Erratum to AR-4:

Bone-headed again! Of course there's a mistake in the direction of the dust devil described in the beginning of AR-4. Any idiot should be able to tell that it's heading straight for the hole in my head. Ivor

Let's talk about the Coefficients now. By dividing the Force observed in a particlar test (or primary calculation) by 1/2 (Rho)SV*2, the observor arrives at (Cf) which applies to that section shape at that particular angle of attack. Checking the full range of useful angles of attack, for Lift, she is able to draw a curve, like your Lift curve, plotting (Cl) against (alpha-0). She may also measure Drag. She may plot that against (alpha-0) also, or alternatively she may plot Drag against Lift, in a "Lift-Drag" curve. All these methods are used to present data for useful shapes for which the parameters are given. For a given foil, tests will be made and plotted run for a range of Reynold's Numbers, usually, 3, 6, and 9 million.

One more coefficient you may see, and don't need to worry about too much, is the "Moment Coefficient." It is a measure of the turning effect of all forces on the section, (taken about the quarter-chord point) where the positive sense tends to increase the angle of attack. This turning effect around the quarter chord point is usually small for normal airfoils, meaning effectively that the centre of pressure is close to 0.25c. Some mean lines, with deep chord well aft (deliberately done in some foils to distribute pressure differences uniformly fore and aft) may have the centre of force almost at mid chord. This is good in water-propellor blades, to spread out and minimize low pressure, and so reduce "cavitation" (the tendency of water to boil and "ping" destructively in areas of low pressure.) The effect may also be seen in a sail with a baggy leech, so producing a "hard-mouthed" steering effect.

Lee helm is rare , but if it is a serious problem for you, this could be a way to cure it. Half seriously, go buy a bagged-out sail. At least it won't cost you much to try it, and it will be easier than sawing off your bowsprit, raking your mast like a skipjack, or installing a mizzen. Some high-performance wings have generically similar meanlines. That sense of "horrible drag," with the wind being turned "up into next week," is nowhere near as severe as intuition and a bad press might suggest.

Otherwise, our generic Force coefficient may take the form of a Lift Coefficient (Cl), or a Drag Coefficient (Cd). Again, the lower-case descriptors (subscripts) are the conventional way of identifying these as SECTION coefficients. Wing Coefficients, which we'll meet in Part 5, take the upper-case forms (CL) and (CD).

Now, we come to one more bit of AHdynamic serendipity. It has to do with the "slope" of the Lift-coefficient curve, plotted against angle of attack (alpha-0). By pure theory, it should be about 0.11 [change of (Cl)] for each degree [change of (alpha-0).]But, because of viscosity effects (that we'll look into soon), the steepness of the curve is reduced slightly in real fluids. Here, within the linear range, (Cl) changes by 0.1 per degree incremement of angle of attack (close as damn is to swearing) for almost any moderately decent foil, including flat plates).

So, for example, a symmetrical SECTION, presented at 6 degrees angle of attack, has a Lift coefficient of 0.6. [Even I can remmber this.] It is a powerful bit of information that we can use directly in many ways.

Here is a a "generic notion" of how we will be using this information: Say we have calculated that sailing to windward at 6 knots in smooth water, the rig and sails develop a Side force of 1000 pounds. We want to know the keel area required to limit leeway to 2 degrees. [We'll take into account the high-lift effects of flapped keels, and skegged rudders in a later Part.(6?) Here we're assuming that the foils are symmetrical and in line.] We substitute 0.2 for (Cl) in equation (9) (correponding to 2 degrees), substitute for the other parameters and solve the equation for S. Now, having S, say we want to know what leeway we will make in storm conditions to windward, when speed is reduced to 3 knots, and side force has increased to 1500 pounds. We substiute our value of S in the equation, and the new speed, and solve for (Cl). The leeway in degrees is ten times that number. A little later on, I offer ways to simplify the calculations.

