Hull Asymmetry and Minimum Wave Drag

Valid for a monohull vessel which is symmetric side-to-side. Very analogous to thin airfoil theory for symmetric, non-lifting wings as pointed out by Newman. But Michell's simplification for source strength doesn't satisfy the boundary conditions for multi-hull vessels, or a monohull which is not symmetric side-to-side (such as a sailboat heeled). In those cases if the longitudinal slope of the hulls are sufficiently small, then using the same logic as Michelleach hull can be modeled with a distribution of sources and doublets (or array of vortices in place of the doublets) on its centerplane, with the strengths of the singularities determined by the solution of the integral equations obtained by applying the boundary conditions. Again, this is directly analogous to thin wing theory applied to multiple wings and/or lifting wings with thickness.

 

Not suggesting "cherry-picking" for agreement. Rather noting what the 2nd-order effects are which might be the cause of the apparent discrepancy between the results from thin ship theory of lowest resistance shape being symmetric fore/aft, and the assymmetry fore/aft is generally shown as design guidance. Fundamentally different.

 

It was not a criticism of you, David.

There are other theories in which the boundary condition on the hull is
satisfied exactly, but the free-surface is still linearised. Some claim that
those (inconsistent) "Neumann-Kelvin" methods are better for stubbier
hulls.

I haven't looked at them for 20 years, but the conclusions of Doctors and
Beck were fairly pessimistic. It wouldn't surprise me if recent work has
improved the situation and clarified the role of a troublesome "waterline
integral" that must also be introduced.

 

Nicely put! I have no fundamental disagreement with that.

 

Without a free surface applying the boundary conditions at the surface location is "exact" for incompressible, inviscid flow, even for bluff bodies. For real incompressible flow without a free surface it provides a very good solution as long as the flow doesn't separate, and can also alse be used to model the attached portion of flows if the separated regions are suitably modeled. But with a free surface then I can understand that if the "hull" is sufficently bluff that the small longitudinal slope approximations are not satisfied then at least in the near field the waves may be large enough that the assumptions needed for the linearized free surface boundary condition might not be satisifed.

Where does the waterline integral come from in the "thick" ship with linearized free surface solution? Does it depend on the formulation of the integral equations? My preference for potential flow is to use a Green's function singularity forumulation which requires that the singularities satisfy the adjoint field equation and adjoint boundary conditions. For incompressible flow Laplace's equation is self-adjoint but I'm not sure about the linearized free surface boundary condition.

 

I'm assuming that what you described are some variant of Hess-type panel methods (without free-surface).

The troublesome waterline integral (from what I remember) arises from the
linearization of the free surface in the Neumann-Kelvin problem. There have
been several methods to try to work around it, or to calculate it accurately.
See, for an early example, Noblesse's paper:
http://www.iwwwfb.org/Abstracts/iwwwfb04/iwwwfb04_36.pdf

As I said, it's not something I have much experience with. I'm happy to
work with things that are simple, consistent and fast to evaluate.

 

Not really. It's a method of formulating the integral equations which can then be discretized and solved by panel methods, etc.

The Green's function is the solution of the adjoint of the governing equation is set equal to the Dirac delta funcition, and the adjoint boundary conditions. A convolution integral of the governing equation and it's adjoint is set equal to the value of the velocity potential in the field. Integration by parts is used to go from the integral over the entire domain to a boundary integral. Then the appropriate limits of the velocity potential and derivatives are taken as the boundary is approached to obtain the velocity potential and its derivatives, ie velocity components, as a boundary integral of the velocity potential and it's derivatives. The singularities are the Green's function and derivatives. Advantages of this method are that mixed boundary conditions are treated in a straight forward and "exact" manner, and the singularity strengths are the velocity potential and velocity components. I developed it as a straightforward extension of the methodology Mike Greenberg's Application of Green's Functions in Science and Engineering and wouldn't claim it to be original.

I'm confident that Michell's thin ship integral could be derived using this method in a very straightforward manner.

I need to find time to study the brief paper by Noblesse you referenced and try to understand the formulation he uses. Any suggestions of other references?

 

Am I reading this correctly....being an (AssocRINA) has really made you into a true thinking NA.....trends, quick...easy...wont be long before you're a full MRINA

 

I get all the benefits of being a crypto-NA, without the mountains of legal
mumbo-jumbo, and the responsibility for other people's lives.

And what else could I join?
I had ideas of joining my local Senior Citizens Club with a view to mounting
a youthful take-over and maybe winning some money hustling snooker and
billiards, but they raised the joining age to 60 a week before my 55th
birthday.

 

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).

 

How would you vary the beam but not the length with sources and sinks one wavelength apart - a transverse row of sources near the bow and a transverse row of sinks near the stern?

Increasing beam better if displacement held constant or ???

Are your statements based on a heuristic understanding of hydrodynamics, analysis, experiments or what??

 

Yes

Yes, that's what I had in mind. But I suspect it would also be true if displacement/beam approached zero.

No experiments. Heuristics and analysis have different meanings to different people. I am guessing that you would consider my reasoning to be more on the heuristic end of the spectrum. I consider the "result" of the optimality of symmetry to be paradoxical enough to require an exploration of the assumptions that lead to it, with a bias toward seeing where the assumptions break down rather than to understanding and accepting the validity of the result. It is more efficient initially to probe with heuristic methods rather than to get bogged down with analytical formulations.

 

Very wise words, yet so rarely done these days.

If the philosophy and logic is not sound, makes no difference how many equations one can produce, all of which is suspect to begin with. Just becomes my dog is bigger than your dog debate...hardly progressive and instructive

 

Good approach to begin with. It can also be productive to seek what such a result means or doesn't mean, not just seek reasons to disregard it.

In thinking about sinks and sources and the waves they generate, you might want to look into how a single Havelock source/sink produces the Kelvin wave pattern seen behind any vessel moving on a free surface with deep water. Related is the way deep water free surface waves are dispersive, compared to 3D sound waves and electromagnetic waves (including light) which are not dispersive.

 

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.