Working out hull drag

jdc

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I've trying to work out the drag on my boat while driving it through the water. Doubtless standard stuff for naval architects armed with 3D CAD programs and tank models like there are at the Wolfson unit in Southampton, but the data seems never to be released by boat builders or designers.

I think I now have some rather simple approximations which nonetheless give quite good results. Any naval architects out there who can comment?

My primary purpose was to work out what my fuel consumption would be, but for added spice I've contrived to subvert it into an anchoring related topic as well! Again, I'd actually be interested in actual data confirming or contradicting the results.


Underlying Theory

The drag is due to both friction, aka viscous forces, and to wave generating resistance. After some reading of papers (in particular a shamefully treated Australian called Mitchell seems to have done the fundamental work in this area), and trying to digest "Practical Ship Hydrodynamics" by Volker Bertram, I think that these are reasonable approximations:

drag_eqn1.PNG


drag_eqn2.PNG


The formula and parameters in the wave drag equation are an empirical fit I made to the Mitchell triple integral, see figure 2 from this paper: http://journals.cambridge.org/downl...29a.pdf&code=3b0d45f2a0482099a22b52fd9ff5084a . The viscous drag is a standard result referenced all over the place.

For my boat, 42' monohull with longish keel and rather traditional shape, they give drag like this:

range_under_pwr_1.png


or, normalising to just the coefficients, which are likely applicable to other people's monohulls.

range_under_pwr_5.png




Actually I found the forces quite surprisingly low - so has anyone got measured data, eg from towing real boats?

What certainly stands out is why I can rather quickly get to 7 kts or so under sail, but getting above 9 takes some doing.

Range Under Power

Now we have drag related to speed, it can also be related to engine rpm knowing the prop pitch and diameter and making an assumption about efficiency. I chose 60% at full revs, which is quite low but I think my Gori 3 blade is not terribly efficient.

I couldn't find any useful equations relating thrust to shaft rpm in the literature: plenty for airscrews but not for water. So after some thought I tentatively used:

R = alpha * (beta * rpm - U);

R is the thrust,
U is the speed through the water
beta is related to pitch, and
alpha is related to diameter and water density.

This seems intuitively correct, but anyone know a more sophisticated one?


The result is plotted here. The red 'x's are values of speed vs rpm I measured last weekend, and the blue line is what the model predicts; I was rather pleased with the fit.

range_under_pwr_2.png


Also plotted is engine fuel consumption in miles per litre and thus range given a 300 litre tank. If true, this really does say that generally we motor too fast - up 'til now I've done between 1800 and 2000 rpm, but it's clear that 1200 - 1400 would be much more economical.

To convert from thrust to shaft HP I again had to make up the relationship, proposing
HP = gamma * rpm * ((beta * rpm - U) + epsilon);

gamma is a simple scaling factor I should be able to work out exactly, and is not worrying
epsilon is there to account for the losses of the real propeller, chosen to make it 60% efficient at full power.

To try to get some extra validation I also plotted HP and the calculated fuel consumption in litres per hour. The red 'x's are the values given by the Beta 50HP data sheet:

range_under_pwr_3.png


The fit is quite good; certainly the shape is spot-on, and if I find the data sheet a bit optimistic that's in accordance with my observation.

Subverting the thread to Anchoring

Since we already know how to relate horizontal force on a boat to wind speed, and we now can relate drag on the hull to speed through the water, we can calculate the effective increase in wind speed that a tide will give. This is plotted below. No dependency on anchor type or scope etc, so I hope not controversial in principle.

range_under_pwr_4.png


I was surprised at how small an effect the tide has if aligned with the wind. My conclusion is that the undoubted effect of tide when at anchor is much more to do with the sheering, and thus dynamic forces it creates, than due to a simple increase in static load. Not exactly an earth-shattering conclusion.

The last thing I bothered plotting was the bollard pull I'd get, when motoring in reverse. This is less than the forward pull due to the different gear ratio, but in my case of a Gori not to do with propeller shape since it's symmetrical between forward and reverse.

range_under_pwr_6.png


It should tell me how many revs to apply when making sure my anchor is well dug in.

