Hull Asymmetry and Minimum Wave Drag

A linearized analysis of the flow past a hull is possible using the linearized free-surface boundary condition along with the zero normal velocity condition at the hull surface, applied over the wetted area of the hull surface below the static waterline. This flow can be formulated as an integral equation over the hull wetted area which in general cannot be solved directly. My somewhat hazy recollection is this linearized problem will result in the same wave resistance for inviscid flow if the flow direction is reversed even if the hull shape is not symmetric fore/aft.

If the hull shape is sufficiently "smooth" the result for the linearized problem described above can also be be generated by a distribution of singularities over the centerplane profile of the hull which should have the same form as Mitchell's thin ship integral but the singularity strength will not be exactly the same as in Mitchell's integral.

Another way to look at this is Mitchell's integral will provide a flow the zero normal flow condition satisfied exactly at same suface away from the center plane. This surface with zero normal velocity will be similar in shape to, but not exactly the same, as the shape which was used to generate the singularity strength in Mitchell's integral.

 

Are these statements based on experiments or calculations? If experiments how was the wave resistance obtained: total resistance - calculated residual resistance or by wave measurements? If calculations what type of calculations?

 

I should emphasize that both Mitchell's thin ship theory and the linearized problem I described in post #136 have two assumptions:
1) Trim is set a-priori and does not change with the flow.
2) Only the hull surface below the nominal flat waterplane is used.

 

To prove that the resistance is the same forwards and backwards it's probably just as easy to do an analysis with a source-sink pair of Havelock sources.

BTW, it's pronounced Mitchell, but spelled Michell.

 

I see. The logic of the argument would be that every hull can be decomposed into matched pairs of sources and sinks, and the completion of the proof would be by assuming that superposition can be applied. But I am not convinced of the general validity of superposition. Reflection, diffraction, and/or absorption effects can make the wake pattern radically different from that predicted from using superpostion, and different in a way that effects total energy. We already know this from studying the difference between flat-side-in catamarans vs. flat-side-out catamarans, where in each case the demihulls are thin (partial reflection toward the convex side of each demihull).

I do agree that superposition is reasonable for thin ships that are symmetric athwartships.

But perhaps you were thinking of a proof which did not rely on superposition?

 

Why isnt more CFD analysis used in boat design? It is by far the most accurate tool available in the modern era we are in today... as great as michells thin ship theory was for its time, its now quite dated and it seems rather antiquated trying to make predictions and/or comparisons of predicted wave patterns using it? Why not use the most advanced/modern/accurate tools and have a discussion as to whether assymetry makes a difference?

And to what end are we having this discussion? surely their are many more important considerations in designing a hull besides whether the wave pattern is the same forwards or backwards? - the end would seem, is trying to simply prove a theory... a theory which is now possibly antiquated... i havnt heard any discussion of the many other CFD theorems which could be applied to this problem?

 

Superposition is not possible in non-linear theories. It is only applicable when
waves are small, i.e. when the free surface can be linearised.
Essentially, if waves are small, their slopes are small, and then sine(slope)
is approximately equal to the slope (i.e. sin(a) ~ a).

The asymmetric demihulls you mentioned require vortices (or dipoles) as well
as sources and sinks. If hulls are thin so that the waves they create are
small (by sources and/or vortices), then superposition is allowable because
the free-surface boundary condition can be linearised.

Tuck wrote a small paper about the effect of demihull asymmetry using
his maths and results from my programs, and found that for small camber
(and a few reasonable(?) assumptions concerning induced drag) that
asymmetry can reduce the total wave resistance, but only when the demihull
spacing is less than "optimal".

By "optimal" we mean in the sense outlined in the second attached paper.

Of course, if theere are any non-linear effects (e.g. steep waves,
wave-breaking, etc) then superposition cannot be justified, unless the
violations occur for small regions of the flow field when they can be ignored,
or accounted for in some semi-empirical way.

