# Prismatic Coefficient and Sinusoidal "thinning"

Discussion in 'Hydrodynamics and Aerodynamics' started by Mike Inman, Oct 28, 2018.

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### Mike InmanJunior Member

I have been casually researching hull design for a few months in my spare time, and stumbled across an interesting relationship that I haven't seen mentioned anywhere (and Google has started steering me back to the same articles over and over again as I search...)

Regarding the Prismatic Coefficient, which apparently indicates an optimal hull profile at a given ratio of hull speed, Cp of ~0.63 or 0.64 being most efficient around hull speed (1.34*sqrt(LWL) kts), higher Cp up to 0.7 better for driving displacement hulls beyond so-called hull speed, and lower Cp down to the 0.55 range being more efficient at lower speeds (all else being fairly shaped and sensible...) so far, so good.

So, I'm thinking about how I might like to "canoe end" my hull profile and I'm thinking that a "sinusoidal extinction" might be the way to go, whereby the maximal cross section is reduced as a function of sin() where the tip-ends are at sin(0)=0 area, and the central fullness of the hull is at sin(90)=1x the area. This got me pulling out the old calculus and integrating the area under a sin curve, which, when normalized with sin(0-90) drawn filling a 1x1 box, the area under the curve is 2/pi leaving 1-2/pi for the area over the curve. Interestingly enough, 2/pi is 0.636619772, just about exactly the Cp for maximal efficiency at hull speed. I'm feeling like this is not a coincidence of the universe, but in my (admittedly limited) reading, I've never stumbled across anyone mentioning the relationship, nor even suggesting sin() as a potential thinning function.

Stating the potentially? obvious: the Cp of 0.6366 is the same whether you apply a = amax*sin(0-90) across the whole length of the hull resulting in a block-flat maximal cross section transom, or if you apply sin(0-90) from the stem to any point amidships, then sin(90-0) across the balance of the hull resulting in "canoe ends", whether symmetrical or not.

I see lots of references to people using parabolas, ellipses and other conical sections for their hull shapes, but so far no-one mentioning a sinusoidal shape, nor shaping function - not good, not bad, just no reference of using this apparently quite convenient function to achieve optimal canoe profiles.

Is anyone here better read on the topic and familiar with the use of sin() in shaping hull area profiles?

To drop the dry, theoretical perspective for a moment, I'm looking to apply this to a ~34' catamaran layout with ~17' beam, ~11:1 or higher slenderness ratio in the hulls and ~24.5" bridgedeck clearance at maximum loading, with displacement ranging from ~7000lbs to a max of ~18500lbs, and I'm trying to get a feel for what a "slippery" double-ended canoe hull profile will look like, particularly if the Cp of ~0.63 is preserved through the range of displacement. I was a little surprised to find that the optimal profile involved taper across the entire length of the hull, of course various profile manipulations could allow for the same 0.63 Cp but with a constant cross sectional area in a mid-section, but it does go a long way toward explaining some of the early cruising catamaran hull shapes, like: SailboatData.com - SNOWGOOSE 35 (PROUT) Sailboat https://sailboatdata.com/sailboat/snowgoose-35-prout

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### philSweetSenior Member

Mike Inman likes this.
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### Mr EfficiencySenior Member

Point of this exercise is what ?

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### Mike InmanJunior Member

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### HeimfriedSenior Member

Other points, you didn't mention, may have a far more significant influnce to efficiency, than the "sinusodial extinction" of the area of the hull sections. Think about the following examples, all of the same lenght (means same hull speed):
1. a square shaped section at main frame with constant width from stem to stern. Only the immersed depth is governed by the sinus (0 - 90 - 180). The cp is 0.6366.
2. a square shaped section as above, but with main frame at stern, sinus 0 [bow] to 90 [stern]. Same cp
3. same hull, but the transom stern is defined as stem. and the stem as stern, sinus 90 [bow] to 0 [stern]. Same cp.

The shape of the hull - which can be very different in spite of the same cp - will do more, than the difference between sinusoidal and elliptic or circular runs of the hull section area. Hundred years ago a average boat wright didn't have a table of sinus values I think, but a beam compass.

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### Remmlingerengineer

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### Mike InmanJunior Member

Thanks for the reference, though that's less a discussion than a pot-shot. From here: Colin Archer - Wikipedia https://en.wikipedia.org/wiki/Colin_Archer#Archer's_Wave_Form_Theory I would take away that while the stated theory regarding wave formation is wrong, it isn't an altogether bad design approach:
"With more undercut forefoot and the displacement curve extending the designed waterline, the lines became fuller and Archer's boats became the seaworthy boats he is known for.

Even to this day, people consult his work when designing new boats."

As is so easily demonstrated, single parameters or design approaches like Cp or the displacement curve are just part of what makes a 3D hull, it's not so much that one can design anything they want and then confirm it with a couple of simple tools, but rather that the simple tools can serve to point out a bad or perhaps overly compromised design.

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### tspeerSenior Member

If you also want to be able to position the center of buoyancy at somewhere other than midships, you might use this approach.

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### Mike InmanJunior Member

Thanks, some nice ideas in there, but we all need to learn to draw more pictures and show what our variables correspond to in the pictures.

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