Working this out amused me for a few mins on a Friday lunchtime anyway. Have a good weekend.
 

Laminar Flow

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I've trying to work out the drag on my boat while driving it through the water. Doubtless standard stuff for naval architects armed with 3D CAD programs and tank models like there are at the Wolfson unit in Southampton, but the data seems never to be released by boat builders or designers.

I think I now have some rather simple approximations which nonetheless give quite good results. Any naval architects out there who can comment?

My primary purpose was to work out what my fuel consumption would be, but for added spice I've contrived to subvert it into an anchoring related topic as well! Again, I'd actually be interested in actual data confirming or contradicting the results.


Underlying Theory

The drag is due to both friction, aka viscous forces, and to wave generating resistance. After some reading of papers (in particular a shamefully treated Australian called Mitchell seems to have done the fundamental work in this area), and trying to digest "Practical Ship Hydrodynamics" by Volker Bertram, I think that these are reasonable approximations:

drag_eqn1.PNG


drag_eqn2.PNG


The formula and parameters in the wave drag equation are an empirical fit I made to the Mitchell triple integral, see figure 2 from this paper: http://journals.cambridge.org/download.php?file=/ANZ/ANZ30_04/S0334270000006329a.pdf&code=3b0d45f2a0482099a22b52fd9ff5084a . The viscous drag is a standard result referenced all over the place.

For my boat, 42' monohull with longish keel and rather traditional shape, they give drag like this:

range_under_pwr_1.png


or, normalising to just the coefficients, which are likely applicable to other people's monohulls.

range_under_pwr_5.png




Actually I found the forces quite surprisingly low - so has anyone got measured data, eg from towing real boats?

What certainly stands out is why I can rather quickly get to 7 kts or so under sail, but getting above 9 takes some doing.

Range Under Power

Now we have drag related to speed, it can also be related to engine rpm knowing the prop pitch and diameter and making an assumption about efficiency. I chose 60% at full revs, which is quite low but I think my Gori 3 blade is not terribly efficient.

I couldn't find any useful equations relating thrust to shaft rpm in the literature: plenty for airscrews but not for water. So after some thought I tentatively used:

R = alpha * (beta * rpm - U);

R is the thrust,
U is the speed through the water
beta is related to pitch, and
alpha is related to diameter and water density.

This seems intuitively correct, but anyone know a more sophisticated one?


The result is plotted here. The red 'x's are values of speed vs rpm I measured last weekend, and the blue line is what the model predicts; I was rather pleased with the fit.

range_under_pwr_2.png


Also plotted is engine fuel consumption in miles per litre and thus range given a 300 litre tank. If true, this really does say that generally we motor too fast - up 'til now I've done between 1800 and 2000 rpm, but it's clear that 1200 - 1400 would be much more economical.

To convert from thrust to shaft HP I again had to make up the relationship, proposing
HP = gamma * rpm * ((beta * rpm - U) + epsilon);

gamma is a simple scaling factor I should be able to work out exactly, and is not worrying
epsilon is there to account for the losses of the real propeller, chosen to make it 60% efficient at full power.

To try to get some extra validation I also plotted HP and the calculated fuel consumption in litres per hour. The red 'x's are the values given by the Beta 50HP data sheet:

range_under_pwr_3.png


The fit is quite good; certainly the shape is spot-on, and if I find the data sheet a bit optimistic that's in accordance with my observation.

Subverting the thread to Anchoring

Since we already know how to relate horizontal force on a boat to wind speed, and we now can relate drag on the hull to speed through the water, we can calculate the effective increase in wind speed that a tide will give. This is plotted below. No dependency on anchor type or scope etc, so I hope not controversial in principle.

range_under_pwr_4.png


I was surprised at how small an effect the tide has if aligned with the wind. My conclusion is that the undoubted effect of tide when at anchor is much more to do with the sheering, and thus dynamic forces it creates, than due to a simple increase in static load. Not exactly an earth-shattering conclusion.

