula to apply aerodynamic data. It comes in more convenient forms, called coefficients.
But we're not yet quite finished with the meaning and value of Circulation. Lift can be seen in yet another way. Skilled and specialized aero-mathematicians can transform the field of vector sums of Circulation and Free-Stream velocities around a cylinder into an image of the flow around an airfoil. This allows them to calculate the velocities, and hence the additive and subtractive pressures (applied to free stream pressure) at every point on the surface of the foil. Summed over the entire foil, these pressure differences represent the force on the foil. These mathematical predictions are so precise that (I quote from a long-ago memory of a statement in Abbott and von Doenhoff) "extreme accuracy and precision is needed in the making and testing of models to produce results of comparable reliability."
The coefficients of force that I referred to are "non dimensional," meaning that they contain no physical units themselves. That way they can be applied to systems of different size, velocity, and fluid density, without corrupting the equality between them. There are several sorts of coefficients. The one of most interst to us here is the Coefficient of lift (Cl) (Lower case L for sections of wings.) Using the coefficient idea, Lift in pounds is given as:
L = |
1/2 * (rho) * (Cl) * S * V**2 Where the right side represents the product of: (rho) the mass density of the fluid (Cl) the section lift coeficient S the plan area (Surface of the foil section) V**2 represents the square of the magnitude of the free-stream Velocity |
Now I hope that this final bit will help to nail down the reality and value of circulation. Let's equate the two expressions for Lift, getting:
(rho) * V * (Gamma) = 1/2 * (rho) * (Cl) * S * V**2 |
(Gamma) = 1/2 * (Cl) * S * V |
Now, isn't this wonderful! We have been talking about the circulation of a fluid, and we find that the measure of its strength is expressed in cubic feet per second (in the FPS system), analagous to a discharge of air being delivered by a fan to circulate air around your heating system. How else would one describe the time rate of fluid motion? For me, this is the final nail in the coffin of doubt about circulation.
Well, there it is. When I offered to explain Aspect Ratio, I never expected to get so deeply involved in circulation. However, I remain grateful for Bill Heinlen's reaction that prompted this. It forced me into an examination of ideas that I haven't looked at for a long time, and then caused me to tidy up my thinking. I apologize for the length. I have re-examined and rewritten this Part 3 several times, pruning, and trying to clarify. If it's still muddy, let me know. If it has truly brought you an insight into circulation, my effort is well repaid.
How the qualities of a "foil" cause circulation and determine its strength (and hence Lift), is the subject of AR-4.
Ivor Slater (slater@fvc.bc.ca)