Preparing The Boat 2

1. 2
   Adjust for the wind. Sailboats cannot sail directly into the wind. As shown below, the red zone in the diagram indicates a "no go" zone when under sail. To sail to windward, a sailing vessel must sail about 45-50 degrees off the wind and change direction by tacking (or zig-zag).
   
   • Turn the boat to the left (port) or right (starboard) so it's about 90 degrees off the wind. This is known as a beam reach.
   • Pull on the main sheet (trimming) until the sail is around 45 degrees away from straight back (aft). This is a safe place for the main while you trim the jib.
   • You will start moving and tilting (heeling) away from the wind. A heel of more than 20 degrees usually indicates that you're being overpowered. Releasing the mainsheet momentarily (breaking the main) will lessen the amount of heel, and you will return to a more comfortable sailing angle of 10 to 15 degrees of heel.
2. 3
   Trim the jib sheets. Although the mainsail is hoisted first, it is the jib that is trimmed first. There are two jib sheets, one for each side of the boat. Pull on the jib sheet on the the side away from where the wind is coming from (leeward side). This is the active sheet while the other is called the lazy sheet.
   • The jib will form a pocket; trim the sail until the front edge just stops luffing. Keep your hand on the tiller (or helm) and stay on course!
3. 4
   Trim the mainsail. Let out the main sheet until the front edge just starts to luff, then pull it back just until it stops.
   • If you or the wind hasn't changed direction, this is the most efficient place to set the sails. If anything changes, you have to adjust them in response.
   • You have just entered the world of the sailor, and you will have to learn to do many things at once, or suffer the consequences.
4. 5
   Watch the front of the sail edge on the main and jib. If it starts to luff, you have two choices: tighten the sail sheet until it stops luffing, or steer away from the wind (bear off). When the sail luffs, it means that you are heading too much into the wind for your current sail setting. If you bear off slightly, (away from the wind) your sails will stop luffing.
5. 6
   Watch your wind indicators (telltales). If you see it change so that the wind is coming from a direction that is more behind you, you will be wasting energy. Let out the sail till it luffs, and tighten again till it stops. You will be doing this constantly; watching the sails, the telltales, and trimming sails if for no other reason than to see where you're at.
   • When the wind is at your back and side (aft quarter), it's called a broad reach. This is the most efficient point of sail as both sails are full of wind and pushing the boat at full force.
   • When the wind is at your back, you are running with the wind. This is not as efficient as reaching, because the jib is covered by the mainsail and not filling with air.
   • When running with the wind, you can sometimes you can pull the jib over to the other side of the boat where it will fill. This is called wing-on-wing, and you have to maintain a steady hand on the tiller to keep this sail configuration. Be sure to be vigilant of obstacles and other vessels, as having both sails in front of you blocks a significant portion of your view.
   • Be careful—when the boat is running, the sails will be way off to the side, and because the wind is basically behind you the boom can change sides suddenly (jibeor gybe), coming across the cockpit with quite a bit of force.
   • If you have a wind direction indicator at the top of your mast, do not align the boat so that the wind indicator points at the mainsail. If it does, you are sailing with the boom on the windward side (sailing by the lee) and are at high risk of an accidental jibe. When this happens the boom can hit you with enough force to knock you unconscious and out of the boat (overboard).
   • It's a good practice for beginners to pull the sail in a bit when running so it doesn't have far to go if it jibes.
6. 7
   Close reach. Turn the boat slightly into the wind, maybe 60-75 degrees off the wind. You will have to pull the sheets tighter so the sails are more closely in line with the boat. This is called a close reach. Your sails are acting like the airfoil of an airplane: the wind is pulling the boat instead of pushing it.
7. 8
   Close haul. Continue to turn into the wind (head up) and tighten the sheets until you can go no farther (the jib should never touch the spreaders on the mast). This is called close-hauled, and is as close as you can sail into the wind (about 45-60 degrees off the wind). On a gusty day, you will have all kinds of fun with this point of sail!
8. 9
   Sail into the wind to a destination. Sail a heading that is as close to the wind as you can, close-hauled. On most sailboats this will be about 45 degrees to the wind.
   • When you've gone as far as you think prudent, suddenly turn the boat through the wind (or changing direction by tacking), pulling the jib sheet out of its cleat or straight up off the winch drum as the front of the boat (bow) turns through the wind.
   • The main and boom will come across the boat. The mainsail will self-set on the other side, but you will have to quickly pull in the jib sheet on the opposite side on its cleat or winch, while steering the boat just to the point where the mainsail begins to draw again.
   • If you do this correctly, the boat won't slow down much and you will be sailing to windward in the other direction. If you're too slow tightening the jibsheet again and the boat bears off the wind too much, don't panic. The boat will be pushed sideways a little until it gains speed.
   • Another scenario would be to fail to put the bow of your boat through the wind quickly enough and the boat comes to a complete stop. This is known as being in irons, which is embarrassing, but every sailor has experienced it. Being in irons is easily remedied: when the boat begins moving backwards you will regain steerage.
   • Point the tiller in the direction you wish to go and tighten the jib sheet to windward, (backwinding the sail). The wind will push the bow through the wind. Once you've completed your tack, release the sheet from the winch on the windward side and pull in the sheet to leeward and you'll be on your way again.
   • Because speed is so easily lost when tacking, you'll want to perform this maneuver as smoothly and quickly as possible. Keep tacking into the wind until you get to your destination.
9. 10
   Go easy when learning. Understand that it's best to practice on calm days, and so, for example, learn to reef your boat (make the sails smaller). You will need to do this when the wind is too strong and you're being overpowered.
   • Reefing almost always needs to be done before you think you need to!
   • It's also a good idea to practice capsize procedures on a calm day too. Knowing how to right your boat is a necessary skill.
10. 11
    Sail safely. Remember that your anchor and its cable (rode) are important pieces of safety gear and can be used to stop your boat from going aground or can even be used to get the vessel floating again should a grounding occur.
    
    
    

Hoisting The Sails

Attach the sails. Secure the bottom front (tack) of the mainsail and jib to their respective shackles on the boom and the bow of the boat.

• There will be a small line (outhaul) attaching the back of the mainsail (clew) to the boom and its cleat. Pull it hand-tight in the cleat. This tightens the foot of the sail.
• Hoist the mainsail by pulling down on its halyard all the way until it stops. It will be flapping around (luffing) like crazy, but thats ok for a short period of time. (Excessive luffing will drastically reduce the life and durability of the sail).
• The leading edge of the sail (luff) must be tight enough to remove folds, but not so tight as to create vertical creases in the sail.
• There will be a cleat in the vicinity of the halyard where it comes down from the top of the mast. Cleat the halyard. Using the jib halyard, raise the front sail (jib, genoa or simply the headsail), and cleat the halyard off. Both sails will be luffing freely now. Sails are always raised mainsail first, then the jib, because it's easier to point the boat into the wind using the main.

Preparing The Boat

1 - Perform a detailed visual check. Inspect all the standing rigging—the cables and ropes that support the mast—including the turnbuckles and cotter pins securing the rigging to the hull. Many sailboats have dismasted because a 15-cent cotter pin was missing!

