# Centerboards and Rudders

## Contents

Introduction
Lift and Drag
Quick Example
Boundary Layer
Reynolds Number
Comparison of Foil Thicknesses
Trailing Edge Thickness
Extra
SD8020
Parallel Sided Foils
New Cross Section

Updated: January, 2010

Creative Commons

### Introduction

Some years ago, I wrote Design and Construction of Centerboards and Rudders. This was a compilation of information, comments and suggestions from various experienced people. At that time, I stated:

Determining a "good" foil shape requires either experimental models or a rather large computer program to determine the lift and drag as a function of AOA for a variety of candidate shapes. Then repeat that process for a range of speeds. Final select a shape based on the expected conditions and your sailing style. The problem is just too complex to have a computer program that solves Newton's Laws of Motion and cranks out the "best" shape.

Since that time, computers have become more powerful and several programs for analyzing fluid flow have become available. One such program is XFOIL, developed by Professor Mark Drela at MIT. XFOIL is widely used for subsonic air plane design. It can be downloaded from http://web.mit.edu/drela/Public/web/xfoil/ and a brief but good tutorial can be found at http://cobweb.ecn.purdue.edu/~aae333/XFOIL/Tutorial/Tutorial%20for%20XFoil.htm

The same laws of fluid mechanics apply to the airfoils (wings, rudders, and elevators) on airplanes and the foils (centerboards, dagger boards, and rudders) on sailboats. A lot of experimental and theoretical work has been put into developing better airfoils. A foil shape that is useful for an airplane wing is potentially useful for sailboats.

### Lift and Drag

In airplanes, lift is the vertical force, perpendicular to the wings, that allows the plane to fly. In sailboats lift is the sideways force against the centerboard that makes it possible to sail to windward. Drag is the force retarding the forward motion. Lift and drag can be computed from the formulas

FL= ρ * v2 * A * CL /2
FD= ρ * v2 * A * CD /2

where

FL is the force of lift
FD is the force of drag
ρ is the density of water
v is the speed through the water
A is the area

Looking at these equations shows that the forces go up with the area. Doubling the size of the centerboard doubles the forces.

The forces increase as the square of the velocity. Doubling the speed, increases the forces by 4 times.

It is not too difficult to decide on the speed of the boat, compute the area of the centerboard, and look up the density of water. However, CD and CL are much more difficult. They depend the shape, the angle with respect to the flow and even the velocity. What makes CD and CL useful is that they make it possible to compare different shapes. For the same size and speed, a centerboard with lower CD will have less drag, and a centerboard with high CL will have more lift. The rest of this paper will focus on CL and CD as computed using XFOIL.

### Quick Example

Given the description of the cross section, XFOIL can plot the flow and from that compute the lift and drag.

 Figure 1. Example output from XFOIL

Figure 1 shows a typical output from XFOIL. The right side of the screen shot lists the foil name, and important parameters. Re is Reynolds Number, which will be described in the next section. The Greek letter, alpha, is the angle of attack, the angle between the center line of the foil and the flow.  CL is the coefficient of lift. CD is the coefficient of drag. L/D is the ratio lift to drag.

At the bottom is a cross section of the foil. The blue and red lines show how the water separates and flows around the top and bottom side.

The large graph shows the pressure at different places along the top and bottom surfaces. The red line shows the pressure pushing against the lower side of the foil. The blue line shows the reduced pressure "pulling" on the upper side. The pressures are combined to compute the coefficient of lift. The graph makes it obvious that, in this example, most of the lift is created by the flow over the top of the foil.

Describing how the flow separates around the top and bottom brings us to the topic of boundary layers.

### Boundary Layers

The term, boundary layer, applies to the region around the foil where the water flow is slowed and / or deflected by the foil. Close to either the top or bottom surfaces, the water speed is speed is very slow because of friction. Speed increases with distance from the surface until it reaches the average speed of the flow of water around the foil. In between is the boundary layer.

In Figure 1, the angle of attack, is 10 degrees. The flow divides at a point a little below the front of the foil. The red flow is pressed against the bottom surface until it reaches the trailing edge. The boundary layer is thin and follows the shape of the lower surface. The boundary layer is said to be "laminar" because it can be analyzed as thin parallel layers.

The blue flow is somewhat more complicated. First it must curve around the leading edged then turn to flow back toward the trailing edge. Although the boundary layer near the trailing edge is thicker, it is still laminar.

The flow is also said to be "attached" because the entire boundary layer follows the overall shape of the surface.

