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Following on from the last technical post on induced drag, this week i’ll give an overview of e, that quantity i left provisionally defined as 1 in the equation for induced drag;

.

is the aspect ratio.

Planform is the word that describes the shape of a foil when looking at it’s broad side, for instance profile view of a boat’s keel.

As you cannot have discontinuities in the fluid medium , the lift produced does not stop at the end of the wing abruptly, rather it tapers off gradually. It follows that if the planform is rectangular (leading and trailing edges parallel to each other) the pressure will be less at the tip than over the root of the foil. In other words the tip will be underloaded due to the fact that the lift must taper off but the chord retains it full width all the way to the end.

Similarly, for each planform with its varying rate of taper, there will be a corresponding lift distribution curve. For a triangular wing, as expected, the lift distribution tapers down more quickly than the lift distribution for a rectangular wing. However, it does not taper down as quickly as the surface does, meaning the tips are relatively *over*loaded.

One way to even out the lift coefficient along the entire span is to twist the wing along its length so that each section of wing is at a different angle of attack. In the case of a rectangular wing, one would have to twist it such that the underloaded tips are twisted to greater angle of attack than the root, in such a way that each section of wing is loaded the same.

For triangular wings, the tips are relatively overloaded, so the wing would have to be twisted the other way, with the angle of attack reducing towards the ends.

The problem with this is that the amount of twist needed depends on the overall coefficient of lift, which varies. Thus there is no way to create a twist that will correctly compensate for the variations of loading at all global lift coefficients.

Furthermore, Max Munk determined that for the least possible induced drag for a given span, the downwash angle has to be constant across the whole span, so that the air stream immediately behind the wing is deflected in a perfectly uniform way.

Without getting too far into it, for uniform planar flows, an elliptical lift distribution curve will result in a constant downwash angle across the whole span. Also it will give the best possible value of

It turns out that there is a planform somewhere between a rectangle (big tips) and a triangle (vanishing tips) that has a lift curve that matches the area distribution curve, thus making each piece of the wing work at the same coefficient of lift, or loading if you will, and that, at all global coefficients of lift. In this case the local and global lift coefficient would in fact be equal at every position along the span.

An untwisted elliptical planform will produce the required elliptical lift distribution.

Another factor of planform is sweep; the foil can be swept back, or forwards, rather than being at right angles to the flow. This will also affect the value of e. Note that what is important in determining the sweep back angle of a foil is neither the chord midline, nor the leading edge, nor the trailing edge. It is the *quarter chord line*, about which one can consider as being the aerodynamic “center of lift” of any foil. Explaining the rather complex effects of sweepback and sweepforward will have to wait for another post though.

For now suffice to say, that in general, any sweep is a deviation from the optimum.

An early example of all this being put into practice is the wing of the british Spitfire. Observe not only the eliptical planform but also the practically straight quarter chord line.

There is however another issue to take into consideration and one which may not evident at first. A pure elliptical wing, despite having the optimal area distribution, has a trailing edge that blends smoothly to the tip and on to the leading edge. Where one starts and the other ends is quite ambiguous, and this has the effect of pulling the tip vortex in towards the wing root. The tip vortex tends to follow the radius of the tip around towards the trailing edge until the angle becomes too great, forcing the vortex to break away. This makes the vortex separation unnescessarily messy and means the tip vortices are closer together than the actual extremity of the foil, thus reducing the *effective* span of the wing.

This would force a substitution of for with

In order to get the tip vortex to peel away as far outboard as possible and neatly, requires a sharp corner between the tip and the trailing edge. This then requires a little manipulation of the quarter chord line near the tip.

In the above image i have manipulated the quarter chord line in such a way that the trailing edge straightens out at 0.8 of the span, this is close to being an optimum planform for planar (non twisted) foils. It loses a tiny bit of but maintains , ie , as high as possible.