• Check the lines (running rigging) that raise and control the sails (halyards and sheetsrespectively). Make sure that they are separated, not wrapped around each other or fouled on anything else, and that they all have a figure-eight knot or other stopper knot on the free (bitter) end so they cannot pull through the mast or sheaves.
• Pull all lines out of their cleats and off their winches. There should be nothing binding any line; all should be free to move and be clear at this point.
• If you have a topping lift—a small line that holds the back of the boom up and out of the way when the sail isn't in use—let it out until the boom sags downward freely, then re-tie or re-cleat it. Watch out for the boom; it's just swinging around at this point; being made of aluminum, it will cause a painful "clunk" if it happens to hit you or your crew. The boom will return to its normal position when you hoist the mainsail.
• Attach thetiller. Be sure that it is properly attached to the back of the boat (transom). Your sailboat is now prepared for you to hoist the sails!                                                       
• 
  
• 2 - Determine the wind direction. If your boat doesn't have some kind of wind direction indicator (windex) at the top of the mast, tie a couple nine-inch pieces of old cassette tape, VHS tape, or oiled yarn to the shrouds—the rigging cables that hold up the mast.
• Place them on each side, about four feet up from the sides of the boat.
• To sail effectively, you will need to know the apparent direction of the wind .
• Some sailors find cassette tape to be just too sensitive for this purpose. If that's the case with you, try using VHS tape instead.

• 3 - Point the boat into the wind. The idea is to have the minimum amount of wind resistance when raise the sail, with the sail straight back. In this position, the sail won't be snagging on any shrouds or any other hardware, either. This isn't always easy. The boat won't turn readily because it's not moving (under way). Do the best you can, but be prepared to work for it!

• Here's a handy tip: if the water is not deep at your dock, or if you have no side pier, walk the boat out away from the dock and anchor it into the sand, and the boat will automatically point itself into the direction of the wind!

• 
  
  

How to Sail a Boat ?!!

You've always wanted to learn to sail, but the different parts of the boat, the unfamiliar jargon, and maybe even the mystique of sailing have left you looking for a flotation device just to keep your head above it all! This article is for you: it will cover the hardware present on most small sailboats, as well as common sailing techniques, terms and definitions, the names of the different pieces of hardware, and much more. This will get you started, but be sure you spend time with an experienced sailor and your boat before you venture out on the water on your own.

470 Aerodynamics

470 Aerodynamics

We have seen in events like the America«s Cup or the Admiral«s Cup that sailmakers are taking a more and more scientific approach in their sail development-programs. WB-Sails has studied 470-sails with the help of the most modern computer tools. This article first appeared in "470 Times" in June -92. WB-Sails chose the 470-rig as one of the first testing grounds for their aerodynamic analysis-program MacSail for several reasons: Firstly, with our long experience of the 470-class we were confident that the shapes of these sails represented well today«s state-of-the art. Also, we had accurate shapes available through normal photography as well as photogrammetry. Another important factor was that the level of racing in the 470-class represents (both internationally & nationally) the highest standards. The MacSail-calculation program is based on a system called "vortex-lattice method". In this method, the sails are replaced by a 3D-grid, dividing both mainsail & jib into approximately 100 elements. Flow velocity is calculated in the center of each element, on both leeward and windward sides of the sails, taking also into account the effect of the jib wake (backwind) on the mainsail. From flow velocity, the pressure can be deduced, and by adding the windward side pressure to the leeward side suction, the pressure difference in each element can be obtained. This pressure acts perpendicular to the element surface, forming a small force vector. Summing up all these small vectors in each element gives the magnitude and direction of the total force acting on the jib & the mainsail. An example of the results can be seen in the illustrations below. These examples are calculated in a wind of approximately 6 m/s (13 knots): Jib Main Both Driving force in kp 10,5 11,5 22,0 Heeling force in kp 28,5 55,5 84,0 Heeling moment in kpm 58,5 174,0 231,5 As you can see, the jib seems to contribute much more (relatively) to boatspeed than the mainsail, but don«t get it wrong: it only does this because it is working in conjunction with the main. In fact, the jib is receiving a strong "lift" from the main, while the mainsail is experiencing a bad header due to the jib (the jib, after all, is in a "safe leeward position"). Take the main out, and the jib will lose its magic. Note how very small the actual driving force pulling the boat forward is, some 22 kilos. In this light, it is easy to understand, how important every little detail, from aerodynamic clothing to an open bailer or dragging spinnaker sheet, not to mention hitting a wave badly, is to getting the maximum out of your boat. 470 Main & JibPressure difference at apparent wind speed 8 m/s apparent wind angle 23 degrees For the purpose of the calculation, the sails are divided into approximately 100 elements each. Different shades/colours represent a different surface pressure according to the table on the left. You can tell from the colours that the pressure is far greater in the upper part of the sails, and even more so in the jib than the mainsail. This is partly due to the triangular shape of the sails, and partly to the intensional shaping of the sails to minimize vortex induced drag.The jib represents 40% of the total force of the sails, although it is only 30% of the area. Nearly 50% of the forward drive comes from the tiny jib. Flow velocity on the leeward sideThe difference in flow velocity on the windward & the leeward sides creates the pressure difference. The pressure difference is the cause of the driving & heeling forces excerted by the rig onto the hull. Here we can see the calculated flow velocity on the leeward surface of the sails , in a free stream velocity (apparent wind speed) of 8 m/s. A flowchart such as this gives the saildesigner a wealth of useful information. You can see, how the flow is accelerated on the surface of both sails so that the speed of the flow is nearly everywhere greater than the wind speed. Near the head of the sails the speed is almost doubled, ie. 14-15 m/s. The influence of the jib can clearly be seen on the main surface - the flow is decelerated in the slot between the two sails, not accelerated, as you often hear claimed. The velocity on the surface of the jib is larger than on the main, and even in most of the leech area it is higher than the wind speed. As a result, the flow is less prone to "separate", and thus the jib can produce a greater drive in conjunction with than mainsail than it would alone. The flow separates, if it is forced to slow down too much too fast. The jib maintains higher velocities all the way down, because the foot lies on the deck, while in case of the mainsail, the pressure difference tends to even out under the boom, thus making the lower part of the sail less efficient. The flow has to slow down to the free stream speed in the leech of the mainsail, which makes it prone to separation. Separation always occurs first in the head of the sails, where acceleration & the following deceleration is big. In the front part of the main, flow velocity is nearly the same as wind speed - there«s hardly any pressure difference, the sail is close to backwinding. Although calculations such as these give a wealth of useful information to the saildesigner, it would be wrong to expect a miracle shape to come out of the computer as a result. We can only optimize the sails for one given wind condition at the time - and although we can simulate mast bend and forestay sag in the computer, the true art and skill of the sailmaker comes out in the intricate balance between the seam-induced, moulded shape and the luff curve shape, combined with the rig characteristics and the fabric strech, to produce a suit of sails versatile enough to power the boat from the lightest zephyrs to the choppy seas in heavy breezes. So, even in this computer-age, the ultimate proving ground of a new suit of sails remains the race course, the way it should be.

Draft position

Draft position

I was interested to read what Frank Bethwaite had to say on the position of maximum draft in his "Tasar" dinghy class -- he was comparing maximum draft at 25% against 50% of chord:

"My authority for [my] statements are experiment and observation but primarily model aircraft glide tests with different wing sections. The dynamics of sailing to windward at max VMG in lighter wind are the same as the dynamics of a glider which is looking for minimum sinking speed. A test model with a wing with a thin section and max camber at 25% from the leading edge was efficient at one particular speed. With a wing of identical shape and weight but with max camber at 50% (ie. a circular arc section) it was as efficient as the “knuckled” section [ie max draft at 25% of chord] at its best point, but it maintained this efficiency over a wide speed range. In smooth air the two sections could glide and lose height as well as each other. In rough air the knuckled section became unstable and the model sank faster, while the circular arc section continued to fly steadily in the rough air, the model remained stable and sank more slowly."

