Hull Section Shapes

[PI Design]

I had an idea last night, and I'd like to hear your opinions.

Does it make sense to have the underwater hull section shapes in the form of catenaries? For those that aren't sure, a catenary is the shape that a rope takes up when suspended between two point e.g. a washing line. Wikipedia and Mathemtaica both have good descriptions with more detail.

I believe there are several benefits of using a catenary for the underwater section shapes:
1. It is most efficient shape at resisting an upward force (buoyancy) and is therefore the lightest/strongest shape to have (could use a thinner core).
2. The shape is a good compromise between minimising wetted surface area (semi-circular being best) and having a flat bottom for planing. Most modern dinghy hull sections look catenary shaped.
3. Being a mathematical shape it is perfectly fair, and is easy to produce.

A lot of modern hulls do seem to look like catenary sections, but I suspect that they are not exactly so. My thought is that if the section lines were drawn by a mathematical equation, the process could be automated, making it quicker and more accurate.

What do you think?

[marshmat]

If I recall correctly, the catenary shape is very, very close to parabolic (if not exactly so). Applying it to the undersides of a boat, you get what is known as a Wigley hullform. Although mathematically 'perfect' (thus many calculations, such as hull drag using Michell's theory, are exact), my understanding is that pure Wigley hulls generally aren't that great once you start including waves in the picture.
That's not to say that parabolas are bad.... just that they need to be combined with other shapes to produce nice hull forms.
Indeed, the section lines of a large number of boats now are drawn by mathematical expressions which are somewhat self-fairing: NURBS spline curves.

[PI Design]

You're right, catenaries are similar to parabolas (but not exactly the same). Granted the catenary is unlikely to be perfect for a hull form, and the ability to 'personalise' the curve would be great. But the trouble with NURBS is that you don't know what you're getting until you draw it. With a catenary (or parabola) the computer can calculate what you need to fit some other predetermined values (Such as beam, draft, section area etc). But it sounds like this Wigley chap might have beaten me to it...

[PI Design]

Anyone know a good link that explains more about Wigley hulls?

PS Although there is hardly anything in it, I think a catenary is a theoretically better shape than a parabola, due to its structural efficiency.

[marshmat]

Are you familiar with the work of Leo Lazauskas? http://www.cyberiad.net/hydro.htm if you're not. He's got an excellent free program on there called Michlet, that can evaluate Michell's equations for an arbitrary hull form as well as optimizing a hull defined only by mathematical parameters, for given load and speed conditions. There's also a few papers there on Wigley-shaped multihulls and other mathematically interesting nautical ideas.

[PI Design]

Thanks for that, I shall have a look. Is the Wigley hull equation only suitable for slender bodies - my very scarce understanding is that that is what Mitchell's equations are for?

[marshmat]

My understanding is that for Michell's method to be accurate, the hull must be relatively slender. Length/beam ratios as low as 4:1 sometimes work but best to stay over 6:1 or so for this method. My interpretation is that with fatter hulls, the code cannot account for the inevitable flow separation as you approach the stern; don't quote me on that though. As far as I know, Wigley shapes can exist in any dimensions, although they're not considered to be very practical. Rather, they're a sound mathematical starting point from which to begin experimenting with different codes.

[Tim B]

The whole point of NURBS is that you can define pretty much any surface and it is totally mathematically defined.

It is an extension of the simple b-spline formulation, "C=SUM(i=0 to n) (PTi * Ni)"
For NURBS this becomes, "C={SUM(i=0 to n) (PTi * WTi *Mi)} / {SUM(i=0 to n) (WTi * Mi)}"

I don't really see what you'd gain by automating a design using parabolas, as opposed to NURBS.

Tim B.

[Leo Lazauskas]

Originally Posted by PI Design

I had an idea last night, and I'd like to hear your opinions.

Does it make sense to have the underwater hull section shapes in the form of catenaries? For those that aren't sure, a catenary is the shape that a rope takes up when suspended between two point e.g. a washing line. Wikipedia and Mathemtaica both have good descriptions with more detail.

I believe there are several benefits of using a catenary for the underwater section shapes:
1. It is most efficient shape at resisting an upward force (buoyancy) and is therefore the lightest/strongest shape to have (could use a thinner core).
2. The shape is a good compromise between minimising wetted surface area (semi-circular being best) and having a flat bottom for planing. Most modern dinghy hull sections look catenary shaped.
3. Being a mathematical shape it is perfectly fair, and is easy to produce.

A lot of modern hulls do seem to look like catenary sections, but I suspect that they are not exactly so. My thought is that if the section lines were drawn by a mathematical equation, the process could be automated, making it quicker and more accurate.

What do you think?

Here is a graph of mid-sections for some of the hull shapes
under discussion. I hope that it clears up some confusion
about parabolic hulls and Wigley hulls.