Notice too that in all this, that the triangular planform is about the worst possible area distribution. There exist empirical tables for the value of and the further one deviates from an elliptical area distribution the lower becomes, increasing induced drag. Yet it has been generally considered over the last fifty years or so that the bermudian (triangular) is naturally the best shape for going to windward etc. Marchaj did a number of wind tunnel tests on this and confirmed that the triangular planform is actually quite poor. This belief mainly stems from whatever is the current trend in raceboats, setting general ‘idealizations’ about what makes a boat ‘fast’, when in fact raceboats have to optimize to arbitrary rules just as much as actually go fast. This conflict inevitably produces design distortions that are completely innapropriate for sailboats that do not have to conform to any race rule.

To be fair, sailboat rig span loading is actually considerably more complex than that due to the fact that a sailboat rig is not *span *(height) constrained but rather *heeling moment *constrained, which imposes optimization along somewhat different lines than as described above. But we’ll come back to the finer points of optimum rig lift distribution in due course.

The value for b or span is taken as the distance separating the separation points of the tip vortice**s**. But what if the lower tip is closed off completely as pictured on the jib of the boat above? This would effectively eliminate the lower tip vortex. How to measure b? In that case, mirror theory states that one can model the three dimensional flow as being one half of the aerodynamic geometry and its mirror image as reflected through the fluid boundary, which in this case is the surface of the water. So b becomes from the upper tip vortex down to its mirror image (underwater).

In simple terms this means that closing off the gap doubles the effective span and thus also doubles the effective aspect ratio, halving induced drag.

This is a vast increase in aspect ratio and one that comes at *no cost in heeling moment*, so eliminating this gap is the single most effective way of improving a sailboat’s rig performance. This applies to not just jibs, but every sail. Of course, there are plenty of other reasons that may make it impractical to close off the gap completely, but if performance is high on the list of priorities, every effort should me made to reduce the gap as much as possible, for even if the gap is not reduced to the point of having a significant effect on induced drag, it still increases aspect ratio and sail effectiveness at no cost.

I remember being on a neighbour’s boat when i was nine or ten and the owner of the boat showed me a boomerang he had sculpted. I had never seen one of these things so he satisfied my curiosity by explaining what it was for and how it worked. He then went on to explain how lift is produced by an airfoil; the air gets split into two parts, the part that goes over the top of the wing and the part that passes below. Since the top is curved (and therefore longer than the bottom) and the two flows need to meet back up again at the same time at the back the top flow needs to go faster. Further, since the faster the stream the lower the pressure, the pressure over the top of the wing decreases, thus “sucking” the top of the foil up.

Unfortunately, this often repeated explanation for lift is rather misleading, so i’ll set out here to give a more thorough explanation.

Basic aerodynamics 101;

Lift is defined as being the total force experienced by an object in a fluid flow orthogonal to the far field flow direction. In other words, the total force at ninety degrees to the relative motion between the undisturbed fluid and the object.

In order to have a force one must have a counter force (Newton’s third law; action and reaction) so the air that is pushing the foil up, means that the foil is also pushing the air down.

Furthermore, we have by Newton’s second law,

Force is equal to the time rate of change of momentum.

This means that as the foil moves through the fluid it is constantly deflecting the flow downwards, imparting new momentum to it continually and it is this that creates the lift force.

So whatever happened to the longer path and the pressure drop?

Well, this is also happening, however it is entirely false to assume that the fluid particles have to meet up again behind the wing. They never do, unless there is no lift and no viscocity and no turbulence. In fact the flow over the top of the wing is much faster than necessary to make up the difference in upper and lower path length.

This actually makes sense when you think that the passage of the wing has deflected air downwards behind it, thus pulling back over the top of the wing the required extra air and similarly, slowing down the air that is passing underneath.

Now we also have Bernoulli’s equation for incompressible flow;

where is fluid density, U is flow velocity and P is pressure.

This is a conservation of energy equation since pressure is a form of potential energy and the second term is the so called “dynamic pressure”, which is the kinetic energy. Therefore, it is indeed true that there is a lowering of pressure over the top of the wing, although there is also a raise in pressure over most of the underside as well.

In the above image the fractional distance along the chord (line connecting leading and trailing edges) is on the x axis and the y axis is negative coefficient of pressure. I generated a Naca 64209 foil and ran Javafoil with the foil at ten degrees to obtain this graph.

Of course the total force experienced by any object in a fluid flow is the integral of the pressure (normal) and shear (tangential) forces upon the wetted skin.

——–*———

How it all works in practice.