This was interesting on two counts.  One was that Graham Bantock uses an aerofoil section on his rudder which he tells me is pretty much the same as that used on competition (full size) gliders.  Both model yacht and full-size glider operate at similar Reynolds numbers.

The other was that Frank Bethwaite was clearly saying that draft forward was not advised.

This is only half of the story, however.  One of the key things about draft amount and position is that they affect the entry angle -- and it is the entry angle which is so important while beating.

Entry angle depends upon draft amount and position

Will Gorgen sent me some comments on entry angle and its importance:

However there is another principle that you need to keep track of. That is the incidence angle at the leading edge.  As you pull the draft forward, you change the angle of the leading edge relative to the incoming air flow. This causes your angle of incidence between the leading edge and the incoming flow to decrease.  In the extreme, this will cause the front of the sail to luff and will effectively force you to sail lower (foot) in order to keep your sail from luffing. When you sail lower, you increase the overall angle of attack of the sail and move yourself closer to stall.

Thus, moving your draft too far forward can cause your sail to stall, and moving your draft too far back can cause your sail to stall. There is an optimal draft position in between these two competing forces that will determine your optimal draft position.  This is often determined by your individual sailing style. For people who like to point high, you generally need the draft of the sail further back in order to keep the entry of the sail at a positive angle of incidence and prevent the sail from luffing.  For people who like to foot, you can tolerate the draft further forward and sail with a higher overall angle of attack and thus carry more force in the sail.  Of course optimal VMG depends on carefully balancing the lack of windward progress with the extra speed that you get when you foot.

Moving your draft further forward allows you to generate more drive in the sails and tolerate a higher angle of attack, but at the expense of pointing.

An Explanation of Sail Flow Analysis Introduction Stanford Yacht Research (SYR) is currently doing a study in performance analysis on yacht sails through experimental and computational methods. This research is being done to study the flow around sails in a wind tunnel and to validate computer results against experimental results. Those involved with the present research include: Dr. Margot Gerritsen, Stanford Yacht Research Dr. Andrew Crook, NASA Ames Research Center Tyler Doyle, Ph.D. student at Stanford University Sriram Shankaran, Ph.D. student at Stanford University Steve Collie, Ph.D. student at University of Auckland Jean-Edmond Coutris, Graduate student at Stanford University Brendan Abbott, SYR intern - undergraduate at Webb Institute Daniela Hanson, SYR intern - undergraduate at Webb Institute Center for Turbulence Research (CTR) Aero-Astro Department NASA Ames Research Center Sail Design Working toward designing more and more efficient sails throughout history, many methods have been employed. From the beginning of sail design, intuition and experience have been the primary means of designing sails. Intuition and experience still play a major role in sail design because presently no database exists to tell a sailboat designer what sail will work best with a given hull shape! From intuition, prototype sails can be built to test a design and measure its effectiveness. Also, wind tunnel testing can be done with model sails to observe and study the flow around sails on a smaller scale. Today, computers have become a means of calculating a lot of information in short amounts of time. Computer programs that employ the Navier-Stokes fluid mechanics equations can provide answers to very complicated and long flow equations in relatively short amounts of time. Through experimental testing, flow results calculated by computers can be validated. In the near future, it is possible that computers will be able to fully simulate flow on a sail. Today, the power and speed of computers limits what work can be done to analyze flow. How Does Air Flow Past a Sail BASIC FUNCTION OF SAILS Sails are instruments that use the wind to propel a vessel through the water. Trimming the sails differently allows a vessel to sail at different angles to the wind. AIR FLOW AT DIFFERENT SAIL ANGLES Upwind Sailing - In upwind sailing, sails act similarly to foils. The forces generated by the sail result in lifting forces generated by the keel, and forward motion is produced. (A FURTHER EXPLANATION OF THE BASIC PHYSICS OF UPWIND SAILING CAN BE FOUND IN THE PHYSICS OF SAILING ) Reaching Downwind Sailing - Sails are used to catch the wind. The wind is used to "push" the boat along. Twist To describe how air flows past a sail, we must describe how air flows over the water. To simplify things, let's first assume that the water is not moving. At the water's surface, the air is moving at the same speed as the water. Thus, the air cannot be moving at the water's surface. However, if there is wind, we know that the air must be moving as we move away from the water. Because the true wind speed is greater further up the mast, the apparent wind created by the boat's forward velocity causes a "twisting" phenomenon: Upwind Twist Diagram The effect of twist is more apparent in downwind sailing. An International Americaحs Cup Class (IACC) yacht, because of its ability to sail downwind close to the speed of the wind, never really sails downwind at all. The IACC yachts are always experiencing apparent wind angles that simulate reaching conditions (about 90 degrees), rather than downwind conditions. Downwind Twist Diagram Difficulty of Modeling Actual 3D Sails Modeling of sails is difficult because the conditions that a sail experiences fluctuate tremendously. Trim, wind speed, boat speed, heel angle, and weather all change over time. These factors change the way a sail will perform. Also, flow patterns due to twist are hard to model in 3-D computer applications because of the complexity of eddies shedding from the head and foot of the sail. In This Study, SYR is Using 2D Sections Rather than modeling 3-D sections, a 2-D section of a sail is used to reduce computational cost. 2-D sections are representative of flow at a given height on the sail. The 2-D sections are optimized with computer programs. The flow patterns generated from a 2-D section, although simplified, are not completely irrelevant. Sail-makers still use optimized 2-D sections to create a 3-D sail. Generally, ten to twenty 2-D sections are used in making a sail. Computational Fluid Dynamics The computer application that SYR uses to analyze flow is a Computational Fluid Dynamics (CFD) program. CFD is a method that employs fluid mechanics equations to describe flow. A shape, such as a sail section, is created in the CFD program. It is the computer's task to calculate the flow field around the shape. However, there are an infinite amount of points around the shape. Calculating the flow in an undefined region would take decades. Therefore, a grid must be created around the shape to break down the computational process into a finite number of calculations. The grid is a crucial part of the solution process. If it is not dense enough, the flow solution will be lacking. If the grid is too dense, calculations will be too computationally expensive. This is the challenge of producing good CFD results.   After a satisfactory grid is created, the CFD program can run. A flow analysis on a 2-D section takes about 5 hours to run on a high quality computer (such as a Pentium 4). The results obtained from CFD are velocities at all grid points, pressure distributions, and forces acting on the sail section. Forces acting on the sail are determined by integrating the pressure distribution over the sail area. Velocities at All Grid Points Pressure Distribution and Streamlines Forces Derived from Pressure Distribution Wind Tunnel Testing To validate the CFD results, 2-D sections are also being tested by SYR in the 7 ft x 10 ft wind tunnel at NASA Ames. To create a 2-D model for the wind tunnel, a "slice" of a sail is taken. This slice is scaled down to a chord length of 2/3 of a foot; then, it is stretched to a height of 10 feet. The model is created with a high aspect ratio to try to eliminate vertical flow. A real sail has a flow pattern like this: We only want to analyze the sail flow in two dimensions: The basic wind tunnel setup looks like this:   Model Construction Data Acquisition in the Wind Tunnel Pressure-Sensitive Paint (PSP)Particle Image VelocimetryOil Film InterferometrySmoke Flow Visualization What Does This Tell Us About Sailing? The use of CFD and wind tunnel testing gives insight into flow around sails and improves CFD codes. Better understanding of flow around sails will hopefully produce more efficiently designed sails in the future. Results from sail modeling tell us, for example, about the importance of trim, staying in "clean air", and how the main and foresail interact. Importance of Trim: If the sail is not trimmed correctly, it is not efficient. Over-trimming causes excessive turbulence on the leeward side of the sail. Under-trimming greatly reduces the forces produced by the sail. Ideal trim produces the best flow distribution around the sail. Effect of Upwind Boat and "Dirty" Air: The downwind boat cannot point as high as the upwind boat because of the effect of the flow around the sails of the upwind boat. Main and foresail interaction: The flow around the mainsail speeds up the air at the trailing edge of the foresail, making it more efficient. The air flowing past the mainsail helps the jib sail at a smaller angle of attack. Limitations of Modern Sail Research In computer and wind tunnel simulations, simplifications are made to the sail's environment. First of all, the sail is analyzed independently of hull and water. A rigid sail is used in an upright position. To completely model a real sail it would be necessary to use a flexible 3-dimensional sail in a heeled position, and to consider the effects of the hull and sea state. However, modeling something like this would be far too computationally expensive. In the future, with a growing amount of power and speed in the computer, the possibility of analyzing more complicated systems will be greater.