Note that the Wigley hull (here in red) has a sharp keel.
Note also that the curve is vertical at the waterline (i.e. at
z=1 in this graph).
Catenaries (here in green) can also be used in this orientation
to give a sharp keel.

The two parabolae forming the Wigley hull are rotated 90 degrees
relative to the "true" parabolic hull (the blue curve) which
is rounded at the keel, as is the "true" catenary (the pink
curve). Note also that these two sections are flared at the
waterline.

The advantage of using Wigley hulls in hydrodynamics research
is that it allows considerable simplification of many very
complicated expressions in Michell's thin-ship theory. Some
useful output quantities from Michell's integral are exact for
Wigley hulls which means that it is a very useful hull to check
that computer calculations are correct.
For some calculations, eg far-field wave elevations, the actual
details of the hull might be irrelevant. From a great distance, all
hulls look the same as far as the hydrodynamics are concerned.

Michell's theory is a very good approximation provided
the longitudinal hull slope (i.e. from bow to stern) is small.
The cross-section shape does not affect Michell's approximation,
for example SWATH hulls are handled as easily as the sections shown
in the graph.

Some waterplane shapes clearly violate the small slope assumption.
For example, an elliptical waterplane does not have a small
longitudinal slope at the bow and stern: the longitudinal slopes
there are infinite. Michell's theory will give very poor estimates
of the wave-making for such a waterplane shape, particulary at small
length-based Froude numbers. Unusually, at high Froude numbers
Michell's approximation gives quite reasonable answers even for
elliptical waterplanes which so clearly violate the small slope
assumption. Very roughly, the reason is that at very high Froude
numbers, the waves made by the body are (generally) much longer
than the body. The region where Michell's theory is invalid is thus
small compared to the body. Remember that not all of an elliptical
waterplane violates the small slope assumption: near the centre of
the waterplane, the slope is small and so Michell's theory is valid
there.

There also seems to be some confusion regarding slender ship theory
and Michell's thin-ship theory. The slender ship approximation can
be thought of as a subset of Michell's thin-ship theory. There are
certain elements in Michell's theory, for example exponential decay
of waves with depth, that are absent in the slender-ship approximation.

Someone mentioned the inability of Michell's theory to predict
boundary layer separation near the stern. This has nothing to do
with Michell's theory. However, it is possible to augment Michell's
theory with viscous effects by, among other ad hoc methods, adding
boundary layer displacement effects, boundary layer detachment layers
etc. Some of these have been included in Michlet.

Happy (Cricket) Season Greetings to all!
Leo.

[marshmat]

Someone mentioned the inability of Michell's theory to predict
boundary layer separation near the stern. This has nothing to do
with Michell's theory. However, it is possible to augment Michell's
theory with viscous effects by, among other ad hoc methods, adding
boundary layer displacement effects, boundary layer detachment layers
etc. Some of these have been included in Michlet.

Guess I need to brush up on that theory a bit, Leo... I was under the impression that part of the reason for the difference between theory and tests with fatter ships was that the flow wouldn't follow the hull very well on the aft sections of a fat ship... my bad!

[Crag Cay]

Happy (Cricket) Season Greetings to all!

Good post Leo - then you went and ruined it. It's a mediocre season at best and I fear, it will only get worse.

[Leo Lazauskas]

Originally Posted by marshmat

Guess I need to brush up on that theory a bit, Leo... I was under the impression that part of the reason for the difference between theory and tests with fatter ships was that the flow wouldn't follow the hull very well on the aft sections of a fat ship... my bad!

You are absolutely correct. I was only pointing out that Michell's theory doesn't try to predict separation. It is an inviscid theory.

Best,
Leo.

[marshmat]

So I'm not as clueless as I thought
(on that note, also... if it is necessary to predict how and where separation will occur, the only code I'm aware of that can predict both the laminar-turbulent transition and the separation point is Fluent's V2F model.... a pricey addition to what is already an expensive code.)

[PI Design]

Thanks for all the replies, I greatly appreciate all the words of wisdom. I found a reference for Wigley hulls and did some similar playing. It seems to me that the catenary is a simpler equation, but maybe it can not generate such a variety of shapes? Still, on balance, it seems there is some merit to the use of catenaries.

Tim B - I guess you could use NURBS that way. The problem is that the CAD software I have makes you draw your curves, virtually freehand. Using NURBS, you may get a 'fair' shape in terms of rate of change of gradient etc, but it still seems to require a certain amount of skill to get a shape that is what I want. A catenary is a simple curve, with no points of inflexion and is easy to use. It may be possible to use NURBS in this way (I don't know much about them), but the catenary seems easier (if less flexible).

The cricket is only made worse by the fact I have an Aussie sat about 3 feet to my left. To think I subsribed to Sky for this...

[Tim B]

The problem is that if you remove the skill from drawing yachts, we're all out of a job.

Tim B.