Again, the last paragraph is only a generic example, having two deficiencies. The first is that a real (finite) keel, wing, sail, "gives away" some of the "geometrical" angle of attack, in a way determined by Wing Theory. Section Theory applies to what is left, called the "Effective angle" (alph-e). We'll be looking at that in Part 5.

The second thing missing above was a knowledge of the proper dimensions and units for the various parameters in Equation (9). Physical equations are only valid when the units of both side are equal, as well as the numbers. Historically, there have been many systems of units used for dealing with reality. The great development work of AHDynamics was done in the variously-dubbed, English, engineering, or foot-pound-second system. That is what the bulk of available data refers to. It is the system we follow here.

In this system, the unit of length is the foot, the unit of force is the pound, and the unit of time is the second. This makes several of our parameters easy to define. Area (S) must be expressed in square feet. Velocity must be in Feet per second. The coefficient (Cf) relates to angles. They are defined in terms of length divided by length (L/L) and so are "dimensionless," as are our coefficients.

That leaves (rho). Several times I have referred to it as "mass density." That suggests, correctly, mass per cubic foot. But what IS mass and what are its dimensions and units? Well, briefly, mass is matter, which clearly differentiates it at once from the idea of force. In all coherent systems, the unit of mass defines "that clump of matter" that will be given "unit acceleration" by "unit force." To avoid undue difficulties for non-technical readers, let's first define acceleration. It is the rate of change of Velocity (itself having the dimensions of Length divided by Time.) So acceleration has the dimensions V/T, meaning (L/T)/T. We short circuit this "double division," and say that the dimensions of acceleration are Length divided by Time-squared (L/T*2). The FPS units are feet per second-squared.

Now all this is to introduce a rather strange unit of mass that (unless you're an English-speaking engineer) you've probably never heard of (and to help you understand it.)

For a consistent FPS system, the unit of mass must define how much of it will be accelerated one foot/second-squared by a force of one pound. In the English speaking world we have grown accustomed to an anomaly involving force and mass. We buy a "pound" of butter and think of it as a "quantity of matter," in other words, a MASS. We also think of the effect that gravity has on it, namely its "weight" as a FORCE of one pound. But if we drop this mass of butter, it doesn't accelerate at one foot/second-squared. It accelerates at 32 feet/second-squared. Either the unit of force is too high, or the unit of mass is too low, to create a consistent dynamic system. Old physicists, when they deigned to deal with such a "common" field as the English system, resolved the dilemma by inventing the "poundal." This force unit was just 1/32 the "weight" of our pound of butter. Under its influence, the block of butter would then accelerate one foot/second-squared - say on a level frictionless surface.) That was a consistent and workable system. But engineers were already well accustomed to the "pound force," and had found the (pound mass - poundal force) system too dinky. So they went the other way, and invented a unit of mass that would be accelerated one foot/second-squared by a FORCE of one pound. Obviously, it must be the equivalent of 32 blocks of butter. They called that mass the "slug."

So the "mass density" of a material is the number of slugs of it in a cubic foot.

The "weight density" of sea water is 64 pounds per cubic foot. Now you see that its mass density is 64/32 = 2 slugs per cubic foot. For sea water, that's the value of (rho) to put in Equation (9). [If you need to be picky, you can divide 64 by 32.2 and use 1.98]

Air, in average sailing conditions has a volume of about 12.5 cubic feet per pound. Running that through a calculator turns up (rho) for air = 0.OO25 slugs per cubic foot.

Velocities in Equation (9) must be expressed in feet per second. But we'd rather deal in knots. One knot = 6080 feet per hour, which we can reduce to 6080/3600 = 1.6889 feet per second, which is identical to velocity in knots, (Vk) = 1.

We can simplify future arithmetic by calculating values for air and water for S=1, (Cf)=1, and (Vk)=1. Under these conditions, for sea water:

     F(sw, S=1, Cf=1, Vk=1) = 0.99379 pounds (10)

For air:

     F(air, S=1, Cf=1. Vk=1) = 0.00125 pounds (11)

Using (10) we can calculate that the Lifting force on 9 square feet of keel, at a lift coeffficient of O.3, moving at 6 knots, is:

     9 (for area) times 0.3 (coeff.) times 6*2 (for V) times .99379 = 96.6 pounds.