Sails shape & aerodynamics

Out of popular demand, we give some aerodynamic background for the sail shapes described in "The Quest for the Perfect Shape". This is by no means a complete treatise of sail aerodynamics - we are merely scratching the surface - but should shed some light on why sails look the way they do.

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The light air case

In light air, a sailboat will almost always go faster if sail lift can be increased. As long as there is no significant flow separation, drag increases as the square of the lift. As lift continues to increase, drag rises even more rapidly until we reach maximum lift and the sails start to stall. Until near maximum lift, sail drive will increase and the boat speed ditto, in spite of the quickly rising drag and the related heeling force. This is because in light air the heeling force (or moment) is not an issue, at least up to very high values of the side force, when the induced drag of the underwater hull gets excessive.

So, in light air we need to look for shapes that maximize lift:

• round headsail entry to allow for large angle of attack
• maximum overlap to allow bringing the boom to the centerline and over, inducing lots of camber into the sailplan looked as a whole
• maximum fullness of both sails separately
• possibly a rounded leech of mainsail to increase rear loading
• twist as needed to prevent stalling

The headsail enjoys a beneficial lift from the mainsail behind it - after all, it is sailing in the "safe leeward position". To prevent flow separation on the leeward side, we want a round, full entry. For best performance in light airs, boats need to sail low and fast, at large apparent wind angles.

Overlap helps to maximize lift in two ways. Firstly, the genoa benefits as its leech is in the high speed airflow on the leeward side of the main. The highly accelerated airflow on the leeward side of the headsail, creating most of the suction that drives the boat, does not need to decelerate to apparent wind speed at the leech of the genoa. Instead, deceleration will happen gradually over the leech of the main.

Ironically, the overlapping genoa leech benefits the main at the same time: it prevents the airflow from accelerating too much over the forward part of the mainsail. Consequently, the airflow does not have to slow down as much as it approaches the mainsail leech, and is not as prone to separate. Separation is caused by the deceleration of the flow, and the flow has to decelerate close to apparent wind speed at the leech of the mainsail (the trailing edge of the wing as one), unless it separates already earlier.

Camber (fullness) will increase lift, but up to a limit: as the airflow on the leeward side accelerates to more than twice the apparent wind speed, it will stop to a halt at the windward side, or even start to flow against the wind, and lift will be decreased. This happens in practice at around 20% depth.

Rear loading, achieved by curving the mainsail leech, may increase the lift a touch more by adding to the positive pressure on the windward side of the sail. Care must be taken that separation will not be increased too much on the leeward side - leech tell tails are a good indicator.

Twist is used to regulate the size of the flow separation areas. Maximum lift (and drive) is achieved with significant areas of separated flow, but there is a limit to it. The flow separation is controlled by the amount of twist. Due to their triangular shape, sails are "tip stallers": separation always starts at the head spreading down along the leech.

Here's the reason: as the angle of attack is increased (the boat bears off or you sheet in), the narrower top part of the sail is progressively more and more loaded (per unit area) than the lower, wider part of the sail. This is nature's attempt to minimize the harmful lift induced drag, by spreading the loading more evenly over the sailplan.

By virtue of their triangular planform, sails twist the airflow in front of and over them as much as 30-40 degrees (!) up along the luff. In the lower part, the sail is headed (effective wind angle is decreased), while towards the head the effective angle increases rapidly (see sketch).

The end result is flow separation starting at the top, while the actual mechanism of this massive bending of the airflow lies in the trailing edge vortices and the strong tip vortice shed at the leech of the sail, at high lift connected with large apparent wind angles (see illustration of the airplane on the right).

This is why in light airs we want sails that are flatter down low (less loading per unit area) and get fuller towards the head (more loading per unit area). A more even vertical distribution of loading means less induced drag and more drive - the famous elliptical loading being ideal but not achievable with triangular sails. The deeper profile fits better for the twisted flow, but you also want to move the maximum camber forward towards the top, further increasing the entry angle, to ensure a smooth flow without separation bubbles at the luff.

Wind gradient

Wind is slowed down close to the sea surface because of friction: windspeed is usually several knots more at the mast top than at the deck level. When this wind speed gradient is combined to the boatspeed, we also experience a twist in the apparent wind in the order of 5 degrees or so (close hauled - downwind the twist can be double as much).

The increase in the apparent wind speed and the wind angle both contribute to loading more the upper part of the sail plan, rendering the triangular sailplan more efficient than it would be in uniform flow.

The case for heavy air

When we reach the design wind and more, maximum lift and the associated flow separation ceases to be an issue. Instead, the heeling moment starts to dominate the performance of the sailboat. We saw in the Quest for the Perfect Shape that drag not only reduces the driving force of the sails, but also increases the heeling force. Therefore, minimizing drag and maximizing the lift to drag ratio become important.

Flattening and twisting the top part of the sails helps keeping heeling moment under control. So does the (often undervalued) triangular shape of the sails: As the helmsman starts to pinch to prevent excessive heeling, the sails are set at a narrower at angle to the wind. The opposite of what was discussed in the light wind case occurs. The upper part of the sails is unloaded, and the effective angle of attack is more even from foot to head. With the reduced lift coefficient the leech vortices are less powerful than in light air, and the sails do not twist the wind nearly as much as before.

This is a property of the triangular planform (side view) of the bermudan sails, something that the hip elliptical leech sails do not enjoy. The sails automatically adjust their loading when the apparent wind angle changes so that the sailboat can cope with a larger wind range more easily.

To avoid backwinding, we want to flatten especially the mainsail in its front part. The feathered top of the sail needs to be even flatter, moving the point of maximum draft towards the leech.

