Hull Asymmetry and Minimum Wave Drag

[DCockey]

Originally Posted by Richard Pitblad

To continue with my heuristic observations, I propose that reflection phenomena at the waterline of a single hull are pervasive, and need to be considered simultaneously with the modeling of sources and sinks. A source located at the starboard waterline will displace water, creating a wave that can freely propogate to starboard but is inhibited in propagating to port, reflecting to some degree off the topsides. If the hull is symmetric and thin, this is of no consequence, because any reflection is mirrored on the port side so the result is the same.

Now think of the sources at the shoulder of a wider ship. We do observe a small pressure wave crossing under the hull across to the other side, but it is relatively small. Most of the wavetrain created by a starboard source will propagate toward the starboard side. That makes a shoulder source located a half wavelength behind a bow bulb more completely cancel the starboard component of the bow bulb wake, especially if the bow sections are hollow for the first half wavelength.

Now consider sticking on a stern to this ship that has a hollow for a half wavelength, with an underwater bulb at the stern (faired to a point for form). Most of the starboard shoulder sink emanates to starboard due to the aforementioned partial reflection, and the same on the port side. The stern shoulders exhibit much poorer cancellation with the stern bulb than occurs at the bow. The stern hollow is counterproductive, a flat run aft would be preferable. I do think that further innovation in stern design is possible, but I would start with something more akin to a swallow-tail or twin side-by-side transoms, almost opposite of a bulb and hollow.

Thin hulls which are asymmetric athwartships are also prone to similar considerations. Hulls that are flat to port and convex to starboard will create larger waves to starboard.

I realize nothing I am saying here has anything to do with hovercraft pressure distributions, so I am ready to concede for the time being that these distributions give the same resistance forward and aft.

You may be using the term "source" in a somewhat different way than has developed in hydrodynamics over the last century.

Reflection of waves by the hull does matter, and a proper analytical model will take it into account

Perhaps you could provide some sketches of your ideas?

[Leo Lazauskas]

Originally Posted by Richard Pitblad

Here's what I think is going on here. Without constraints on beam or consideration of viscous resistance, there is no optimum at all. The beamier the better, bow and stern alike, with perfect cancellation in the limit as beam approaches infinity, with the "free" surface becoming two-dimensional (so long as sources and sinks are one wavelength apart fore-and aft).

I'm not sure if I have already mentioned Ward's Optimum Symmetric Ship.
These are the result of analytic minimisations of wave resistance within the
confines of thin-ship theory.

See the post and attachments:
theoretical displacement hull shape for min drag

The strange bumps and hollows are clearly not practical, but the measured
wave resistance was very low at their design speeds, as predicted.

Another "zero wave drag" assembly of theoretical interest is Krein's Caravans.
These can be shown to have zero wave resistance, but they have an infinite
number of hulls joined bow-to-stern. The individual hulls are not fore-aft
symmetric, but the assembly as a whole is symmetric. The individual hulls
have cusped waterplanes.

Again, they are clearly impractical, but there are insights to be gleaned from
their characteristics. For example, it is interesting to examine the free-wave
spectrum of the caravan with different (finite) numbers of sub-hulls, and to
see at which wave angles the most energy is being shed, and at which angles
it is very low.

[Richard Pitblad]

Originally Posted by DCockey

Reflection of waves by the hull does matter, and a proper analytical model will take it into account

Perhaps you could provide some sketches of your ideas?

The Calypso II hull design happens to incorporate many of the elements I am pondering (see the video in http://www.cousteau.org/about-us/calypso2). It could be a practical ship in either direction, with appropriate fairings to address form drag. In deep, flat water I would put my money on less resistance in the original design direction at hull speed. A math or cfd analysis that models the various sides of the ship as reflecting barriers is beyond my own resources until I retire from my day job.

[Leo Lazauskas]

Those who are interested in theoretical zero wave-drag vessels can experiment with Krein's caravans in Michlet. See the manual.

The first 4 vessels in the sequence are attached.
The 1st has no "satellite" hulls. It is fore-aft symmetric with cusped waterplanes.
The 2nd has one satellite hull at each end. The satellites have cusped waterplanes, but they are not fore-aft symmetric.
The 3rd has two satellites at each end.
Repeat unto infinity

[Richard Pitblad]

Originally Posted by DCockey

Reflection of waves by the hull does matter, and a proper analytical model will take it into account

Perhaps you could provide some sketches of your ideas?

Another example: An arrow trimaran with laterally asymmetric amas can have very different wave resistance going forward and backward, when reflection and diffraction are accounted for.