Using (11) we can compute that the Lift on 360 square feet of sail operating at Lift coefficient 1.2 in 16 knots of wind is:

     360 (area) times 1.2 (coeff.) times 16*2 (for V) times 0.00125 = 138 pounds.

Remember that we are not yet taking into account wing effects.

** Real and "Ideal" Fluids

Have you noticed that there is a certain dichotomy in fluid dynamics? Often we can approach systems in two different ways. Lift, for example. We saw that we could equate it to a force generated by a circulation in a free stream, and get a direct answer, L=(rho)V(gamma) at right angles to stream and spin. (Actually, it takes fairly powerful vector mathematics to predict the direction, sense, and magnitude of the lift this way.)

Alternatively, we can vector-add to the free stream the circulation velocity components parallel to the foil. So we may determine the pressure distribution around the foil. We can sum pressure differences times bits of area over the whole foil and get total force, or LIFT. We can (also) this way determine the turning effect, or Moment of the force, tending to increase or decrease the angle of attack.

After that review, you probably won't be too surprised to hear that the essential ideas of fluid dynamics employ two different versions of the fluid, it's "real" and "ideal" forms. The real fluid is the one we wave a fan (or oar) through, feeling resistance as we move air (or water.) The ideal fluid is in all ways identical, except for the subraction of the visocity. To repeat something said in an earlier discussion, viscosity is the fluid-frictional resistance of a fluid layer to sliding over its neighbouring layer. Without viscosity, the ideal fluid is like a world of micro-miniature ball bearings (of the same density as the real fluid) but entirely without friction, able to slip around obstructions in any direction, at any speed. A fan waved in ideal air feels no resistance, and moves no air. And rowing in an ideal fluid will never get you home. However, if somebody will give you a starting bunt, you'll skim along forever.

This sounds like Alice in Never-Never Land doesn't it? (From 1200 miles away, I can feel Bill Heinlen squirming in his seat, ready to spring.) And yet, except for certain "starting" conditions, and a very limited area close around the foil (called the boundary layer that we'll examine soon) the fluid acts, and we treat it as, an ideal fluid. Outside these areas the real fluid moves along without developing any viscous forces, and in that sense it is "ideal." The ideal fluid has density and it can correspondingly develop dynamic energy, corresponding to variable pressures that can be transmitted through the boundary layer to the foil.

Now let's see how the contrast of the ideal and real fluids helps us to understand both the start, and strength, of the circulation, and the development of profile drag.

Imagine a circular cylinder in a flow of ideal fluid (always from our right of course.) The flow divides around a line at the middle of the upstream edge. (In a section we think of that line as a point, called the "stagnation point.") It is convenient to think of this as the end of a stream-line. But in fact no fluid actually stops at the stagnation point because the width of a line is zero, and there is no fluid in it. The fluid divides around the cylinder, and meets itself at the after stagnation point. There it parts from the surface as smoothly as it split on the leading surface. All velocities and accelerations are balanced around the surface, so no forces are developed.

Now let's put an airfoil section in an ideal flow. Let's set up a noticable angle of attack. We are accustomed to thinking of the forward stagnation point as being on the lower-leading edge, with fluid flowing smoothly off the trailing edge. But the ideal fluid says, "I don't have to do that. That's just an out-of-shape cylinder. I'm going straight for its middle (well abaft the leading lower edge.) I'll divide there, and scurry at equal speeds about both sides, and leave from the middle of the back (well forward of the trailing edge." And of course, without any stickiness, the ideal fluid can do just that, turning round the sharp trailing edge with ease, and no loss.