In extreme conditions, it is advantageous to allow the top "reverse" completely a part of the time, providing a negative heeling force (lift to windward), supporting the boat with a large lever. This allows to produce much more lift down low and you end up with more drive for a given heeling moment. However, before the sail starts to backwind you want to flatten it as much as possible. Backwinding not only pushes the boat back but also tends to increase weather helm, usually a problem in heavy air.

Minimum Induced Drag of Sail Rigs and Hydrofoils

Part 1: Planar Planforms

Abstract

It is widely believed that an elliptical span loading produces the minimum induced drag for a lifting surface. In general, the optimal span loading is quite different and the elliptical loading is only optimal for the restricted case of an isolated planar wing in uniform flow. The correct conditions of optimality, as derived by Munk for minimum induced drag for a given span and by Jones for minimum induced drag with a moment constraint, are applied by means of an inverse lifting line technique. Validation of the method by comparison with test data and classic theoretical results is shown.

Design charts showing optimal span loadings, planform shapes, induced drags, and centers of effort are derived for isolated surfaces, single sails in ground effect, twin sail biplane rigs, and hydrofoils operating at high speed near the water surface are presented. The application of the method to the design of optimal keel winglets is also shown.

Introduction

The efficiency of a sail rig is determined by its ability to generate a large amount of aerodynamic lift with a minimum amount of drag. Both the maximum lift and the drag are determined in part by viscous effects in the boundary layer and the surrounding inviscid flow - especially the wake.

Figure 1 presents a schematic view of the flow around a sail rig. The lift created by the sail results in deflecting the wind sideways, and this deflection takes the form of a trailing vortex sheet which wraps up into two large counter-rotating vortices. For rigs of moderate to high aspect ratio, the aerodynamics can be computed with reasonable accuracy by separating the three-dimensional flow problem into two separate two-dimensional flow problems: the flow about a section of the sail cut by a plane perpendicular to the axis of the sail, and the computation of the local flow conditions at the section which are determined by the wake. It is the second problem that is the subject of this paper.

The effect of the trailing vortex sheet shed into the wake is to induce a shift in the relative wind at the sail which reduces the local angle of attack and also rotates the lift vector (which is defined to be perpendicular to the local apparent wind) aft, which results in a component of the local lift which is in the free stream direction. This component is known as the induced drag. The velocities induced by the wake, and thus the local angles of attack for the sail, can be determined by looking at the configuration of the trailing vortices in a downstream plane perpendicular to the apparent wind called the Treftz plane.

Section design is primarily concerned with viscous effects. When measured relative to the section's zero lift line, most sections produce much the same amount of lift at the same angle of attack. So the art of section design consists of controlling the boundary layer through the pressure distribution so as to increase the maximum lift of the section and decrease the viscous drag. This leaves controlling the inviscid flow to the design of the planform.

The design of the planform, through its effect on the wake in the Treftz plane, seeks to maximize lift by ensuring that some portions of the sail do not operate above their maximum angle of attack while other portions operate well below their limit. Planform design also determines the lift-induced drag, which is the major drag component of a well designed rig at high angles of attack. Accomplishing these two goals simultaneously is one of the few happy coincidences in yacht design that doesn't require reaching a compromise between conflicting requirements, since minimizing the induced drag tends to lead to more uniform conditions over the sail, which aids in achieving maximum lift.

Several authors have used optimization techniques to determine the best sail plan. Greely et al (1989) used quadratic programming with a lifting line analysis to generate spanloads which maximized the net thrust, including the effects of viscous drag. When using Lagrange multipliers to do the optimization, they experienced difficulties with the technique trying to use negative lift at the head to meet the heeling moment constraint.

Day (1991) used simulated annealing with a lifting line analysis to determine lift distributions which maximized the performance of a sailboard. Both induced drag and a simple viscous drag estimate were used to model the sail. He found that due to a constraint on heeling moment, the optimum spanload distributions had a negative lift at the head. Day's optimum distributions were shown to be similar to those of Wood and Tan (1975). Day (1995) also used a genetic algorithm to optimize the parameters of a sloop rig having triangular sails, in conjunction with various hull models.

However, the author has not found any source which handles the optimization of sail planforms in a comprehensive and general way, or that provides insight into the difference between the contributions of the inviscid vs. viscous portions of the drag and lift. All of the papers cited used a hybrid approach, in which an analysis method was coupled with an optimization technique in order to vary the geometric parameters and arrive at the best performance.

An alternative approach is to use an inverse method, in which the desired aerodynamic characteristics are specified first, and then the shape required to satisfy them is computed. Inverse methods may not always yield a shape which can be practically constructed, but they are very useful for determining shapes that are unconventional, and for defining what performance is possible. Armed with this knowledge, the designer can assess the impact of a more practical configuration which is close to the optimal shape. Often the performance degradation is minimal.

Minimizing Induced Drag

Day (1995) states that "If the rig is of high aspect ratio and low sweep angle, then the application of this criterion [lift to drag ratio] will lead to the well known result of an elliptical distribution of lift in the spanwise direction." As will be shown in this paper, this statement is not correct for the case of sail plans with a gap between surface and foot, when heeling moment is constrained, and for vertical hydrofoils.

Munk's 1923 paper established a number of important theorems regarding lifting surfaces and their induced drag. One, known as Munk's stagger theorem, states that the total induced drag of a system of lifting surfaces is independent of the streamwise position of the surfaces, provided that their loading is unchanged. This makes it possible to project all the surfaces into one transverse plane for analysis.

Munk's criteria for minimizing the induced drag is less well known: "Let the dimensions of a system of aerofoils be given, those in the direction of flight being small in comparison with those in other directions. Let the lift be everywhere directed vertically. Under these conditions, the downwash produced by the longitudinal vortices must be uniform at all points on the aerofoils in order that there may be a minimum of drag for a given total lift." "If all lifting elements are in one transverse plane, the component of the velocity perpendicular to the wings, produced by the longitudinal vortices, must be proportional, at all lifting elements, to the cosine of the angle of lateral inclination." Having the induced velocity normal to the surface be constant and proportional to the cosine of the dihedral angle corresponds to the wake leaving the surface as though it were a solid sheet sliding linearly with no distortions.

Notice that he says nothing about the shape of the spanloading. Most aerodynamic texts use a Fourier series to represent the spanloading and solve for the minimum induced drag of an isolated planar wing in a uniform flow. The result is the classical elliptical loading and this is said to result in a constant downwash. In fact, Munk shows that the converse is true! The induced drag is a minimum because the downwash is constant, and the elliptical loading is, for this particular case, the loading required to generate the constant downwash.

Jones (1950) later showed that when the wingroot bending moment is constrained, the induced drag is minimized when the downwash distribution varies linearly across the span. Note that the moment about the wingroot for a wing corresponds to the heeling moment of a sail rig when coupled with its image system. The planform shapes and lift distributions of Jones' results were more tapered and weighted toward the center of the wing than the elliptical loading. Although the resulting lift distribution produces more drag for a given span than the elliptical lift distribution, the lower moments allowed the span to be increased which reduced the induced drag approximately 14% while keeping the same moment as for the classic elliptical planform.

Lissaman has shown that the a lifting line analysis performed in a nonuniform flow, such as a wind gradient, can be transformed into an equivalent problem in a uniform flow. He also shows that the minimum induced drag is obtained when the induced velocity distribution is a constant. So for the sake of simplicity and clarity, all the results in this paper will be computed for uniform flow.

