# Calculating friction resistance, drag coefficient?

Discussion in 'Hydrodynamics and Aerodynamics' started by dustman, Jul 11, 2019.

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### DolfimanSenior Member

I have continued the exploitation of the Univ. of Southampton models tests to address the upright catamaran issue (as tested with the 2 hulls indentical the water and spaced S between the hulls axis), then to propose an empirical formulation to also address the sailing catamaran issue (i.e. taking into account the differentiation between the leeward and the windward hull) and finally, to check as to demonstrate the process, I did some numerical examples with giving the step by step computation : for a 16m cruising catamaran (L/B ~ 9,7 ) and 3 speed & heel cases, and for a 9m racing catamaran (L/B ~ 22,7 ) and 1 speed & heel case. Main results concerning the residuary drag / friction drag proportion are :
Case 1 : 16m cruising Cata, with speed 7 Knots (Froude ~0,30) and heel angle of 2° >>> 60% residuary / 40% friction
Case 2 : 16m cruising Cata, with speed 12 Knots (Froude ~0,50) and heel angle of 4° >>> 69% residuary / 31% friction
Case 3 : 16m cruising Cata, with speed 18 Knots (Froude ~0,75) and heel angle of 7° >>> 62% residuary / 38% friction
Case 4 : 9m racing Cata, with speed 12 Knots (Froude ~0,64) and heel angle of 4,5° (for which the windward hull is above the water) >>> 30% residuary / 70% friction

Limit of this process done from a series of 10 models sharing the same Cb ~ 0,4, covering Lw/Bw = 7 to 15 and Bw/Tc = 1,5 to 2,5, S/Lw = 0,3 to 0,5 :
>>> for racing catas, we are usually over Lw/Bw = 15 by far, for the example above I took by default the Lw/Bw = 15 drag value for the estimation to be conservative a priori
>>> when the windward hull is at low displacement in the water, its Bw/Tc tends to be over 2,5 : again, I took the 2,5 value when it occurs, and again a priori conservative
>>> the Cb of modern shape hulls are more in the 0,42 to 0,48 range, instead of 0,4 for the models
>>> and of course, no tests results of heeled catas for the residuary drag of each hull, I tinkered a formulation paying attention to the continuity when the cata is upright or when the windward hull is over the water.

Next step is to plug all that in a VPP, but not easy to translate the set of figures in relevant formulations …, with a lof of "IF" due to the above limits.

If a University with a towing tank facility envisage a new serie of model tests (and to publish the results ), I would recommend :
- models sharing same Cb = 0,48, Lw/Bw from 9 to 22, Bw/Tc from 1,5 to 3,0, with more elliptical/U shape for the sections
- tests in monohull mode, in cata with S/Lw from 0,3 to 0,5, with various heel angles
- tests with Froude up to 1,1 (based on Lw of the leeward hull)

#### Attached Files:

• ###### Residuary drag of sailing catamaran.pdf
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### fastsailingJunior Member

I think you need to do your work again, although mathematically valid due to limited data it is misleading. This time replace this:
with this:
The results you will get are not the same as in the analyses you posted, and the result will be by far better than in the analyses you already did.
That is because greatest effect on residuary drag is caused by Froude number, Lwl and displacement, not Bwl neither Tc.
For example your conclusion in pages 23&24/30 that in case 2 leeward hull (3.34%) has less Dr/Mg as a monohull than windward hull (3.51%) is wrong, the reality is just the opposite. If you do as I suggest, I think that you will see that being the case and that Lwl/D^1/3 is the dominant parameter having more effect on Dr(Mg than Lwl/Bwl.
Also in the real world when designing hull shape for catamaran, lwl and total displacement is fixed, and Bwl and Tc can be varied independently when Cb is also changing. If Cb is fixed, the Bwl and Tc are related, since displacement and lwl is fixed.
In the data you used displacement is variable and differences in both drag and Dr/mg is mostly due to that, not nearly as much due to differing hull shape as you assumed in your analyses causing the error.

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### DolfimanSenior Member

Ok, I did this alternative analysis of the Dr/Mg Monohull by using the length displacement ratio Lw/D^1/3 as reference : effectively, for each triplets of models sharing the same Lw/D^1/3 , that tends to uniformise the Dr/Mg curves, it is like erasing the influences of the 2 others ratios Lw/Bw and Bw/Tc (again, within this set of hulls sharing the same Cb ~ 0,4). So it seems justify to average the curves sharing the same Lw/D^1/3, which lead to a simple synthesis figure (Fig. 1B in the revised document attached) with just Lw/D^1/3 ratio for the graduation of the curves. I named this approach B, the approach A being based on Lw/Bw (Fig. 1A) + a correction for Bw/Tc (Fig. 1B). What can be disturbing is that Lw/Bw seems no longer at stake in the approach B (actually masked), although it is a familiar ratio for the cata designer. The question is : can approach B be considered more reliable than A for an estimation when Cb is not equal to 0.4 (in absence of test results with another Cb value)