 

Superposition works to the extent that the equations and boundary conditions are linear. Linear is not restricted to constant coefficients in the equations nor to straight and flat boundaries. The governing equations for inviscid, incompressible flow are linear, and with the assumption of irrotational flow become simply Laplace's equation. Similarly the boundary conditions for solid surfaces, such as a hull surface or bottom, of zero (or a perscribed) normal velocity are linear.

The exact forms of the free-surface boundary conditions, location dependent on the solution with normal velocity of the water equal to normal velocity of the surface, and constant (atmospheric) pressure at the surface, are non-linear. There are linearized free surface boundary conditions, and these have been used in various methods including thin ship theory.

Wave reflection and diffraction occur with linear solutions using the linearized free surface boundary conditions, whether the solutions are achieved using "superpostion" or some other method.

Also note that thin ship theory is a special case of linear wave theory, and the hull slopes being small and applying the hull boundary condition on the surface are not required for linear wave theory.

 

The free surface and resulting waves considerably complicate the problem, including grid generation, and significantly increases the computer resources needed. It would be great for someone with access to a suitable code and resources to study this question using a more sophisticated analysis methodology.

But sometimes fundamentals are best discovered by looking at a simplier problem.

Reason for this discussion is not to guide design per se. Rather it's to obtain some insight into one aspect of how hull shape affects wave resistance.

 

I agree, groper, but it depends on what is being analysed.
CFD can produce excellent results (if experiments are available so that
the code can be tweaked).
http://www.boatdesign.net/forums/design-software/flotilla-hovercraft-hydrodynamics-36344.html

CFD produces very good results for these cases. So can linear theory, but
4 million times faster.

Leo.

 

In many cases, too, some CFD codes resorts to using linear theory in the
far field. Incorrect formulations of viscous effects in CFD over-dampen
waves to the extent that waves are almost zero, or the computer
resources necessary to calculate waves far from a ship are prohibitive.

Gopher, some CFD codes can take days on multiple processors to calculate
the resistance (or waves) at one speed. Imagine how long it would take to
estimate the wave patterns of fleets of different ships, including all the
interactions. There are many scenarios where CFD won't finish before the
heat death of the universe

And who told you that CFD is the most accurate method available?
A CFD salesman?


As David said, sometimes we are looking for insights into the behaviour of
ships and other hydrodynamic scenarios. These can be gleaned from simple
theories, but not so readily from codes that try to include many different
effects such as non-linearity, viscosity, turbulence, splash and spray,
unsteadiness, etc. They can mask the underlying features of interest.

 

I pulled the above quote from another thread...

So why is it, that an assymetric shape is modelled with the least resistance in Michlet using thin ship theory, was better than anything godzilla could come up with using symmetric shapes with equal displacement...?

 

I'd have to know a lot more about the parameters used in Godzilla and the
straight Michlet file. Users sometimes forget that boundary layers are
included, or they don't use sufficient sections and waterlines, or they use
too few theta intervals, or they run the program only once, etc.
If you could give me the two files I could pin it down.

Remember, too, Godzilla is a "stochastic" search routine. It is not
guaranteed to find an optimum in all situations, as I mention in the manuals.

In some problems (in maths, as well as in real naval architecture) an optimal
solution might lie in a very wide. narrow basin. A short hull with semi-circular
cross-sections might be almost as good as a slightly longer hull with
different cross-sections. If the difference in total resistance is tiny, then
it might be wise to look to other factors to act as a tie-breaker.

As I wrote in the (first and subsequent) Godzilla manuals:
...carbon-based Naval Architects are probably quite good at this sort of thing!

 

So it wouldnt be because the front section of the hull created a wave that the rear section of hull served as a sink for? - which was the intended purpose of the designed shape...

 

I'm not sure what you mean by that. There is no single "wave" made by a hull.

In thin-ship theory, there is a spectrum of waves from "pure" transverse to
extreme diverging. Minimising wave resistance within the confines of the
theory means minimising over the entire spectrum. At some Froude numbers
that will require reducing waves transverse waves more than the diverging
end of the spectrum. At high Fr, diverging waves are more important.