 Figure 2: Laminar flow

Figure 2 shows another example of laminar flow. In this example, the speed is much less than in the previous examples. (Actually the speed corresponds to a boat that is barely moving. The purpose is to give examples of the variety of flows that can happen with the same shape.) The force pressing the water to the board are reduced and so the boundary layers are thicker. The pink "whiskers" are actually mini-graphs showing how the speed increases with distance from the surface.

Laminar flow is a topic of much interest to air plane designers. With calm air at high altitudes it is quite possible to actually have a thick, laminar boundary layer. Water is a different situation. Bubbles, waves, small motions of the boat can upset the laminar flow with the result shown in Figure 3.

 Figure 3: Turbulent, Attached Flow

The only difference between Figures 2 and 3 is that a parameter of XFOIL was changed to magnify the effects of irregularities in the flow. The result is turbulent flow. The flow in the graphs only shows the average flow over time.  In reality, the flow is really churning over and over along the upper side.

Many texts on laminar and turbulent flow suggest that laminar is "good" and turbulent is "bad". Certainly inside a water pipe, laminar is quieter and takes less energy from the pump. In the marine environment, things are different. In some circumstances, it is hard to avoid turbulent flow in the water around a boat going at any speed, especially if there are waves.  Silicon polish and 1600 grit abrasives make a slick surface which can help with laminar flow, but they can only do so much.

In Figure3 the flow is still attached to the upper surface. The lift is greater and the drag is actually less for the laminar flow.

For the rest of this paper, unless otherwise noted, if the conditions could have either laminar and turbulent flow, XFOIL will be set for turbulent flow.

 Figure 4: Separated Flow

In Figure 4, the angle of attack has been increased to 30o. The flow on the upper surface is so far from the surface that it is described as "separated". There is a huge turbulent boundary layer. The turbulence extends for a long distance in the wake. The lift is reduced while the drag is almost as large as the lift.

This is what happens when the rudder is turned too hard. The boat turns, but there is so much drag that it is like putting on the brakes. Experienced helmsmen know to avoid turning the tiller further than necessary.

### Reynolds Number

The last of the mathematical concepts in this paper is Reynolds number. Reynolds number combines the size of the object with speed and the characteristics of the fluid (water or air) to give a dimensionless number.

 RE = V * L / kv where V is the velocity L is the length (fore and aft) of the foil kv is kinematic viscosity kv = ~10-5 ft2/sec or 10-6 m2/sec for water

The key to understanding Reynolds number is this: if two different situations have the same Reynolds numbers the fluid flow will be the same. This allows one to take results for airplane wings and apply them to center boards and rudders. It also allows comparing big boats with small boats and fast boats with slow boats. If the dimensions are unchanged, then increasing RE can be interpreted as a simply a change in speed. The following graphs show effect of Reynolds Number with the same shape and same angle of attack.

 Figure 5: Flow for NACA 0010; RE 30,000 (Small boat going very slow) Figure 6: Flow for NACA 0010; RE 300,000 (Small boat at modest speed; or large boat going slow) Figure 7: Flow for NACA 0010; RE 3,000,000 (Large boat going fast)

Figures 5, 6 and 7 are for Reynolds Numbers of of 30,000, 300000 and 3,000,000 respectively. This covers the range from small boats sailing very slowly to larger boats going fast. Each graph shows the behavior of the CL and CD different angles of attack. CL uses the scale on the left. CD uses the expanded scale on the right.

Referring to Fig 5, as the angle increases, the lift increases reaching a maximum for AOA around 9 or 10 degrees. For greater AOA, the lift decreases. Drag also increases with angle, and increases rapidly above 10o. Considering both lift and drag, the optimum performance will be in the range of 6o to 8o. Higher angles generate more lift, but the increased drag slows the boat. In a very over-simplified way, this is the difference between sailing on the "best" close-hauled angle and "pinching" to a higher angle, but going slower.

Figures 6 and 7 show what happens as Reynolds Number (or boat speed for a fixed size) increases. The increased forces press the water closer to the surface of the centerboard. The angles for maximum CL increase. For any angle, the higher RE has lower CD.

If we compare all of the graphs, there is an interesting result. CL is essentially independent of RE. For example, for AOA of 5o, CL is 0.5. The significant effect of RE is the angle for maximum CL. Of course, the actual force force increases as the speed squared.

One more comment. XFOIL is strictly a 2 dimensional simulation. In the 3D world, there are a number of effects, including the surface of the water, the presence of the boat hull, the centerboard being relatively short, which made the drag greater than the numbers shown here.