We need to introduce the concept of circulation. Circulation is just what it sounds like; the air circulates around, in this case the foil. It is a key concept to understand. Imagine a vortex spinning around the foil such that the axis of rotation of the vortex runs from wingtip to wingtip. Now imagine adding, superimposing, the straight line flow to this vortex; what happens is that the flow is accelerated on one side, decellerated on the other, and deflected one way ahead of the foil, the other way behind.

Mathematically this is;

Which is the closed line integral (enclosing the foil), V is the velocity of the fluid element and ds is the differential length along the loop.

One may object that this seems to imply that the upwash (air being sucked upwards towards the low pressure zone above the wing) cancels out the downwash behind the wing thus leaving no overall momentum change to the fluid stream, but in fact the downwash is always twice as great as the upwash.

To understand why this is so, one must go back to the beginning of the path of motion of the foil.

The following images show the initial conditions when the wing commences its path;

What happens is that as the foil begins to move forward at an angle of attack, the air tries to make up the short path difference by curling around the trailing edge, initiating a vortex which quickly gets left behind but which is the exact reciprocal of the wing’s bounding vortex and which forces the establishment of the bounding vortex itself. Only like so can the overall circulation in the total airmass remain zero, as it must be according to the law of conservation of angular momentum.

Therefore, the downwash is actually due to the sum of the two vortices, whereas the upwash ahead of the wing is proper only to the wing’s bounding vortex. And when tacking a sailboat, the circulation decays to zero, only to re-establish itself in the opposite direction by shedding a new starting vortex behind.

——*——

So, to recap;

The lift created by a foil or other lifting body always involves imparting by means of an effective angle of attack of the lifting body, new momentum to the fluid flow, in a direction orthogonal to the stream and opposite to the direction of the lift force.

The mechanism for achieving this invariably involves circulation.

This circulation imposes a pressure distribution on the foil; low pressure above and high pressure below.

—-*—-

Some astute readers may object to me using an equation for incompressible flows (Bernoulli’s eqn) when air is in fact compressible. This is a very good point, but the reason that it is valid is because for small mach numbers (the ratio of flow speed to the speed of sound) , less than ~0.3, the amount of compression or expansion air undergoes is so small that it can be taken out of the equations with negligible loss of accuracy, greatly simplifying the mathematics.

As promised, here is a short article about scaling laws. Scaling is a topic that does not get much coverage outside of specialized engineering forums so i think it is worthwhile to explain the concept here more generally.

The basic concept is that you cannot simply take an existing design and scale it up or down without affecting the design in other ways than the linear dimensions. That may seem obvious, but apparently it wasn’t obvious to the not insignificant number of people who have attempted to do just this and ended up with boats that were floating disasters.

We will introduce a variable here, the linear scaling factor;

Such that

Where is the new linear dimension Length Over All new (say) and is the old corresponding linear dimension Length Over All old, and similarly for all other linear dimensions.

In other words ;

Now, the sail area is times longer and times taller so times as much

so

And, the volume of displaced water is times longer, wider and deeper so more voluminous

so, displacement;

Therefore, sail area increases as the square of the linear scaling factor, whereas displacement increases as the cube of a.

———–*———–

Relative stability.

Furthermore, righting arm; since GZ is linear

and righting moment; where m is total boat mass (same materials means same densities but times volume, which scales to the cube, so mass also scales to the cube)

So

i.e.

On the other hand, heeling moment is heeling arm times heeling force; HA x equation of lift, V is wind speed

.

([SA] is sail area) or

So

i.e.

So we see that righting moment increases to the fourth power of the linear scaling factor, whereas the heeling moment increases to the cube. This means that in a given wind a boat which is scaled up by a factor of two is sixteen times as stiff, but is subject to only eight times as much heeling force; so would heel approximately (since the righting moment curves are not linear but at small angles can be taken as approximately linear) half as much, for a given wind.

This also means that when scaling down, boats become excessively tender, so draft needs to be proportionately increased, and/or greater ballast ratios used and/or other means employed to offset the disproportionate loss of stability.