These criteria suggest a new way to design a sail rig:

1. Establish the constraints on heeling moment (height of center of effort) due

the stability of the hull and any constraints on the height of the rig, such as the need to pass under fixed bridges.

2. Vary the induced velocity distribution so as to minimize the induced drag while

observing the heeling moment constraint, using an inverse method to determine the

spanloading from the induced velocity distribution.

3. Chose section designs based on viscous considerations.

4. Based on the downwash distribution, zero-lift angles of attack of the sections, and

their maximum lift coefficients, establish the combination of twist and planform

shape which results in the most stall margin along the span.

5. Modify the optimum planform to get a shape which can be practically constructed.

6. Analyze the final shape for comparison with the optimal performance.

Since most sections have similar lift curve slopes in their linear range, it is possible to define the twist distributions in terms of the zero-lift angles of attack, and to determine the planform shape independent of the choice of section shape. This enables the designer to examine many design tradeoffs in a very general sense, as will be shown below.

This approach requires both an inverse method and an analysis method. The inverse method is used to compute the optimal spanload and planform shape from the induced velocity distribution. The analysis method is used to analyze the optimal shapes at off-design conditions, and to examine the performance of practical, suboptimal configurations. The lifting line method can satisfy both requirements, whereas lifting surface techniques, such as the vortex lattice method, are almost exclusively analysis methods only.

Lifting Line Method

Lifting line basics. The lifting line method is based on the classical assumptions:

- the lifting surfaces are thin,

- the oncoming flow is steady, uniform, and inviscid,

- trailing vortices are straight and extend to infinity parallel to the apparent wind,

- all velocities are kept small such that small angle approximations suffice for the induced angles of attack.

Figure 2 shows the overall geometry of the lifting line and trailing vortices. Looking upstream, the sail rig appears edge-on, with the lift distribution acting to one side and the induced velocities acting in the opposite direction. The key input parameters which define the geometry of the problem are shown. By appropriate selection of the coordinates for head and foot, any planar configuration can be specified from an upright sail rig to a horizontal wing. For both sails and hydrofoils, the Z axis extends away from the surface - upwards in the case of sails, and down for the case of hydrofoils.

The lifting line is broken up into a number of discrete segments. Each bound vortex segment trails two vortices, one at each end of the segment. The trailing vortices rotate in opposite directions, and all three vortices for a given segment have the same strength.

The Trefftz plane is located at infinity in the downstream direction, so the bound vortices have no effect there. However, the strengths of the bound vortex segments, and their trailing vortices, have to be determined such that there is no flow perpendicular to the segment. The induced velocities in the Trefftz plane are twice those induced at the origin of the wake, since the trailing vortices extend to infinity in both directions from the Trefftz plane, but only in one direction from the sail planform.

The detailed geometry in the Trefftz plane is shown inFigure 3. A control point is located on each segment, and the normal components of the induced velocities from all the trailing vortices 

  

 

are summed there. The process is repeated for each segment and the simultaneous equations are solved to determine the vortex strengths.

The induced velocity at point i due to vortex j is given by

1)

where vij is the velocity induced at point i due to a vortex at point j, rij is the radial distance from vortex j to point i, and Gj is the strength of the vortex at point j. wij is the induced velocity component normal to segment i:

2)

The relationship in equation 2 is applied for each trailing vortex line. Each bound vortex segment actually has four trailing vortices when the corresponding image vortices are included. These images are located at the same distance from the surface, but on the opposite side, and share the same Y coordinate.

The boundary condition for the solid surface is that there be no flow normal to the surface. This requires that the direction of rotation for the image vortices be in the opposite direction for the case of the solid surface. This also approximates the linearized free surface condition in the limit at zero Froude number, when the effects of gravity far outweigh the forces in the flow.

At high speeds, the linear free surface condition is that the pressure on the surface be a constant, which requires that the sum of the tangential velocities induced there must be zero. The surface will have a vertical velocity due to the passage of the hydrofoil. This approximates the behavior of the free surface in the limit at infinite Froude number, when the effects of gravity are negligible when compared to the forces in the flow.

For the case of the solid surface, the matrix elements, mij, are assembled into the overall matrix, M as shown:

k = 1: vortex at beginning of segment (+z, +G)

k = 2: vortex at end of segment j (+z, -G)

k = 3: image vortex of k =2 (-z, +G)

k = 4: image vortex of k = 1 (-z. -G)

3)

4)

Inverse problem (wi known). At this point, it is easy to determine the optimum span loading (G,) required to minimize the induced drag. According to Munk, to minimize the induced drag for a given span, one need only specify a unit value for all the wi. Since the problem is linear, once the lift coefficient, CL, is found for a unit induced velocity, the induced velocity required to achieve any desired CL can be found by taking the ratio of the lift coefficient for unit velocity to the desired lift coefficient.

5)

6)

Once the lift distribution is known, the twist distribution, ti, is selected, taking into consideration the section zero lift angle of attack, a0, and lift curve slope, ai. With these values, the local angles of attack, ai, can be computed and the planform shape determined.

7)

8)

9)

Analysis problem (ci known). The analysis problem begins with the same formulation as the inverse problem. In this case, the chord and twist are known, the downwash velocities are eliminated in order to solve for the lift distribution in terms of the chords:

10)

Validation of Lifting Line Method. 

  

 

Results

The inverse lifting line method was applied to determine the characteristics of sail planforms with minimum drag, sail planforms which had reduced heeling moments, and vertical and horizontal hydrofoils operating at high speed. Since the classical semi-elliptical planform is so well known, most of the results are referenced to this planform for comparison.

Some standard definitions:

AR is the aspect ratio, P is the span of the sail rig (e.g. luff length), and S is the total sail area. Di is the induced drag (the net aerodynamic force component due to the lift, acting parallel to the freestream direction), CDi is the induced drag coefficient, r is the fluid density and V is the freestream velocity. L is the lift - the net aerodynamic force component that is at right angles to the freestream - and CL is the lift coefficient.

All of the cases were computed with 20 segments in the lifting line, distributed with cosine spacing. The control points were also distributed using cosine spacing, so they were offset slightly from the center of the segments toward the ends of the span.

Minimum Drag Sail Rigs. For a given span, the minimum drag sail rig has the spanloading which results from a uniform downwash distribution. Figure 4 shows these minimum drag span loadings for gaps between foot and surface that range up to 20% of the span. For zero gap the result is the classical semi-ellipse. Even for very small gaps, however, the optimal loading is quite different. The spanload distribution with gap starts out somewhat egg-shaped, with the point of maximum loading near 40% of the length of the surface. The loading rapidly approaches a full ellipse as the gap is increased.

Assuming that the surface is untwisted, the planform shape will be similar to the spanloading shape. Owing to the constant downwash, the untwisted planform will also have a uniform lift coefficient across its span. Assuming the same section is also used throughout, this will aid in achieving maximum lift, since all parts of the sail will approach the maximum angle of attack together.

The induced drag of the surface is given by

E is the Oswald efficiency factor, and is a function of the planform geometry and the surface effects. The value of e for the minimum drag planforms is shown in Figure 5. For zero gap, e is two, which is in agreement with classic wing theory. With as little as a 1% gap, however, e drops by two-thirds to 1.36. As the span is increased while keeping the gap fixed, E increases slowly. E is actually a function of the gap divided by the span, as can be seen by the fact that surfaces of different size have identical values of e when the gap is sized proportionately.