### Comparison of Foil Thicknesses

The most asked question has been how thick a center board should be. The NACA 00XX formula for foils is a general purpose formula which is intended to be scaled for different thicknesses. The last two digits represent the thicknesses as a percentage of the chord length. For example, NACA 0010 has a thicknesses that is 10% of the (fore and aft) length. The is the formula:

where:

x is the position along the chord from 0 to 1
y is the thickness at a given value of x
t is the maximum thickness as a fraction of the chord

Here is the same formula formatted so it can be copied and pasted to your spread sheet or other program.

y = (t / 0.2) * ( 0.2969 * SQR(x) - 0.1260 * x - 0.3516 * x2 + 0.2843 * x3 - 0.1015 * x4 )

 Figure 8: Lift and drag for different thicknesses Figure 9: Lift and drag for different thicknesses

Figures 8 and 9 show the lift and drag for NACA 0006 (blue), 0008 (green), 0010 (yellow), and 0012 (red) for different angles of attack. The solid lines are for CL with the scale on the left. The dashed lines are CD with the expanded scale on the right.

For low AOA, thickness makes little difference. As the angle increases, the NACA 0006 has maximum CL at AOA=8o. The slightly thicker NACA 0008 has maximum CL at AOA=11o. The thicker boards higher angles for maximum CL.

Figure 9 shows the same sections at a higher Reynolds Number. The stall angles are higher, but overall there is a similar pattern. For AOA=5o, CL is approximately 0.5. For AOA=10o, CL is approximately 1.0. With the exception of NACA0006 at RE of 300,000, which stalls out at AOA=8o.

Aside from the low stall angle for NACA0006, the graphs don't show much difference. Looking closely at the numeric values shows that the NACA0008 has 3% less drag in one condition, while the NACA0010 has 3% more lift in another. These differences are not very important, especially when 3D effects will probably be much greater.

### Trailing Edge Thickness

Foil shapes are typically shown with a sharp trailing edge. Accordingly people making centerboards and rudders try to make the trailing edge very thin. This leads to a fragile foil. Builders have devised all manner of reinforcement including metal, fiber glass and carbon fiber. I once made a board using thin pieces of electronic circuit board fiberglass inlaid into the wood to reinforce the trailing edge. Over time, sharp edged centerboard can actually cut into the centerboard trunk!

Is this really necessary? What is the effect of making the trailing edge a little thicker? XFOIL makes the effect easy to compute.

 Figure 10: NACA 0010 with original thin trailing edge.

Figure 10 shows the flow over the NACA 0010 shape and 10o angle of attack, which is within a fraction of a degree of the stall point. The following figures show the effect of a slightly thicker trailing edge.

 Figure 11: NACA 0010; trailing edge padded to .01 thickness Figure 12: NACA 0010; trailing edge padded to .02 thickness

Figures 11 and 12 show the NACA 0010 shape with the trailing edge made thicker. The "padding" was a maximum at the trailing edge and reduced to zero at the midpoint of the foil. A trailing edge of .01 and a chord length of 12 inches would result in trailing edge of 1/8 inch. The thicker foil would be approximately 1/4 inch thick at the trailing edge. These thicknesses would be possible to make out of wood without special reinforcements.

As for performance, the lift and lift to drag ratio are actually a tiny bit better with the thicker trailing edge! Also the angle of attack for maximum lift is also increased by a fraction of a degree.

 Figure 13: NACA 0010; stretched for trailing edge .01 thick Figure 14: NACA 0010; stretched for trailing edge.02 thick

Figures 13 and 14 show the effect of mathematically stretching the foil shape and then cutting it off to achieve the desired trailing edge thickness. This causes a slight distortion along the entire length of the foil. This process also slightly increases the lift, but the drag and lift to drag ratio are worse.

So the suggestion for the trailing edge is to first decide on the thickness that is practical to make with the desired materials. Then add a little thickness all along the back half of the shape when computing the profile to make. Don't go to excessive efforts to make the edge extra thin.

### Extra

The Extra shape has the leading edge in the shape of a ellipse. The trailing edge is a straight wedge. This section has been popular with makers of model planes because it is easier to make than the more complex curve of the NACA or other sections. Home builders of centerboards might also find it easier to build.

 Figure 15: Extra foil section
Figure 15 shows the flow over the extra section.  There are bumps in the pressure where the curved and straight sections meet.  I double checked to make sure that there was not a bump in the surface.  The cause of the pressure bumps is more subtle. Along the leading section, the surface is curving away from the flow, reducing the pressure and making it "easier" for flow to continue.  When the surface changes to straight, this effect stops.