In the case of functional model sailboats – as opposed to scale model sailboats – this translates to what appears to the novice as absurdly disproportionate keels as well as extremely high ballast ratios. On the free sailing models i used to make standard dimensions were; 30 cM Loa , 15 cM draft and a ballast ratio that was around 95%.

————–*—————

Speed to length ratio.

The equation for the Froude number is (the Froude number is the dimensionless speed/length ratio) where U is boat speed, g is the acceleration due to gravity on the surface of Terra and L is waterline length.

So in order for the boat to operate at the same speed to length ratio, the speed needs to scale too, in such a way that the Froude number remains the same.

i.e.

so

but

so

Thus

——————*——————

Period of rolling and period of pitching.

The mass moment of inertia ( ) is dependant on mass – which is proportional to volume – and distance *squared* so scales to the fifth power.

The equivalent equation for in rotation is where is torque and is heel or pitch angle (the corresponding values of RM and I need to be used depending on whether one is looking at pitch or roll motion).

Substituting we get

.

.

so

i.e. the angular acceleration scales to the reciprocal of a or the larger the boat the lesser the g forces and vice versa.

For small angles, where is the slope of the righting moment per heel angle curve near the origin.

so

or [1]

which we recognize as the standard differential equation for simple harmonic motion and which can be solved

by putting

which gives

and which, by inspection can be seen to indeed satisfy eqn [1] and gives an arbitrary starting time point and a roll frequency of and an amplitude A. Both A and depend on initial conditions.

Now scales to the , while scales to the

so

giving

and

giving

So we can see, that, as determined earlier, the magnitude of the acceleration is inversely proportional to the scaling factor a, and we see that the frequency scales as the inverse root of a. Or that the period scales as the root of a, since period is the inverse of frequency.

At larger angles the approximation of linearity is no longer valid. However, the dimensional analysis still holds up the same.

————–*—————

Apparent speed

Time taken to travel one boat length is

so

thus, substituting appropriately;

giving

which is

so if a video of the times scale version of a boat is replayed at times the original speed it will simulate the appearance of speed of the original boat. And, happily, since roll and pitch period scale at a rate of as well, the same time stretch factor will also give the correct visual impression of the boat’s oscillatory movements.

Film producers everywhere; you’re welcome.

———————–*————————

Scale wind.

This brings us to the potential utility of scale models for prototyping full scale boats. There is certainly immense value in these kinds of trials which can be done at a fraction of the cost of finding a flaw in the full sized boat. However, as i hope is now pretty clear, considerable attention needs to be given to the subtleties of scaling laws .

The first consideration is to establish the scale wind; the amount of wind that will affect the model in a comparable way to the original boat.

Wind force is proportional to wind speed squared, as given by the equation of lift;

.

We need to find the scaling relation between the new wind speed and the original wind speed such that

.

heeling arm is

so

eliminating

thus

So we see that in order for the boat to behave similarly the wind speed needs to be scaled by root . This explains why large sailboats have greater difficulties in light winds. If a is 10, say, then the wind needs to blow ~3.16 times as fast to have the same effect.

————–*—————-

Reynolds number.

There is still the issue of the Reynolds number, which is the dimensionless ratio of inertial to viscous forces in the fluid stream. This number is important to distinguish what type of fluid flow regime the boat is operating at and which governing fluid flow simplifications can be reasonably applied.

. where is the fluid’s kinematic viscocity

Which can be seen to scale to the so it is impossible to keep both the Froude number and the Reynolds number correct as one scales a boat, unless different special fluids are used, but this topic is beyond the scope of the present essay…

————————————————*————————————————-

Feedback on if the maths symbols do not parse, along with which formula is giving problems would be highly appreciated , this is the first time i try “Latex” mathematical notation typesetting in wordpress.

For those who may be curious as to what it’s like to grow up on a sailboat I will write now.

Broadly, there was life at sea and there was life at anchor;

#### At sea

Leaving a harbour would be a rather sad affair for me leaving friends and sometimes little girlfriends behind, in the dimly fainting grey landmasses astern.

Sailing itself tends to be rather monotonous, despite weather changes, and life becomes very routine. Meals cut up the clock and are also the high points of each day.

If we had been at harbour too long, I would be seasick for the first few days and then the seasickness would gradually disappear.

