yacht design » hull and keel » what is the perfect hull cross section?

This is a bit of a 'how long is a piece of string?' type of question, but let's have a go anyway.

There are several design criteria that the 'perfect' hull should have a combination of:
1. Minimum surface area for the required hull volume
2. Positive form related stability that has a linear onset
3. Enough hull volume to satisfy the purpose of the boat

A minimum surface area for the required hull volume ensures that frictional drag is kept to a minimum. Technically, the cross section curve that best satisfies this requirement is a (semi-)circle, however, opinion appears to be divided as to whether a (semi-)circular cross section has any form stability at all. (By form stability, I mean a shape of uniform density that has a tendency to float the 'right' way up. This happens if the centre of buoyancy moves horizontally relative to the centre of gravity as a shape tips, thus providing a righting force that is in equilibrium when the shape is 'upright'.)

Traditional wisdom refers to the example of a floating barrel or log and how these cylindrical objects exhibit no tendency to consistently float with any fixed 'side' up - hence the claim that a circular cross section exhibits no form stability. The counter argument runs something along the lines of 'cut you barrel/log down the middle lengthways and try it again' - now it will float the 'right' way up. So what has changed? The cross sectional shape of the bit in the water is the same, but the stability has changed.

For the fully circular barrel, the centre of gravity is at the centre of the circular cross section (which, incidentally, corresponds to the centre of rotation as well, see below). But attach a weight to the inside of your barrel (or remove the top half of your barrel/log) and you lower it's centre of gravity, resulting in a stable hull form. Thus the 'half a barrel/log' argument is misleading as the stability results from shifting the centre of gravity (also achieved by attaching a keel, for example) and not from any form stability component.

Another factor affecting form stability is the centre of rotation. Again, there appears to be great debate over where the centre of rotation for a floating hull is. Steve Killing (an America's Cup yacht designer) in his book 'Yacht Design Explained' tells us that this 'metacentre', as he calls it, lies at the intersection of a vertical line passing through the centre of buoyancy and a line perpendicular to the fore-aft centreline of the hull (i.e. parallel with the mast). As the hull heels, the centre of buoyancy on a typical hull will move horizontally, thus the centre of rotation (metacentre) will shift up/down the line that is perpendicular to the centreline (i.e. the centre of rotation moves as the hull heels). On a typical hull this 'metacentre' will be above the deck.

Other sources, however, claim that the centre of rotation lies on the hull's floatation plane (i.e. the horizontal plane corresponding to the surface of the water). Still others claim that the centre of rotation lies somewhere between the floatation plane and the horizontal plane that passes through the centre of gravity (i.e. below the floatation plane on a typical hull).

I have yet to find a definitive answer as to the location of the centre of rotation. The issue of the location of the centre of rotation also impacts on our discussion about the form stability of a circular cross section. If the centre of rotation corresponds to the geometric centre of the circular cross section, then no form stability will be displayed (c.f. the fully round barrel/log). If the centre of rotation lies elsewhere then some form stability will be evident (c.f. the half round barrel/log). The barrel/log examples do suggest that the location centre of gravity does impact the location of the centre of rotation, which in turn casts doubt over any location theory based on geometry only.

A final point about form stability that was touched upon at the top of this section was a 'linear onset' of form stability. By this I mean that the righting moment provided by the hull form as it heels should increase in a linear fashion. An accelerating righting moment results in a hull that is uncomfortable for the crew.

To further my study of how the centre of buoyancy shifts as a hull form heels I have constructed a set of animations. Each frame for each animation was generated to exactly locate the centre of buoyancy of the specific curve at the angle of heel being modelled. The centre of rotation used in each animation is the centre-base of each curve, however, this centre of rotation may not reflect the real world (see above). The illustration is still instructive, but care should be taken when using these animations to assess form stability as a change in the centre of rotation would almost certainly have a great effect on the relative motion of the centre of buoyancy.

My animations model the 0 to 30 degrees heel of each of the following hull cross section curves: Parabola, Sine Curve, Square, Right Angle Triangle. For the parabola I have modelled three different scenarios that correspond to three different immersion depths. More about this below.

Here is a key to the features you will see on each animation:
1. On the parabolas and sine curve, this point indicates the location on the curve where the gradient of the tangent = 1. For the square and right angle triangle, this point is an arbitrary point on the hull. This point is typically related to the waterline at the maximum angle of heel (30 degrees).
2. The axes (graduated at 0.5, 1.0, 1.5, ...) - coloured blue.
3. The line of symmetry of the curve - coloured black.
4. The centre of buoyancy for the curve at this angle of heel.
5. The line of curve being modelled - coloured black.
6. A plot of all centre's of buoyancy for the curve.

One final point about the three parabola scenarios modelled. Each differs in that the 'tangent=1' point has a differing relationship to the waterline at the maximum angle of heel. One parabola animation demonstrates the parabola immersed such that the waterline at the maximum angle of heel passes through the 'highest' of the two 'tangent=1' points - 'Parabola > 1'. Another parabola animation demonstrates the parabola immersed such that the waterline at the maximum angle of heel passes through the 'lowest' of the two 'tangent=1' points - 'Parabola < 1'. The final parabola animation demonstrates the parabola immersed such that the waterline at the maximum angle of heel passes through a point midway between the two 'tangent=1' points - 'Parabola = 1'.

Here is a graphical comparison of the various curves (plus some extras for good measure).