Michell-Lamb Wave Resistance Integral for Thin Ships

This modification to J.H. Michell's famous wave resistance integral for thin ships arises as a consequence of including surface viscosity in the usual linearised free-surface boundary condition for gravity waves.



In the first Figure, the Michell wave resistance coefficient curve (a=0.0) for a standard Wigley hull has large oscillations for Froude numbers less than 0.35. Introducing a small non-dimensional, speed-independent value of the viscosity parameter (a=0.005) into the integral reduces those oscillations and results in better agreement with Mustafa Insel's PhD experiments.

The actual viscosity parameter used in the integral is:
b = 4*n*g/U^3
where g is gravitational acceleration (in m/sec/sec), U is ship speed (in m/sec), and n is the turbulent eddy kinematic viscosity (in m^2/sec). Note that n is not the same as the usual kinematic viscosity of water.

The speed-independent viscosity parameter used in the Figure is:
a = b * Fr^3,
where Fr is the usual length-based Froude number, Fr = U/sqrt(g*L).
Therefore, a = 4*n/[L*sqrt(g*L)]

If we are free to choose the value of the viscosity parameter at each Froude number, (e.g. in an attempt to account for wave-breaking effects evident between Fr=0.5 and Fr=0.65), then we could get almost perfect agreement with the experiments.



The second Figure shows wave resistance coefficients for a Wigley catamaran (with demihull separation equal to half the hull length) for a wide range of the viscosity parameter "a".

An interesting exercise is to find the variation of the viscosity parameter as a function of Froude number that gives the best agreement with Insel's experiments (as I did to find the blue curve in the first Figure).

Some students emailed me asking for a small program to calculate the Michell-Lamb integral for multihulls. I tried to upload a 1.35Mb zip file that contained an executable to do that, but boatdesign.net sat there for a long time and did nothing

I'll try again tomorrow.
Leo.

 

The "surface viscosity" looks like the coefficient in a damping term which has been added to the integral, which has been given a particular physical interpertation. Is that correct?

 

It is more than the simple addition of a term inside Michell's integral.

The free-surface boundary condition has been modified to



When the eddy kinematic viscosity is equal to zero this is the usual Kelvin free-surface boundary condition. With non-zero viscosity it corresponds to the damped equation for plane waves derived by Lamb.

How we intrepret its effects in a wave resistance context is an interesting question.

For example, a large value of the surface viscosity for 0.5 < Fr < 0.65 gives good agreement with Insel's experiments for the monohull. He noted that there was quite a lot of wave-breaking for this range of Froude numbers. In breaking, these normally high waves would have lost height before reaching the wave probes. The effects of breaking might also have increased the turbulence near the surface. That's one (rough-as-guts) way of interpreting the large eddy viscosity.

Whether it is helpful in estimating form factors, or in extrapolating from model scale to full-size, remains to be seen.

Perhaps some naval architects will chime in

Cheers,
Leo.

 

Thanks Leo. It's been over thirty years since I did anything with wave theory and almost that long since I looked at Lamb. But I'm still somewhat skeptical of terms added to the governing equations with coefficients which need to be chosen based on the particular set of conditions or configuration. On the other hand all fluid mechanics requires some amount of hand waving in terms the length scales needed to treat the fluid as a continuim.

 

That's a very healthy scepticism, Mr DCockey!

I'm just trying to provoke a bit of discussion here, and also to see what people think is an acceptable number of coefficients.

For example, in their model of wave resistance for hulls with transom sterns, Molland, Wellicome and Couser vary the length of a fictitious hollow behind the stern in order to get better agreement with experiments. If it is acceptable to choose a different length at each Froude number, is it acceptable for me to vary the surface viscosity at each speed?

CFD often uses a variety of empirical constants to get good agreement with experiments. Sometimes different meshes are also used at different speeds. If they choose several "constants", how many can I use so we have a fair comparison of the capabilities of each model? E.g. it would seem unfair for CFD to use 5 constants, and for me to be restricted to none.

All the best,
Leo.

 

There is a difference between modifying a model to improve agreement with experimental data by introducing additional terms, and assigning particular physical meaning to those terms. A model is not invalidated just because it requires some adjustment; all models are approximations to some degree.

On the other hand the need for additional damping may come from the energy which is absorbed by the wave breaking rather than "eddy viscosity". Perhaps someone should look at the stability of the generated waves and try to correlate the stability to the amount of damping which is needed. Dig deep enough from several starting points and hopefully they will converge.

 

Leo,
It seems clear to me that most everyone prefers a formula (model) which relates identifiable physical parameters and/or properties. We prefer the model to give insight to the physical relationships and functionality.

However, once we have reached the limits of our understanding and ability to model the physical phenomena, our next priority turns to characterizing the phenomena. Once you cross that line between modeling the physical to computationally characterizing, those seeking a model that gives understanding will never be satisfied. The question is then, can we accurately predict the phenomena with a computationally convenient method. Computational convenience is perhaps a mater of usability from the hull designer’s perspective. The number of constants or coefficients may not be significant at all, if the user does not have to enter them in by hand, and the user has confidence in your black box.

(I do not imply that you are presenting at black box at all. You are doing the opposite and it is appreciated.)

 

I agree completely.

The eddy viscosity in the ocean can vary by about 6 orders of magnitude (i.e. from 10^(-5) m^2/sec to 10.0 m^2/sec) so there's plenty of room there. I haven't found any references to how much it varies in towing tanks.

Another complication with Insel's experiments is that his wave probes could not measure wave heights less than about 3mm and that would lead to an under-estimate of the "real" wave resistance. Choosing a surface viscosity to make the mathematical model agree better with those experiments would seem like very dubious practice IMO. But then the experiments are of dubious value too.

Cheers,
Leo.

 

Yes, sometimes it is a matter of how far we can push a particular model before we are on shaky ground (or water).

For example, the attached Figure shows the integrand of the wave resistance integral for three values of the viscosity parameter I described earlier.

The curve for a=0.0 is the integrand of Michell's integral, i.e. without the effect of viscosity.

Waves propagating at theta=0.0 are pure transverse waves, i.e. those travelling perpendicular to the ship's track; waves at theta=90 degrees are those travelling outwards and exactly parallel to the track.

Integrating the curves gives the wave resistance. (Actually half the wave resistance because the curve is symmetric about theta=0).

There are many wiggles in the curve, particularly near theta=90 degrees. In a real flow, viscosity should damp out those high-frequency waves and they will never reach wave probes located abreast of the ship. Including surface viscosity in the model seems to have the desired effect. But is a=0.001 too much or too little? Unfortunately I don't have enough high-fidelity experimental data to decide that yet.

Including viscosity also has some computational benefits you alluded to - you don't need to use as many theta-nodes in the integral to capture all those wiggles near theta=90.

Leo.

 

The attached file "mlwri_v10.zip" contains a short demo program to calculate Michell's Integral and the Michell-Lamb Integral for Wigley monohulls and catamarans.

To install, unzip the file into a subdirectory.

Brief instructions are given in the file "michell_lamb_wigley.txt"

To uninstall, just delete the files in the subdirectory.

Examples are available in four very short batch files.
If calculations are taking too long on a slow computer, press Ctrl-C to terminate the program.

Leo.