Now we put the same airfoil in a real flow. (In this bit, to make a point, we anticipate, a little, the more complete discussion of boundary layers that follows soon.) The layer of molecules next to the surface sticks to it. Every succeeding layer takes force to move it along at a slightly higher speed, until at a short distance from the foil, free stream velocity is reached. The fluid out there, having no need to slide past itself, doesn't show its stickiness (viscosity) and behaves like an ideal fluid. Now, If I may be forgiven for apostrophying the fluid once more, listen to the close-in layers approaching the trailing edge from below, "Hey you guys on top there, I can't turn this sticky mess around this trailing edge to catch up to you in the middle of the top there as if I was some flakey ideal fluid. If you insist on jumping off from that ideal stagnation point, and I let go here, there's going to be a hole in the universe, and you know the boss gets upset about things like that. Let's put in a requisition for something to cure this problem, before the boss dumps on us for breaking Somebody-or-other's law of fluid flow continuity."

Here, kindly Nature lends a hand. She sticks her fingers into the flow and gives it a counter-clockwise snap. Being clever, she sees that this provides a double cure for the problem. It speeds up flow on top, and slows it down below. But it also moves the forward stagnation point counterclockwise (further forward and higher), so reducing the differential of flow-path length on upper and lower surfaces, and hence the need for hurry and slow down. Mother does like economy in her systems! She also thinks "the little punch of downward velocity that I've given the flow behind the trailing edge is another way of explaining Lift. Won't my engineering children have fun finding all the ways of accounting for it, and measuring it!"

I'd like now to lay to rest the ongoing discussion of whether circulation is "real" or just a "convenient mathematical fiction." To say that circulation is an artificial concept is comparable to saying that the true South wind is a mathamatical fiction that you subtract from boat velocity to produce the apparent wind. While you are sailing, you see only a vector combination that we call the apparent wind. To see the true wind, (one of the components in the effect) you must stop the boat. And there it is, a South wind. In my way of looking at the world, it is as real as it needs to be. Did it stop while you were sailing, just because you didn't see it? I think not. In the same way, you don't see the circulation around a lifting foil, because it is added to the free stream (think apparent wind if you like) in a way that creates a re-formed flow about the foil. You must stop the free stream to see circulation alone. But you must look quickly because viscosity drags it to a halt. It's somewhat as though the South wind died down a second or two after you stopped sailing. But circulation is just as real as the true wind.

Viscosity also has two other effects. It sticks air to the section profile, more on the upper side of the trailing edge than elsewhere, so creating a thicker boundary layer, effectively changing the section shape slightly. One result is a slight reduction of the angle of attack. That accounts for the lower slope of the Lift curve than pure circulation theory predicts. (This is the reduction from 0.11 to 0.10 increment of (Cl) per degree change of (alpha-0) that we saw earlier. The corresponding thickening of the trailing edge also produces a "wake" whose effect is added to surface friction and measured as the "Profile Drag" of the section.

Here we can see some of the effects of increasing the Reynold's Number (NR.) In a fixed system, it varies as velocity. Higher velocity wipes off some of that upper-trailing-edge boundary layer, so effectively increasing the angle of attack, and hence Lift coefficient. At the same time, by "thinning" the effective trailing edge, it reduces the width of the wake, and hence the Drag coefficient.

There is one boundary layer that we are all familiar with, although we may not think of it that way. It belongs to the earth itself, in a breeze. Rare is the sailor who has not heard the saying, "The apparent wind is fairer aloft." We relate the idea to the lovely spiral twist in the array of a windjammer's yards as she takes advantage of the phenomenon. We acknowledge it by not getting too excited about some twist in our mainsails. What it means is that the true wind, faster aloft as it is, when subtracted from our boat speed, produces both a faster, AND more favourable wind aloft. (To display this graphically, draw our boat velocity, true wind, and apparent wind triangle, and see what happens as you lengthen the true wind vector.)

Increasing velocity with distance from the surface is typical of all boundary layers. So let's stay with study of the earth and wind for a moment. It will help us understand sailing conditions, as well as provide insight into AHfoil action and stalling, and broaden our base for approaching Part 6.