I did again the numerical examples with the two approaches A and B for comparison : results for the Dr/Mg Monohull part of the computation are not very different, for the example, for the case 2 you look at :
Leeward hull >>> Dr/Mg monohull (%) is 3,7 with approach A (not 3,34, I did a copy/paste mistake in my previous document), and 3,9 with approach B
Windward hull >>> Dr/Mg monohull (%) is 3,57 with approach A and 3,67 with approach B

To keep in mind that the two hulls of a sailing catamaran, having each a proper displacement D and a proper Lwl, have different Lw/D^1/3 but also are functioning at a different Froude.

Another point not yet discuss, is the transom immersion (I add these data in the document) : yes for the leeward hull, but no for the windward hull, so may be we overestimate a bit the wave drag of the windward hull.

#### Attached Files:

• ###### Residuary drag of sailing catamaran_V2.pdf
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### fastsailingJunior Member

I think not so. When a hull has immersed transom when stationary at some displacement, and dry transom at speed with the same displacement, it will have less wave making resistance than either another hullform with no immersed transom when stationary or another hullform with immersed transom at speed.
Seems to me the result is that you may be underestimating a bit the wave drag of both hulls when heeled, because neither hull has as optimum transom as tank test data did at that point.

As another matter, take a look at:
https://www.oossanen.nl/beheer/wp-c..._of_small_high-speed_displacement_vessels.pdf
page 7/12 has averaged series 64 test data, not the original.
That would be more relevant to the hull shapes you seem to be using.
books.google... has the relevant part in pages 500 and 502 of:
I have access by that method for table A3.13 - A3.15 in Appendix A3 Showing the original data of series 64.

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### fastsailingJunior Member

There are more than one method to convert residual resistance into residual resistance coefficient, not sure which one is used in that table above.
Most likely Residual resistance Rr = Cr*dynamic pressure*reference area
But reference area can generally be either wetted area, or (displacement volume)^(2/3). In both cases reference area is not the same for all models in the table, resulting that higher Cr can mean smaller Rr, if reference area is smaller for the hull model with higher Cr value.

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### DolfimanSenior Member

Many thanks, this paper is very comprehensive and clear about the units used, I picked up results from Series 64 and also some of the SSPA's and NPL's ones when relevant, and just redigitalised them in the same format Dr/Mg (%) in function of Froude Fn with L/D^1/3 as unique parameter, and compared with the Southampton series curves (pages 13 to 15 of my revised paper attached). Shapes of the curves are very similar, the new results leads in average to a bit less residual drag that from Southampton models ones, showing also that when L/D^1/3 is > 11, there is no any hump of the drag curve.
Like you, Oonnasen in his paper advocates the L/D^1/3 as unique significant parameter for Froude 0,4 to 0,9 (providing of course that the L/B ratio is high enough, let 'say 7 at least), quotation : "…. Up to Fn value of 0,9, the result for the 3 B/T values are almost identical, leading to the observation that in the speed range between 0,4 to 0,9, the length displacement ratio is the only significant parameter ''.
So I average/smooth all these results to propose an updated Fig. 1B for the approach B (page 17) which can be helpful for an early stage project (when L/B > 7, Froude 0,15 to 1,0) , i.e. giving a good enough order of magnitude and not misleading within the design convergence process.
To note that the proposed empirical method to estimate the residuary drag of two hulls of different parameters mounted in a catamaran mode (i.e spaced S and sharing the same speed) can be used also for a trimaran design issue. By hoping a University will tackle this specific experimental issue, i.e. adressing the residuary drag measurement of each hull (leeward and windward of a catamaran with heel, or central hull and float of a trimaran) and comparison with the drag of each hull when towed as a slender monohull.

#### Attached Files:

• ###### Residuary drag of sailing catamaran_V3.pdf
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### jehardimanSenior Member

Sounds like there is a PhD paper waiting for some one to pick it up... I know SNAME used to publish a list of investigation topics, I wonder if anyone still does that. If I ever go back for an advanced degree I already have my topic...I just have to "develop" it from scratch.

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### fastsailingJunior Member

rxcomposite and Doug Halsey like this.

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### DolfimanSenior Member

Thanks for this very well documented report. I used NPL results for the models having an L/B of 7,5, which complement the Southampton series (L/B 7 to 15,1) and the Series 64 (L/B 8,45 to 18,3). For lower L/B , we are no longer in a cata or tri hull issue subject of this thread.

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