The lift curve slope for the three-dimensional sail planform is given by

Where a is the lift curve slope (per degree) and a0 is the two-dimensional lift curve slope (per degree). Figure 6 shows the ratio of this formula to the computed results, and the results are shown to agree to well within the numerical accuracy of the method.

The design implications of the minimum drag sail rigs are shown in Figure 7. All data and dimensions are referenced to semi-elliptical planform, which has E = 2 and a height of the center of effort above the surface of 42% of the reference planform's span. This chart can be used to assess the design tradeoffs for a wide variation in planform sizes, simply by changing the reference span, so long as the gap remains less than 20% of the span.

Starting from the reference planform, the induced drag increases rapidly with the introduction of a gap. Eventually, the curve at the right hand edge of the carpet plot would become asymptotically vertical at a value of two, as the gap becomes very large and e drops to its classic value of one for an isolated planar planform. If the gap is kept fixed and the head of the sail rig is raised, the induced drag drops and the center of effort goes up. Eventually, these curves would become vertically asymptotic to the Y axis as the flow becomes increasingly two-dimensional.

Reduced Moment Sail Rigs. The heeling moment of the sail rig can be reduced by changing the design induced velocity distribution so that it tapers linearly from a maximum value at the foot to a minimum value (which may be negative) at the head. In principle, there is no limit to one's placement of the center of effort, provided that it is permissible to have negative lift at the head of the sail. It is even possible to have the center of effort at the surface for complete cancellation of the heeling moment. However, there are practical limits to this, as will be seen.

Reduced moment, minimum induced drag spanloadings corresponding to the previous cases were computed using triangular downwash distributions which had a maximum value at the foot, and went to zero at the head. The resulting spanload distributions are shown in Figure 8. Compared with the minimum drag lift distributions, these are more tapered toward the head and weighted toward the foot. The point of maximum loading is at approximately 30% of the span compared to 40% to 50% for the minimum drag loadings.

There is not a unique planform which corresponds to these loadings. If an untwisted planform is adopted, it will have a similar spanloading at all angles of attack. However, it will be very sharply tapered toward the head, and the head will operate at a much higher angle of attack (owing to the tapered induced velocity distribution) than the foot of the sail, making it prone to stall. A better solution is to apply a twist that is parallel to the induced velocity distribution, washing out the head so as to maintain a constant angle of attack across the span. The required twist distribution is shown in Figure 9. The twist (in degrees) for this particular choice of induced velocity distribution can be seen to approximate the following relation:

Where t0i is the twist applied to the zero-lift angle of attack - the aerodynamic twist. For a design lift coefficient of one and an aspect ratio of four, the resulting aerodynamic twist is around seven degrees. This can be achieved with either physical twist of the planform or a change in camber, or both. If the camber is increased towards the head, as is common practice, the physical twist will be greater than seven degrees. With this twist distribution, the sectional lift coefficient is once again constant and equal to the surface lift coefficient, and the planform shape is similar to the spanloading shape. These shapes bear a remarkable resemblance to boardsail planforms, so it can be seen that the rapid, unfettered evolution in board sails has settled onto something very similar to these optimal shapes.

Because of the twist, the spanloading can be broken down into a basic lift distribution and an additional lift distribution. The basic lift distribution is the spanloading that would be obtained if the sail were operated at the angle of attack for zero lift and the twist was held fixed at the design values - like a tin sail in the wind tunnel. This basic lift distribution is shown in Figure 10. Although the net lift is zero, the local lift varies up to over 20% of the design lift coefficient; negative near the head, and positive near the foot. This is an aerodynamic couple in the anti-heeling direction, and it is this couple which supplies the reduction in the center of effort for these planforms.

The additional lift distribution is the change in the spanloading for a change in angle of attack or surface lift coefficient. It can be found by subtracting the basic lift distribution from the design spanloading and dividing by the angle of attack or design lift coefficient. Owing to the taper in the planform, the additional lift distribution is proportionately greater at the head than at the foot. This is shown in Figure 11, which shows the change in sectional lift coefficients that result from a unit change in the surface lift coefficient. These range from less than 80% of the change in surface lift coefficient at the foot to over 150% at the head.

The span efficiencies for these reduced moment design spanloadings are shown in Figure 12. These are approximately 10% or more lower than the corresponding minimum drag spanloadings because of the switch from uniform induced velocities to tapered induced velocities.

As seen in Figure 13, the same expression for the three-dimensional lift curve slope holds, provided that the new values of E are used. Since the aerodynamics are linear, the magnitude of the lift distribution is proportional to the magnitude of the induced velocities, as is the twist distribution. So the value of the lift curve slope holds for other angles of attack provided that magnitude of the twist distribution is also varied with the angle of attack.

The design tradeoffs for the reduced moment sail planforms (triangular induced velocity distribution) is shown in Figure 14. Once again, the data are relative to the semi-elliptical planform, which is also indicated on the chart for reference.

The height of the center of effort isn't the whole story with regard to heeling moment for a twisted planform, however. Such a planform has an aerodynamic center which is different from the center of effort. The aerodynamic center is defined as the location about which the moment is a constant as the angle of attack is changed. The center of effort (or the center of pressure), which is the location at which the moment is zero, will move up and down as the lift changes.

Although the yacht may have adequate stability when the moments are balanced with the design lift acting at the design center of effort, the effect of gusts will be different. The increased loading of the additional lift distribution means that the incremental effects of gusts will be applied at the aerodynamic center, not the center or effort. The dashed lines in Figure 15 show the position of the aerodynamic center for the reduced moment planforms, compared to the centers of effort from Figure 14 (solid lines). The aerodynamic center is always higher than the center of effort for these planforms, because of the anti-heeling moment of the basic lift distribution. For the case of untwisted planforms, like the previous minimum drag planforms, the location of the center of effort and the aerodynamic center are the same.

In effect, the basic lift distribution is acting like water ballast which has been moved to windward to balance the boat, and the lift is effectively acting at the aerodynamic center. So it is important to consider the height of the aerodynamic center when doing stability calculations as well as the height of the center of effort.

Now consider taking the reduction in heeling moment to its practical extreme. Figure 16 shows the resulting lift distributions when the induced velocity distribution is adjusted so that the lift chord distribution goes to zero at the head. This requires an induced velocity distribution which is positive at the foot and negative at the head, varying linearly in between. As with the previous reduced moment planforms, the twist distribution (Figure 17) is set so as to maintain a constant section lift coefficient across the span, but the twist is more than doubled. Any more twist (more negative induced velocities at the head), and the sail will produce negative lift at the head, which is either not physically realizable or would require a separate surface with its own deflection.

Again, since the lift coefficient is constant, the planform shape is similar to the spanload distributions shown in Figure 16. These planforms resemble the conventional triangular sail, except that the maximum chord is not at the foot, but occurs 20% to 25% of the length above the foot.

All of the planforms presented bear some resemblance to conventional sail rigs, except for the position of maximum chord. It would appear that cutting away the area at the clew and redistributing it elsewhere would improve the performance of most sail rigs. The large chord at the foot results in a rapid change in the lift at the foot, which sheds a stronger trailing vortex there than is necessary, increasing the induced drag and making the largest area of the sail comparatively ineffective.

The basic lift distribution shown in Figure 18 is also twice that of the previous case, and the additional lift distribution shown in Figure 19 indicates that the pointed tip will be way over-loaded for any significant increase in angle of attack over the design values.