 Figure 16: Extra for Reynolds Number 300,000 Figure 17: Extra for Reynolds Number 3,000,000

As can be seen from the graphs, the performance is not quite as good as the NACA or SD designs, but it is a reasonable compromise for a for simplifying the construction.

### SD8020

Prof. Michael Selig of the University of Illinois, John Donovan and others have spent a lot of time with experimenting with model gliders. Using the wind tunnels and computer simulations, they developed a number of wing sections for use at the low Reynolds numbers associated with small planes at low speeds. Many of the wing sections are designated "SD". Most are not symmetrical, which is good for a wing that never flies upside down. This is not useful for a centerboard (unless you change centerboards when you go from port to starboard tack.)

The one symmetrical SD design is SD8020. Unlike the NACA formula, SD8020 was designed to be the best thickness for their specific purpose. It really wasn't intended to be scaled to thinner or thicker sections. This doesn't prevent multiplying their data to yield thinner or thicker shapes, but it is not really correct to call these "SD8020".

That said, Figures 18 and 19 show CL and CD for SD8020 and also versions scaled to 6%, 8% and 12% thickness.

 Figure 18: SD8020 for Reynolds Number 300,000 Figure 19: SD8020 for Reynolds Number 3,000,000

For the range of RE of interest to most sailors, NACA 0010 and SD8020 perform similarly. The biggest difference is that the lift of the thinner variations doesn't drop off as fast small increases in AOA above the angle for maximum.

### Parallel Sided Foils

Parallel sided foils are of interest to sailors for several reasons. They are easier to make than the tear drop shapes common in airplane wings. It is easier to shape the trunk that holds a parallel sided board.  Also, rules for some designs of sailboats require boards with parallel sides with shaping on a limited extent on the leading and trailing edge.

In the February 1988 issue of Australian Sailing Neil Pollack wrote, Section Shapes for Foils. He described several shapes, including suggestions for shaping the leading and trailing edges.

y=Tmax*( 8*SQRT(x)/(3*SQRT(XLE)) - 2*x/XLE + x^2/(3*XLE^2) )

where:

y is the thickness at a given value of x
Tmax is the maximum thickness as a fraction of the chord
x is the position along the chord from 0 to 1
Xle is the length of the tapered leading edge

Trailing edge thickness:

y=Tmax*(1 - (3*x^2)/(2*Xte^2) + x^3/(2*Xte^3))

where:

y is the thickness at a given value of x
Tmax is the maximum thickness as a fraction of the chord
x is the position along the chord

IN THIS FORMULA,
x is 0 at the beginning of the taper and
x increases to Xte at the trailing edge of the foil

Xte is the length of the tapered trailing edge

 Figure 20: Parallel Sided board with Reynolds Number 300,000 Figure 21: Parallel Sided board with Reynolds Number 3,000,000

Using XFOIL to simulate these shapes gives a surprising result. For all of the shapes described previously, the 6% thick boards stalled at a much lower angle than the thicker boards. With the parallel sided board and Reynolds Number 300,000, the angle of maximum lift is about 10 degrees for all thicknesses.

The particular parallel sided board analyzed has the leading edge shaped over the leading 20% of the length before it reaches the maximum thickness. All of the other shapes have the maximum thickness at 30%. Therefore, the other boards have sharper leading edges. The water has a hard time flowing around the leading edge and following the contour of the board. At angles above 5o, the flow separates and the lift drops. With the rounder leading edge of the parallel sided board, the water can follow the shape of the board at a larger angle of attack.

### New Cross Section

Originally I envisioned the work with XFOIL as primarily an academic exercise to confirm what had learned previously. When I saw that the thin versions of the parallel-sided board had greater stall angles than any other shape, I got excited because I sail a boat with the center board thickness set to approximately 6% by the class rules. Maybe there was an opportunity to develop a new shape that was not based on airplane wings.

Design for airplane wings is usually about generating adequate lift to keep the plane in level flight, low drag for fuel economy, and well behaved pitching moment. Pitching moment relates to the fore and aft position of the center of lift. If this moves with changes in AOA, it can cause the plane to pitch up or down and be hard to control. On a sail boat, the helmsman must be constantly adjusting for waves, wind puffs, etc. Pitching moment is not a major concern.