This imagery may help to provide a beginning feeling for laminar boundary layers: Think of a large surface like a smoothly-frozen lake, with no obstructions for miles. Fancy that the space above it is empty, until we encounter air 200 feet above it, and higher, which is moving steadily as a block with no "internal" motion. Now, bring that moving air down until its bottom surface touches the earth. The lowest layer of air, molecular thin, latches onto the surface as though it were glued. The second layer, held back by friction from the stopped layer below, slips ahead, but not as fast as the third which is pulling it, and the third is not as fast as the fourth, and so it goes, molecular layer on layer, steadily getting faster, until the speed of the "block of air" (equivalent to our free stream) is reached. Knowing just where full free stream velocity has been reached is difficult. So by aerodynamic convention, the boundary layer (as it is called) is considered to end where velocity has reached 99% of free stream velocity.

What we have just described is called a "laminar" boundary layer, related to the idea of movement in layers or sheets. It can appear in real winds, blowing over miles of very smooth water, not too fast. If you have a fast boat, it can create the impression that you're racing about in a calm. I once had the experience of such a wind on Lake Ontario, sailing my fast little boat C'est Moi 21 miles dead to windward, tacking through 90 degrees hourly, doing 4.3 knots through the water, hard on an apparent wind caused by a true breeze that I could scarcely feel when we hove to, looking for it. Jib drawn away, she'd go racing off again. The lake was like a millpond, with not a ripple on the water. Occasionally, a tiny catspaw appeared and vanished. How can this be?

Well, it has to do with the shape of the "Wind gradient" (the pattern of changing speeds with height.) In laminar flow it is "parabolic." (This is a mathematical analysis, confirmed by observation.) That means it can be represented by an equation of the form h = kW*2, where W is wind speed, and h is height, and k is our ever-ready friendly constant. Wind speeds are usually measured and reported at a height of ten metres, 33 feet. Just to put C'est Moi's story into a credible framework, let's use that parabolic equation to see the general form of that wind. Suppose that at a height of 1O metres (1 metre =3.281 feet) the wind was blowing 2 knots (perhaps a little more, but not much.) We can substitute these figures in the equation to get "k." [As, 10=k(2*2) leads to k=2.50] We can now rewrite the equation with this value of k, getting V = square root of (h/2.5). From this we can make a table of heights and windpeeds, getting:

h meters	W knots	h metres	W knots
0.0	0.00	5.0	1.41
0.5	0.45	6.0	1.55
1.0	0.63	7.0	1.67
2.0	0.89	8.0	1.79
3.0	1.10	9.0	1.90
4.0	1.26	10.0	2.00

You will see from this that, half lying in the bottom of a half-decked daysailer-cruiser, I wasn't likely to feel much of a half-knot breeze when we stopped. Yet the force of the (about) six knot apparent winds on the upper reaches of her sails while she was sailing was adequate to drive her to windward in such still water. (By the way, these are the only conditions in which a so-called laminar-flow keel would give an advantage.)

Perhaps I should explain that although I designed her in 1957 and launched her in 1958, C'est Moi was decades ahead of her time. She was a double-ended centreboarder, with a large, refined rig (rotating, foil-shaped mast, so her mainsail was much like a bird's wing), a light, slender, fair, and soapy-bottle-smooth hull, and the most refined underwater foils I could build. She sailed extremely fast for her 22 feet OA, 16+ foot waterline. (Speed-length ratios of about 2 (8 knots) almost anytime the wind blew, sometimes carrying a fair load. And often she reached speed length ratios of 2.25.) It was she who caused me to discard all traditional notions of limits on sailing speed, and start the investigation that led to the Index of Sailing Speed. (Not published, but so valuable I believe it should be, later.)

Laminar flow within the boundary layer (where all the viscous forces are being developed) is the least "energy expensive" (lowest resistance) way of getting a fluid past a solid. Unfortunately, it is difficult to establish and maintain. Incoming turbulence (roiling) of the fluid, can prevent it. Unfair surfaces and roughness kill it. And, in AH terms, it needs a favourable pressure gradient to drive it along.