Finally, Figure 20 puts together the whole sail rig center of effort vs. induced drag tradeoff. The solid lines indicate the minimum drag planforms. The dashed lines show the centers of effort for the triangular induced velocity distribution planforms, and the dash-dot line shows the practical limit for reducing the center of effort of the rig.

The reduced moment designs all have a lower center of effort and greater drag than their uniform induced velocity counterparts. However, by following up the lines of equal gap, one can see that by lengthening the span of the reduced moment design, one can reduce the drag over the minimum drag/fixed span design while not accepting any additional heeling moment.

But what about the center of effort? Figure 21 shows that there is no free lunch here. The drag tradeoff in center of effort for the reduced moment designs is the same as for the minimum drag designs. However, the picture with regard to gust response is better than this chart would indicate.

By definition, the basic and additional lift distributions assume that the twist is not a function of angle of attack. However, if the twist is varied with the angle of attack, as shown in equation X, then the center of effort and the aerodynamic center will coincide once again, and the drag reductions shown in Figure 20 can be realized even when the gust response is considered. This is basically what is happening with modern fat-headed sails in which the head of the sail is allowed to twist off under increased loading.

Figures 20 and 21 indicate what is possible in planform design. Other planforms will have greater induced drag for the same center of effort because of their non-optimal induced velocity distributions. However, practical planforms can be constructed which come close to these, so these charts are useful for showing the designer what performance is possible.

Vertical Hydrofoils. The same criteria for minimizing the induced drag of sail rigs applies to hydrofoils and hulls. In this case, the moments are not as important because the hydrofoils (such as keels and rudders) are not as long as the sail rigs, and because the span is typically more constrained. Therefore, only the results for minimum drag with a fixed span will be presented.

The present method cannot handle the nonlinear effects of waves in a free surface. However, it can handle the two limiting cases of very low speed and very high speed, in which the surface can be considered flat and the perturbation of the surface shape is small. For the low speed case, the water surface acts like a solid boundary and all the results previously presented for sail rigs apply.

For high speeds, corresponding to chord Froude numbers (V/(g*c)0.5) about roughly two and planforms of moderate to high aspect ratio, the infinite Froude number approximation gives reasonable results (Kuhn and Scragg, 1993). For this approximation, the image system used to enforce the surface boundary condition is the same for the solid surface, but the signs of the vortex strengths are reversed.

The linearized free surface boundary condition was applied to determining the minimum drag spanloads for vertical hydrofoils, as shown in Figure 22. Once again, the planform shape is similar to the load distribution due to the uniform induced velocity distribution and the lack of twist. Compared with the corresponding solid surface shapes, the area nearest the surface is reduced and the profiles have an inverted egg-shape.

With zero gap, the span efficiency factor E, shown in Figure 23, is 0.81 - a 19% reduction from the isolated planar wing case, and only 40% of the value for the semi-elliptical solid boundary case. The efficiency grows and the shapes become more elliptical as the hydrofoil is designed for conditions farther from the surface.

Figure 24 shows the center of effort/induced drag design tradeoffs for the minimum drag vertical hydrofoils. As before, all data are referenced to the geometry and results for the semi-elliptical planform at a solid boundary. Also shown are the results for the solid boundary case and for the elliptical planform in an infinite fluid.

For the surface piercing foil, the design location of the center of effort is not greatly different from that of the infinite fluid case, but the best one can do is to limit the increase in induced drag to approximately one quarter of the latter.

Horizontal Hydrofoils. Flying hydrofoils are also concerned with minimizing the drag of horizontal hydrofoils. There are two types of surface effects to consider - chordwise surface effects and spanwise surface effects. The former is concerned with the change in the section pressures as the hydrofoil approaches the free surface. This tends to reduce the sectional lift curve slope. The spanwise surface effects have to do with the change in induced drag and the three-dimensional lift curve slope due to the change in effective span as the hydrofoil approaches the surface, and these are the surface effects considered here.

Figure 25 shows the minimum drag spanloads for the horizontal hydrofoils computed with the linear free surface approximation and uniform induced velocity distribution. The shapes for all depths are elliptical. Since the design shapes are the same, these results apply for both foils designed for different depths, and for elliptical foils which are moved up and down.

The best span efficiency (Figure 26) for hydrofoils operating at the surface is 0.5, which corresponds to the planing condition. E increases as the depth is increased, and is over 0.9 when the hydrofoil is one semi-span below the surface.

Wings in Ground Effect. For the sake of completeness, Figure 27 and Figure 28 present the spanload/chord distribution for wings designed to operate near a solid boundary. These are also minimum drag planforms, having a uniform induced velocity distribution. As the wing is designed to operate successively closer to the surface, the planform becomes more tapered and the efficiency goes up very rapidly. This occurs because the image system is canceling the trailing vortices shed by the wing.

Summary. The span efficiency factor, E, is shown in Figure 29 for all the planar lifting surface cases considered. All the points in this figure represent design conditions, and do not represent the effects of moving a given planform closer or farther from the surface. The horizontal hydrofoils are the only exception to this.

Once E is determined, the three-dimensional lift curve slope can be calculated from equation Y. The induced drag can be computed from equation YY. Provided the optimal twist is used (if any), the section lift coefficients will be the same as the planform lift coefficient.

Knowing the section lift coefficient and the spanload distribution, the chord distribution can be determined based on the local section lift-curve slope. Note that this makes it possible to take into account such effects as the loss of lift due to separation at the leading edge caused by the mast, by simply making the chord larger where the section lift curve slope is less. Finally, the geometric twist can be determined, based on the optimal aerodynamic twist and the zero lift angles of attack of the sections.

References

1. Day, A. H., "Sail Optimization For High Speed Craft", Transactions of Royal Institution of Naval Architects Vol. 133, 1991.

2. Day, Sandy, "The Design Of Yacht Sailplans For Maximal Upwind Speed", Twelfth Chesapeake Sailing Yacht Symposium, January 1995.

3. Greely, D.S., Kirkman, K.L, Drew, A. L., Cross-Whiter, J., "Scientific Sail Shape Design", Ninth Chesapeake Sailing Yacht Symposium, March 1989.

4. Jones, Robert T., "The Spanwise Distribution Of Lift For Minimum Induced Drag Of Wings Having A Given Lift And A Given Bending Moment", NACA-TN-2249, 1950.

5. Lissaman, Peter B. S., "A Numerical Solution For The Minimum Induced Drag Of Nonplanar Wings", AIAA Journal of Aircraft Vol. 5, No. 1, pp 17-21, Jan-Feb 1968.

6. Lissaman, Peter B. S., "Lift In A Sheared Flow", AIAA Ancient Interface Symposium (year unknown, probably late '70's).

7. Kuhn, John C. and Scragg, Carl A., "Analysis of Lift and Drag on a Surface Piercing Foil", Eleventh Chesapeake Sailing Yacht Symposium, January 1993.

8. Munk, Max M., "The Minimum Induced Drag Of Aerofoils", NACA Report No. 121, 1923.

9. Schlichting, Hermann and Truckenbrodt, Aerodynamics of the Airplane, McGraw-Hill International Book Company, 1979.

10. Wood, C. J. and Tan, S. H., "Towards An Optimum Yacht Sail", Journal of Fluid Mechanics, Vol. 85 Part 3 pp. 459-477, 1978.