On a sail boat, a lot of time is spent going to windward. It might be a serious race, it might be a casual sail wanting to get back to the dock. Either way, a small increase in lift can make a big difference. All foil shapes have similar lift at the same angle of attack. The big difference in foil shapes is that some work at higher angles of attack than others.

 Figure 21: Extra foil, thickest at 0.300 Figure 22: Extra foil, thickest at 0.200

The previous simulations (Figure 21) with the extra shape had placed the maximum thickness at 30% of the length from the leading edge. This was simply because that is where the other wing shapes were thickest. The formulas for were modified to move the thickest place forward and thereby make a more rounded leading edge. As shown in Figure 22, this indeed improved performance. For AOA 9o, the flow on the original design was separated with high drag, the modified design had flow that remained attached to the trailing edge with low drag and higher lift.

The XFOIL simulations were so encouraging that I made a new board. This was the first of many iterations of calculating even better shapes, improving my shop skills in fabricating the shapes, and trying the new shapes in the water. A lot of other things were also learned along the way.

Simply Improving the Centerboard Makes No Difference

Actually my first experiment was a disappointment. Despite the encouraging computer numbers, the boat didn't handle any differently. There was a good reason for this. When the sails are set a certain way for a certain wind speed, they develop a certain force that pushes the boat sideways. The sideways speed builds until the force of the sails is balanced by the lift of the centerboard and hull. The sideways movement (leeway) of the boat combined with its forward motion creates the angle of attack. In short, angle of attack is something that is adjusted by the balancing of forces and not something that the sailor can adjust directly like, for example, the length of the main sheet. To use to take advantage of the higher lift possible at higher angles of attack, it is necessary to make some careful adjustments to the sail trim. This takes some experimenting because it is too easy to just pull in the main sheet too far and have sails not working to their best performance.

Caught in Irons

Being "caught in irons" is a situation where the boat is moving very slowly and the centerboard and rudder are generating enough lift to keep the boat from being pushed back, but they are also generating so much drag that the boat won't move forward. It happens most often when tacking in strong winds. First, the boat looses speed during the tack. Then it is pointed on a heading that would be appropriate, if the boat was sailing at speed. However since the speed (aka Reynolds Number) is reduced, there is less lift from the underwater surfaces. The boat slides sideways until sufficient lift is generated. This happens at a large angle of attack with the accompanying situation of separated flow and very high drag. The boat is then caught in a balance of too much drag to move forward, and too much lift to fall off from the wind. The correction is to force the boat to a greater angle to the wind, build up speed, and then trim for close hauled. All of this is complicated by the frustration of seeing other boats sailing past.

In my fleet, when strong winds are anticipated, it is common to for sailors make adjustments to the mast and sail that sacrifice a little performance at low wind speeds to reduce the likelihood of getting caught in irons during high winds. One day the winds were building. Between races other sailors were making the customary adjustments. I thought about the situation and deliberately made some poor practice tacks without adjustments. Then I went on to race successfully without the changes that I would have made if I was using my old centerboard and rudder. This was definite proof that the new foils were indeed better.
Summary
The sailing season came to an end before I had time to complete my work with new shapes and new sail trim. Often there were enough other factors, such as variable winds and waves that made a properly controlled experiment impossible. However, I have did have a number of occasions of racing against good sailors where I could point just a little closer to wind while going at the same speed. So I have confidence that I have a valid idea to refine before spring.

Here is a comparison of three foil shapes showing the angle and drag associated with a particular value of CL. The last columns are for the new shape I have developed. Note the last entry under NACA 0006. This would correspond to just barely getting CL of 0.9 at a very high angle of lift and a lot of drag. This is what might happen when caught in irons. The new shape can generate efficiently generate that much lift and more.

 Coefficient of Lift NACA 0006 AOA NACA 0006 CD SD8020 (6%) AOA SD8020 (6%) CD New Shape AOA New Shape CD 0.5 4.6 0.012 4.5 0.012 4.5 0.013 0.6 5.6 0.014 5.6 0.015 5.4 0.014 0.7 6.6 0.017 6.7 0.021 6.3 0.015 0.8 7.8 0.029 -- -- 7.2 0.018 0.9 19.8 0.258 -- -- 8.1 0.020 1.0 -- -- -- -- 9.1 0.024 1.1 -- -- -- -- 10.3 0.036

Alas, the summer sailing season has come to an end. Work is now focused on computer optimization and also wood working techniques to accurately make the correct shapes. The goal is to build the best computer designed shape before the next sailing season.

New information will be posted as it becomes available, or send me an email with your questions.