That last idea needs explanation. Between the free stream and the surface of the AH-subject (it could even be a hull) we have this (possibly very thin) layer of fluid struggling to move ahead against the resistance of viscosity. It needs a push or pull to keep it going. The only way we can provide that is to cause a lower pressure ahead of it than behind it. This requires that the local flow (outside the layer) must be accelerating (getting faster) in the flow direction. Then, following Bernoulli's law, the pressure drops in the flow direction, so "pushing and sucking" the laminar boundary layer along. The way to maintain a continously accelerating flow is to create a constantly-diverging (wedge) shape as far back as possible. It can only be carried so far, because the shape and fluid must close again, of course. We'll talk more of this a little later, and the classes of shapes that attempt it.

Incidentally, while we're thinking of "driving" laminar flows, reflect on the slight driving force of that rare laminar wind on Lake Ontario, just described. It rose after a protracted calm, giving seas and swells time to subside (meaning zero water turbulence). If the barometric pressure difference had been large, the wind would have been stronger, causing ripples and waves, and then air turbulence. If a ship had gone by somewhere to windward (even over the horizon) the wake would have disturbed the wind, which would probably then have turned the excitement into wavelets at least. So we had a very slight barometric pressure difference, spread over more than twenty miles of lake (it is more than 30 miles wide in the area) moving the air so gently as to maintain a laminar boundary layer. I think you'll agree - a rare concurrence of events. Although summertime Lake Ontario is prone to this sort of thing, I have never, before or since, seen such a protracted phenomenon. This one seemed like a magical wonderland that I have never forgotten.

Now it's time to consider turbulent boundary layers. They are the ones we encounter more often in our "earth model." The wind isn't moving as a uniform block to be drawn into layers of sliding sheets by viscosity. Instead, it is tumbling and mixing, excited by weather events, land masses, hot spots, swooping over waves, and that so-and-so motor yacht that's passing too close on your windward side. Here, the energy to keep the boundary layer moving isn't provided directly by pressure gradient. Indeed, a pressure gradient isn't needed. There is a "feeding" of the boundary layer by velocity exchange. Low velocity air from deep in the boundary layer migrates up and picks up speed from contact with higher, faster levels, and migrates back down again in all sorts of random patterns to deliver it to lower levels. While more energy is expended, this is a much more dynamic and stable system than laminar flow.

The Velocity gradient of turbulent winds (representing also other turbulent boundary layers) is not parabolic. Indeed, so far as I am aware, it is not susceptible to mathematical prediction. So we are left with empirical observation. Back in the late forties, and early fifties, one of the many things I did, instead of paying full attention to my university studies (I also read hundreds of NACA Reports and Bulletins, which had nothing to do with required studies either), was to spend time reading scientific meteorological reports, looking for empirical formulas for turbulent wind gradients. I'm sorry to say that I don't remember any of them. But that's not a great loss to us. Almost any reasonable sketch we might make has as much chance of being right in some conditions as a formula (which might relate to other conditions.) To get an impression of the "shape" of a turbulent wind, you might do something like this:

Assume the speed at ten-metres height is 10 knots. Take paper, (best type would be squared), and draw a vertical scale for heights (metres are convenient) 0 to 10. Draw a velocity scale to the right, 0 to 10 at least. You know that the wind gradient is going to pass through the points (0,0) and (10,10) for both laminar and turbulent winds. If we go through the same excercise as before for a laminar gradient, (I have just done it for you) we find that laminar velocity at height of 1 metre would be 3.2 knots. At 2 metres it is 6.3 knots. (Here we are making the very large, and improbable, assumption that a wind of such speed could remain non-turbulent.) Now sketch in the parabolic curve. The turbulent gradient would lie between that and the horizontal axis and the vertical 10-knot line, in a "squarer" pattern. Having just made the sketch at my elbow, my guess would be that the turbulent line would go through (7knots, 1m), (8knots, 2m), (9knots, 5m). If you compare the two gradients, and look for sailor's language, you might perhaps say, "In turbulent conditions, the wind comes closer to the water." I couldn't have said it better myself.

