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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;

.                                                $C_{D_i} = \dfrac{{C_L}^2}{\pi{e\Lambda}}$

$\Lambda$ 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.

Lift distribution along span of foils with varying taper ratios.

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 overloaded.

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 $e$

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.

The famous elliptical wings of the spitfire

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 $\Lambda_e$ for $\Lambda$ with $\Lambda_e \leq \Lambda$

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.

Elliptical area distribution with straight quarter chord line modified near tip so trailing edge is straight and tip has vortex shedding corner.

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 $e$  but maintains $\Lambda_e = \Lambda$, ie , as high as possible.

The 1988 US catamaran Stars and Stripes with a vortex shedding tip planform designed by Burt Rutan.
Image from François Chevalier

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 $e$ and the further one deviates from an elliptical area distribution the lower $e$ 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 foot of the jib is in close contact with foredeck, effectively eliminating one of the airfoil tips.

The value for b or span is taken as the distance separating the separation points of the tip vortices. 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.

Non infinite planet

Enough with reminiscing for now, lets look instead at the present and the future. This post is not one I have been looking forward to since the topic invariably raises peoples intellectual defense mechanisms, so I’ll try to keep it short and straight to the point. Besides, there is an enormous amount of information on the subject of oil, energy and modern civilzation on the net already. The point of the post is more of a summary of the situation and its intersection with boats.

It’s no secret that modern civilization relies on petroleum, but it’s easy through habituation to forget at what point.

For instance; there’s a high chance that the clothes you’re wearing are at least partially made from synthetic fibres, made from petroleum. Even if you’re wearing all cotton or linen clothes, those crops were undoubtedly nourished with synthetically made nitrogen, sprayed with pesticides made from petroleum, the fields mechanically tilled by machinery that runs on petroleum. Then the fibres are irrigated, harvested, processed, transported, woven and sewn with yet more machinery all ultimately running on fossil fuel energy.

The computer you’re reading this on is made from metals mined with machinery running on petroleum and plastics made from petroleum. A comparable amount of energy goes into building the computer as it will use during its lifetime.

Cars are much the same except worse.

Of greatest concern of all, food has become extremely reliant on petroleum. Depending on which study you read, between 10 and 14 calories of fossil fuel energy goes into producing one calorie of food. This is a less than parity EroEI (energy returned on energy invested) which is clearly untenable, but which has permitted, temporarily, for us to vastly exceed the earth’s normal carrying capacity.

Indeed, there are very few things left in modern society that does not depend in some way on fossil fuels.

Rate of world population growth
upper graph aligned with
per capita world oil extraction rates
lower graph

A useful mental exercise is visualizing how far your car can go with just one litre of fuel. Then imagine you have to push the same car that same distance, up and downhill. There is a lot of energy in that litre of fuel!

Then think that we burn more than 70 million barrels every day to power our industrial society… This is a staggering amount of added energy. It is only because of this that we have an industrial society at all.

There is a tremendous amount of unrealistic wishful thinking that goes on in the attempt to believe otherwise; that we will be able to transition to alternative energy sources and carry on with all the energy hungry modern technological marvels, but this does not stand up to closer analysis. A typical reaction is “I’m sure they’ll think of something” which is about the least useful response possible. This not only downplays the gravity of the situation but also reveals humanity’s hubris and arrogance.

The main issue is that of exergy. Fossil fuels have remarkably high levels of exergy, as the mental exercise of pushing the car illustrates. There is ample total energy in renewable, but the exergy is nowhere near high enough to be a reasonable replacement for fossil fuels.

Simply put, the only energy that we can use sustainably in the long term, is a small fraction of the total solar energy throughput that falls on our orb. And fossil fuels are nothing other that stored solar energy accumulated through millions of years of minutely imbalanced biological energy flows. Energy that we are using in a blink of an eye compared to the time it took to accumulate. There we come to the crux of the matter; we have become accustomed to a situation which is by definition unsustainable, and no amount of denial will make that fact go away.

Energy can neither be created or destroyed, only transformed from one form to another form and this is crucial to understand. What happens instead is that as any process is performed by high exergy energy the energy gets transformed into different forms always of lower exergy. This is the conclusion of the second law of thermodynamics. In other words, the same amount (almost exactly) of energy shines onto the daylight side of the planet from Sol as our planet radiates away on the night side. The big difference though, is that the exergy of the outflowing energy is always lower than that of the inflowing energy.

Our planet is finite and thus contains finite amounts of every resource. The rate of usage of every non renewable resource (and even renewable resources whenever the rate of consumption is sufficiently greater than the rate of renewal) tends to follow a roughly bell shaped curve. Over half of the world’s oil producing countries have already peaked and have declining production rates. Conventional global crude peaked around 2005 which caused a persistent increase in the price of crude, allowing previously economically non-viable fields to be exploited. This then has created a certain plateau to world oil production, at the expense of future decline rates. Cantarell oil fields in Mexico is a good example of this process at work at smaller scale. They were able to extend the peak using unconventional means, but when the decline finally came it was precipitous.