I would spend time playing lego and if seasickness allowed, reading. A lot of reading. Drawing I very much enjoyed as well, but this could only feasibly be done on unusually calm days, on account of the motion, which jerks your hand around.

My room was in the bows so I had the strongest motion. Upwind in strong weather, I would be nearly weightless as the bow begins its descent, and then at the bottom of the motion, crushed into my bunk and against the side of the boat or the lee-board. Sometimes I would amuse myself by jumping slightly when the bow starts to drop and remaining ‘airborne’ for a few instants more than what seems normal. I had to be careful not to get thrown badly and hurt, though. L’ Artemis had that characteristic pitching motion of boats with a shallow but sharp forefoot, in that when the forefoot is clear out of the water the bow will pitch more or less vertically downwards, but as the forefoot bites back into the water, it tends to follow its own axis pitching down diagonally to windward, following a kind of “j” shaped trajectory. In practice, this would feel like you are very light, then as you get heavier the boat gets sort of jerked sideways under you. This may be a reason why I can sleep anywhere (as long as i’m not cold) and sleep very soundly.

Also I could spend hours at a stretch just looking out from the companionway hatch – until I was 10 or 11 I was never allowed on deck when sailing – contemplating the waves rolling on and on from horizon to horizon. One wave builds, then breaks and sighs back down, only to be consumed by the next wave growing behind,..meanwhile, my thoughts pondering all manner of things great and small.

Inside, the sound of a wave breaking just to windward would predictably be followed by a mighty thump against the side punching the boat bodily to leeward.

Downwind the motion is much gentler naturally; the stern lifting up to an overtaking wave while the boat accelerates onwards down the incline of the wave, eventually the wave passing and the boat balancing down the rear of the wave more slowly only to be picked up and propelled along by the next one, all the while rolling. The rolling downwind goes in cycles; big angles for awhile, then the rolling almost stops, then a wave or other picks the boat up from one side and the big rolling starts up again. I would lie down on the saloon settees and press my ear against the wood and listen to the water sloshing around in the water tanks and the bubbles rushing along under the hull, the centreboard dully clunking in its case.

Upwind, the centreboard does not make any sound as it is down and the whole boat is pressing hard against it. Indeed, so hard that it is quite impossible to move up or down; for an adjustment to be made one must luff up to relieve the pressure. The centreboard case was an imposing central feature to L’ Artemis. It went right up to the cabin top, dividing the boat into right and left halves.

Sometimes at night I would sit in the companionway and marvel at the stars. The air is generally astonishingly clear at sea. The blackest black is not as black as the space in between the stars there, except that upon closer scrutiny that empty space has hundreds of stars too, almost too faint to distinguish individually, so the velvety blackness becomes elusive, like the concept of absolute nothingness filled with infinity.

We had star charts and I came to know and recognize the stars by name and constellation. A mind game would be, if cloudy, to see how quickly I could recognize the stars despite only seeing a few at a time.

In fact the stars are so many and so bright that you can clearly see the boat in the starlight and manoeuvres are made easy. For the ultimate darkness you actually need a moonless and heavily overcast sky, as well as the running lights out and only then does it become truly pitch black. Manoeuvres in this case then have to be done entirely by tact, and this is why one must know one’s boat very intimately.

The full moon is also quite magical. Hours could be spent observing that dancing streak of silvery moonlight going from the boat all the way to the horizon, curiously nearly never directly under, but rather a little to one side or the other of the moon silently illuminating the tumult under, the rushing boat, the never ending roaring wind and shiny black waves.

One time we saw a ship’s illuminated tower bridge. It seemed to be coming more or less straight at us. As it approached it became visibly larger, then we could see it visibly grow – just how fast can a big ship approach us !!? Getting very alarmed, the clouds shifted slightly, breaking up the tower bridge’s symmetry revealing the orb of the rising moon glowing pinkly in between clouds…

At sundown, father would get the lights in and fill them with kerosene, checking that the wick is neat and straight and the globe clean. Then he would light them and place the two white running lights back in their light boxes with their fresnel lenses, and then these on their respective platforms at each quarter taffrail, securely tied down, usually with wedges underneath to keep them level despite the boat’s heel.