Now let's have a look at the idea of "separation." Perhaps we should leave the earth-wind model here, and come down to smaller sizes. If the driving force of a boundary layer diminishes, flow in the boundary slows down. If the drive stops, so soon does the boundary layer flow.

In LAMINAR layers, in the condition of most interest to us, this means that a minimum pressure "peak" (corresponding to highest velocity) has been reached in the local stream outside the boundary. All the outer fluid can do from here on is slow down, thereby increasing pressure in the flow direction. In response, the boundary layer starts to flow away from the higher pressure, against the stream. When this happens, we say it is "separated."

You might gain an insight into separation in general from thinking of "back eddies" in rivers and ocean currents. Here the central flow is not able to deliver enough energy to the "boundary water" (which is encountering resistance from the river bank, or the unmoving bulk of the ocean) to keep it moving ahead. So it stops, and, under the influence of a higher pressure ahead, moves backwards. (Actually, most such separations are "turbulent" which we'll look at next.) I hope that this idea helps you to understand that a "separated" boundary layer has not wandered off into next week, never to be seen again.

Turbulent separation has a different cause and explanation. It is caused by a thick boundary layer building in the direction of flow. This may appear on the surface of a vessel's hull. More particularly, in the sort of AHdynamics that we're looking at, it is a thickening layer on the upper surface of the foil towards the trailing edge. It rises from the combined effect of increasing (alph-0) and (alpha-lo). Practically, the turbulent boundary layer is becoming too thick for the "molecular interference" transport mechanism to deliver enough energy to the lower levels to keep them moving ahead. So the lower levels slow, stop, and move forward, more or less rolling up, forcing the outer shape outwards, effectively changing the body shape. This is the nature of that slow loss of lift, and drag increase, we called the "gradual stall."

In some conditions, a separated flow may "re-attach." Older section shapes, of the thick-leading-edge and fine-tail type, had a particular propensity for this. (This is the sort of foil that one is apt to see in popular illustrations, and in many pseudo-scientific sailing books explaining Lift - often with the Lift vector pointing impossibly far forward.) These shapes had a characteristic early velocity peak on the upper surface (minimal pressure.) Occurring there, as (alpha) increased, the LAMINAR separation commonly "punched up" into the free stream, and, in the suddenly-started turbulence, was "reattached" as a turbulent flow. What this really means is that the free stream washed the boundary layer back down into forward-moving turbulent contact with the body. (Not perfect, but better than "permanent laminar separation," and sudden stall.

Turbulent separation may also be reattached. The cause may be accidental or deliberate. Those of you who fly much may have seen evidence of attempts at deliberate boundary layer reattachment. Have you seen airplane wings with a row of metal tabs rising up a couple of inches on the upper wing surface, at about 50% chord? Did you notice that they were bent over, looking as though were meant to "corkscrew" the air? That's exactly what they were for. By sweeping in some outer air, they were intended to add forward momentum to the boundary layer, keeping it going, preventing excessive thickening, so maintaining lift, and reducing drag. So, at a slight cost in traction power, the airplane was able to carry more load.

Finally, when laminar flow is carried deliberately too far aft, as by wedge-shaped, laminar-flow foils, laminar separation, when it occurs, may be permanent, beyond hope of reattachment.

These ideas may be giving you some insight into why some slow-speed systems, actually benefit from "turbulating" the air. One trick is to spread a thread or a wire ahead of the foil. The slight turbulence so induced minimizes the risk that the otherwise-all-too-probable laminar flow in these circumastance will permanently separate and destroy lift.

Next, in AR-5, we'll take the walls away from our sample SECTION and see what it does as a WING. Finally, we'll shake hands with that mysterious stranger called ASPECT RATIO, and count the fillings in his teeth. So ends AR-4.

Ivor <slater@fvc.bc.ca>