First a building block: Consider a single narrow hull with one flat side and one convex side. Say the curved side is to port. Then the bow wake will be larger on the port side due to reflection of the hull sides. Aft of the stern, any residual wake will cease to be reflected and will diffract to starboard.

Moving on, now consider an arrow trimaran with half of its displacement in the center hull. Assume the thin center hull's length is one quarter of a wavelength at design speed and the amas are each the same length as the cetner hull. Assume each thin ama is flat side out, with a longitudinal distribution of volume proportional to the central hull.

As far as positioning, assume the bows of the amas are one half wavelength behind the bow of the center hull (e.g. out of phase), and placed laterally so the bows form a Kelvin angle.

Except to the extent that waves can transmit through the sides of the amas, the amas will absorb most of the energy of the wake created by the center hull. (Any "leakage" of wave energy under the amas can be countered by designing some curvature into the outsides of the amas.)

If the trimaran were run backwards, the resistance would increase to the extent that diffraction at the sterns of the amas allow the wave energy to spread out, rather than propagating solely in the direction of the central hull for cancellation. (This would be exacerbated in the case that reflection is not 100% and the amas had some outward curvature.)

[Ad Hoc]

Richard

You should read, if you haven't already,

"Fundamental Study on Optimum Position of Outriggers of Trimaran from View Point of Wave Making Resistance" by Suzuki K., Ikehata M, Fast '93.

They have performed experiments doing approximately half of what you are proposing above. It is an interesting read.

[DCockey]

Originally Posted by Richard Pitblad

Another example: An arrow trimaran with laterally asymmetric amas can have very different wave resistance going forward and backward, when reflection and diffraction are accounted for.

First a building block: Consider a single narrow hull with one flat side and one convex side. Say the curved side is to port. Then the bow wake will be larger on the port side due to reflection of the hull sides. Aft of the stern, any residual wake will cease to be reflected and will diffract to starboard.

Are you considering the waves generated by a hull as a single bow wave which is reflected as it travels along the hull and a single stern wave?

Moving on, now consider an arrow trimaran with half of its displacement in the center hull. Assume the thin center hull's length is one quarter of a wavelength at design speed and the amas are each the same length as the cetner hull. Assume each thin ama is flat side out, with a longitudinal distribution of volume proportional to the central hull.

As far as positioning, assume the bows of the amas are one half wavelength behind the bow of the center hull (e.g. out of phase), and placed laterally so the bows form a Kelvin angle.

Except to the extent that waves can transmit through the sides of the amas, the amas will absorb most of the energy of the wake created by the center hull. (Any "leakage" of wave energy under the amas can be countered by designing some curvature into the outsides of the amas.)

Why wouldn't the amas reflect the wave energy rather than absorbing it?

If the trimaran were run backwards, the resistance would increase to the extent that diffraction at the sterns of the amas allow the wave energy to spread out, rather than propagating solely in the direction of the central hull for cancellation. (This would be exacerbated in the case that reflection is not 100% and the amas had some outward curvature.)

What is your intent? Describe a configuration which does not have the same wave resistance when direction of motion is reversed?

[Richard Pitblad]

Originally Posted by Ad Hoc

Richard

You should read, if you haven't already,

"Fundamental Study on Optimum Position of Outriggers of Trimaran from View Point of Wave Making Resistance" by Suzuki K., Ikehata M, Fast '93.

They have performed experiments doing approximately half of what you are proposing above. It is an interesting read.

Thanks for the reference. I had some difficulty accessing the article you cited, but Iwas able to read some of the authors' follow-up work. It was nice to see that experimental results validate wave reflection off the outriggers.

[Richard Pitblad]

Originally Posted by DCockey

Are you considering the waves generated by a hull as a single bow wave which is reflected as it travels along the hull and a single stern wave?

Not really, I had in mind a more gradual distribution of sources and sinks.

Originally Posted by DCockey

Why wouldn't the amas reflect the wave energy rather than absorbing it?

The reflection of the wave generated by the central hull and the wave generated by the sources and sinks of the amas are 180 degrees out of phase and cancel each other, resulting in the net effect of absorption. That's why I put half the displacement in the amas.

Originally Posted by DCockey

What is your intent? Describe a configuration which does not have the same wave resistance when direction of motion is reversed?