The EIA and IEA Have been quietly revising their projected date of expected world peak oil for the last few years, reflecting the dawning realization that the day has come and it is us, rather than the not yet born that will have to deal with this.

Oil production vs price

The above plot should be reminder enough that economics is a subset of environment, not the other way around… No matter how high the price goes, physical laws refuse to concede to economic fanatsies.

When the predominant form of energy peaks it tends to put pressure on other sources of non renewables making them peak sooner in turn.

Nuclear power needs uranium or other unstable elements which itself will peak soon (uranium ores are already becoming very poor) and mining uranium itself requires petroleum. Additionally, nuclear power plants take over a decade to bring online and at enormous monetary and energy cost. In fact, the only way nuclear power is economically feasible is through government subsidies and by working symbiotically with the nuclear arms industry.

Besides, de we really want the next ~100 000 generations curse us for leaving behind impossible to dispose of toxic waste just for the sake of running AC units etc for ours and maybe one more generation?

Nuclear fusion I think can be safely relegated to the aptly named “vaporware” closet, after the better part of a century spent investing enormous resources into researching nuclear fusion, the solving of the immense technical difficulties remain perpetually over the horizon..

Coal will peak too in the not too distant future, and dirtily.

Gas the same but cleaner.

Tidal energy is interesting, although it also requires a large energy investment, and may well prove to be a false avenue once fossil energy subsides are no longer available or are being used on other more urgent applications.

Geothermal can also be useful, but only in places where the subsurface temperature gradient is steep enough for the energy investedto be a sufficiently small fraction of the energy payback over the lifetime of the heat well.

Biomass, wind energy, solar power, hydro and wave energy are all ultimately derived from the energy of the sun. As such, there are very definite limits to how much can be extracted, nevertheless, this is about our only viable long term strategy. Remember that biomass means without any fossil fuel inputs, otherwise it is just fossil energy transformed, and that usually at a loss! In practice, and for most applications, the most efficient use of cropland is to fuel humans with food. Solar panels are massively over rated. Their EroEI is quite poor and can really only be justified for specialized applications, such as boats where space and weight are very limited. Wherever weight and space are not at such a premium (industrial, residential rather than on board transport) it makes much more economical sense to use solar thermal rather than photovoltaic.

It is important not to be fooled by claims to the contrary; there is no way we can maintain present day energy use into the future.

That is why this is a predicament, not a problem. Problem implies solution.

However, that is not to say that there aren’t plenty of worthwhile mitigating strategies that one can apply, and despite the inevitable process of technological triage in the decades and centuries to come, there are useful and low energy technologies which can and ought to be preserved moving forward.

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So what does all this have to do with boat design?

A lot actually.

There are a great many ‘modern’ boats which will be seen as curious relics of a by gone age within a generation or two. The first to go extinct will be the highest energy consuming boats, of course, but a lot of standard design and material choices which are commonplace nowadays will soon feel the squeeze too.

Only with plenty of cheap fuel can something like this exist

Here is a short list of the embodied energies per mass of some common boat building materials in MJ/kg. The ranges represent the spread depnding on source of information and their method of calculation.

_________________________________________________________

Titanium                                                       361 – 745

Aluminium sheet, virgin, sheet                 160 – 217

Aluminium sheet, recycled, sheet            14.8 – 27.8

Epoxy resin                                                         ~125

Fiberglass cloth                                                  ~55

GRP composite  approx                                  100

Bronze                                                                 77

Stainless Steel                                                    57

Steel , virgin                                                    25 – 35

Plywood                                                           10 – 14

Timber, softwood air dried, roughsawn    0.3 – 7.4

Timber, hardwood air dried, roughsawn   0.5 – 7.8

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These numbers agree to a certain extent with the graph of per capita oil above, when one considers when the different materials became commonplace boat building materials.

The other side of energy use is energy used to operate during lifetime.

Here the first line of adaptation will be towards greater efficiency. Huge motors pushing motorboats that have more than a passing hydrodynamic resemblance to household appliances will get rarer with each passing decade. Most cargo ships are already very efficient, but yachts and most ferries can be tremendously improved in terms of efficiency. Another thing will be the trend towards slower speeds. Resistance is roughly proportional to speed squared, while passage time is inversely proportional to speed. Fuel consumption is proportional to power neededwhich is speed times resistance, so fuel consumption is proportional to speed cubed. Therefore a lot less fuel gets used when going slowly, despite spending more time powering. We can already see this trend with large commercial ships; the so called ‘slow steaming’ approach.

It’s pretty much a given that sailing will become more and more used as it’s comparative economic viability improves.

Eventually motors themselves will become rare, although this will take a long time still and potentially heat engines will never completely disappear evn if they are limited to special uses where the energy equations make sense despite there being no more fossilized sunlight to exploit.

Welcome to the new paradigm.