On approaching land, often the first sign is the loom of a large town or city glowing over the night horizon. A large city can have a loom that can first be seen when still 60 or even more miles off. Then there is the smell; approaching places where there is a goodly amount of nature the smell is some variation of humus, damp earth, and rotting vegetation, and this can be smelt ten or more miles away depending on how the wind blows. But approaching a large city the ~~smell~~ stink is overwhelmingly of auto-mobile exhaust with a fainter smell of refuse, and this can be smelt quite far indeed, considerably farther than the musty nature smells. I always wondered how people could willingly live in places that stank so bad that you can detect the stink 50 miles away! But the sense of smell is the sense that most quickly desensitizes itself and so city dwellers are quite unaware of the tremendous stench of their surroundings.

Smell or stink, excitement would build over the imminent new place to visit and make new friends in. Then as land ever so slowly gradually makes itself visible all eyes are riveted landwards. With my keen eyesight I would not only be almost always the first to see land, but also the first to pick out the outermost approach buoys. Then the features become more and more distinct, landmarks then houses, masts in the harbour, then cars and finally people can be seen. And the calm! After days of never ending jostling around, the boat’s motion as it enters more and more sheltered waters becomes surreally serene..

After deciding on a good spot to anchor, we sail in amongst the other boats and round up into the wind, coasting to a stop, while the jib is smartly dropped and mainsheet eased. The anchor goes over and the rest of the sails come down in sequence, from the front to the back always.

#### In a harbour

After taking in our new surroundings and after the usually tedious entry formalities we would go visiting the other sailboats. I was an expert at judging which boats were most likely to have other children on board.

If father expected to stay more than a few weeks, I would be enrolled in the local school. In this way, I have ended up studying in fifteen different schools.

#### School

Father would row me ashore so I could get to school in the morning, and later on when a bit older I would scull or sail my dinghy ashore myself.

Most of my time at school would be spent counting the hours till I would be released from that day’s serving of mind crushing boredom.

I can only remember a few things of note that I learnt at school; one of them being that history books are very far from being objective, and would describe the same events very differently depending on which country one is in.

Almost all the important things I learnt, was from listening to adults converse, and reading. That, and always asking a lot of questions. School was definitely a waste of time. It could be argued that one does learn social skills at school, but that can be learnt anywhere there are people, not necessarily at a school.

Eventually I discovered the thrills of skipping class. I would sit close to a window and when I could take it no longer I would slip my bag out the window and ask to be excused for the bathroom. After recovering the bag I would head out and explore the hills or go bodysurfing.

#### No school

If I was not attending school, my mornings would be spent studying aboard. This is how I learnt to read. Otherwise I would be assigned certain chores appropriate to my age to get done. The afternoons I was free to do as I pleased.

When we were in Faro, Portugal, when I was seven, one of those near daily ‘chores’ would be to walk to the bakery and buy a bread. One day I came back with the wrong change and my Chinese stepmother sternly ordered me to return at once to the bakery and reclaim the missing change. Luckily I was able to convince them to correct the matter, but ever since then I am always careful to check my change.

Up till a certain age my father would always read or tell me a “bedtime story”. When I was six, one of those times, instead of the usual story he explained to me what a loan was and how not only do you have to pay interest in addition to paying back the money, but also if you miss your payments the bank can just take back whatever you had bought, even if you had already paid almost the whole loan back, in fact more so, due to the interest.

I have never taken out a loan.

Starting around nine I started reading the many books we had on board on boat building and naval architecture. When I was ten my father explained to me how to draft a boat’s lines plan. This was much more interesting than the drivel they were teaching at school! Later, I started making model sailboats, eventually making nearly thirty of them, although not all were completed. This proved to be extremely useful in testing out different design concepts as well as providing me with much enjoyment.

When i was thirteen a neighbour lent me Marchaj’s “Aero/Hydrodynamics of Sailing” as well as the “Seaworthiness; the Forgotten Factor”, which were a step above Herreshoff’s, Skene’s, and Uffa Fox’s books in thoroughness, and which i devoured and tried to delay returning for as long as possible. Now i have my own copies and they are the centrepieces of my library. These two books may well have been the real motivating reason i ended up studying mathematics at university.