Yes, this thread is about the optimality of longitudinal symmetry. A necessary (but not in my mind sufficient) part of the argument is that wave resistance is the same forward and backward. I hope I have established that reflection of waves between components of hulls invalidates the entire argument. Only in special circumstances will hull designs exhibit the same resistance going forwards and backwards, and longitudinally asymmetric hull forms and configurations can provide superior results in minimizing wave resistance.

[Leo Lazauskas]

Originally Posted by Richard Pitblad

Yes, this thread is about the optimality of longitudinal symmetry. A necessary (but not in my mind sufficient) part of the argument is that wave resistance is the same forward and backward. I hope I have established that reflection of waves between components of hulls invalidates the entire argument. Only in special circumstances will hull designs exhibit the same resistance going forwards and backwards, and longitudinally asymmetric hull forms and configurations can provide superior results in minimizing wave resistance.

Essentially, all you have done is to invalidate a 1st order theory by
introducing 2nd order effects and using different boundary conditions. The
1st order conclusions still hold. If you now introduce viscosity, you will find
that reflection might not be all that important because short wavelength
waves won't reach the other hulls, or they will be so small as to have an
insignificant effect on the wave resistance after reflection.

"All models are wrong. Some are useful." - George Box.
To which I would add, simple models are much faster and often give
the same results within experimental uncertainty as complicated models in
appropriate circumstances.

[Richard Pitblad]

Originally Posted by Leo Lazauskas

Essentially, all you have done is to invalidate a 1st order theory by
introducing 2nd order effects and using different boundary conditions. The
1st order conclusions still hold. If you now introduce viscosity, you will find
that reflection might not be all that important because short wavelength
waves won't reach the other hulls, or they will be so small as to have an
insignificant effect on the wave resistance after reflection.

In this case, Leo, I consider myself a student of your work, adding a few modifications that I do consider 1st order, not 2nd order. My thinking was particularly inspired by your diamond configuration tetrahull. The viscous effects you are referring to would apply equally to invalidating the wave cancellation for the tetrahull (such a beautiful animation at http://www.cyberiad.net/waketet.htm) . In the absence of viscous effects I have nothing to refute your results for a strutless SWATH or a pure pressure distribution. However, if we are talking about the surface penetrating hulls we are used to considering, the wake patterns would not match your model. There would be wakes generated on the outsides of the outside hulls (due to less than complete transmission of the out-of phase bow hull wavetrain through/under the outside hulls), and less net wake generated on the insides of the outside hulls (due to partial reflection of the bow hull wavetrain), and therefore less than complete cancellation at the sternmost hull.

What I have done is transform these wrinkles into an advantage, that allows the same degree of cancellation by eliminating the sternmost hull and adding curvature to the outside hulls (the degree of inside or outside curvature would depend on the ratio of wave transmission under the outside hulls to wave reflection off the sides, which in turn depends on draft, keels, etc.). Cancellation can therefore occur with less hulls, less complexity, and within a total length less than one wavelength.

Originally Posted by Leo Lazauskas

"All models are wrong. Some are useful." - George Box.
To which I would add, simple models are much faster and often give
the same results within experimental uncertainty as complicated models in
appropriate circumstances.

I agree. For this reason, I would avoid complicating the analysis with interactions between viscous effects and wavemaking, unless data from experiments and sea trials clearly refute the simpler predictions in ways that such interactions help explain.

[Leo Lazauskas]

Originally Posted by Richard Pitblad

In this case, Leo, I consider myself a student of your work, adding a few modifications that I do consider 1st order, not 2nd order. My thinking was particularly inspired by your diamond configuration tetrahull. The viscous effects you are referring to would apply equally to invalidating the wave cancellation for the tetrahull (such a beautiful animation at http://www.cyberiad.net/waketet.htm) .

That's probably a bad example, Richard
The demihulls are unusual: SWATH-like in cross-section and trapezoidal in sideview.
That means they individually shed little energy for large wave propagation
angles (theta). The diamond arrangement cancels waves at lower theta
leaving little at higher thetas (because there wasn't much there in the first
place).

Originally Posted by Richard Pitblad

What I have done is transform these wrinkles into an advantage, that allows the same degree of cancellation by eliminating the sternmost hull and adding curvature to the outside hulls (the degree of inside or outside curvature would depend on the ratio of wave transmission under the outside hulls to wave reflection off the sides, which in turn depends on draft, keels, etc.).

I'd have to see experimental data to judge whether your configuration is
better than a tri with symmetric demihulls, and if so, over what range of
Froude numbers. Cambered hulls can create (wave-making) vortices, so
the wave resistance might be similar to using uncambered hulls.