The last thing I bothered plotting was the bollard pull I'd get, when motoring in reverse. This is less than the forward pull due to the different gear ratio, but in my case of a Gori not to do with propeller shape since it's symmetrical between forward and reverse.

range_under_pwr_6.png


It should tell me how many revs to apply when making sure my anchor is well dug in.

Working this out amused me for a few mins on a Friday lunchtime anyway. Have a good weekend.

Thank you for your interesting post.
Just like you I was trying to determine relative drag for my boat, in particular to try and find out in how much the fairing of the deadwood and profiling of the rudder have reduced it in comparison to total drag. Thus I was able to determine that the combined modifications, apart from dramatically improving her steering, also let me gain o.6 kts at 6kts of speed and about 0.8 kts at 4kts.

I came up with similar curves to yours (pretty obvious really) and from these I also determined that for our boat the optimum motoring speed would be around 0.9 - 1.00 relative speed (Fn 0.51 - Fn 0.576)

Parametres for determining wave (form) resistance are available in "Principles of Yacht Design" by Larsson/Eliasson.
 
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TernVI

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The big factor that's missing is waves.
Even small waves have a big effect on the power needed.
Also a big effect on the loading of a mooring or anchor rode.
Bethwaite did some actual drag measurements vs speed and displacement for some dinghy hulls.
 

Ian_Edwards

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Interesting, they are very similar to what I get in practice.
Southerly 46, Yanmar 4JH4E, continuous rating 36.8kW (50 hp) @ 2907 rpm ... from the datasheet.
The prop is a three bladed Max prop.
I get about 2L per hour at 1,800 rpm and a clean hull in flat water, a speed of just under 7 knots.
Flat out I get just over 9 knots the fuel consumption increases to about 10L per hour.
The fuel consumption figure are approximate, I don't have an accurate way of measuring it, and the tank also supplies the Eberspacher Hydronic 10 and a 5kVA generator. However, I have reasonable estimates after starting from full and motoring all day, then filling up again.
I normally motor in the 1800 to 2000 rpm range, which gives a speed of 7 knots plus in most conditions, but obviously much slower when pitching into a head sea and wind.
The realistic calculated range under power is around 1,100 nm from a 400L tank.
 

Neeves

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JDC,

Good stuff!

The graphs you produce of Boat speed, consumption and range is data that everyone should produce for their own yacht - they will vary with engine, prop and yacht so you cannot rely on data from another yacht. As mentioned the missing variable is the impact of waves or seas.

We have 2 x MD2020s and at 3,000 revs we can achieve 'about' (( don't recall and the data is on the boat) 6 knots and consume 3l/hr with one engine and not much over 7 knots with 2 engines at 3,000 revs and a consumption of 6l/hr. We usually motor with one engine at around 2,500 revs.

I would caution your graph of engine revs vs equivalent wind speed.

If you set your anchor and reverse up (power set) then your yacht develops momentum, or kinetic energy, which lifts the catenary and tensions the anchor. As the catenary tends to becoming straight your momentum reduces and the kinetic energy is converted to potential energy in the rode. You always have the energy developed by the engine - so the catenary is 'straightened' by the energy developed by the engine and the kinetic energy of the yacht moving backwards. This latter decreases and at some point is insufficient to balance the potential energy stored in the catenary - the yacht at some point stops and then moved forward.

You can repeat this horizontal yo-yoing.

The setting of your anchor demands time at tension, to allow the seabed to shear and allow the anchor to move 'forward' and 'dive' more deeply. That yo-yoing effect does not allow much time for the anchor to dive - so you need to keep the tension on the rode for longer than the graphs implies.

The maximum tension on the rode develops during yawing, or sheering, and that maximum is much higher than the average. The tension on the rode will vary with the development of chop, or seas.

I've made some measurements of tension in the rode, I have a 2,000kg load cell, and I'll dig them out and add to the post.

I've only just noticed that Ian has resurrected an , almost, 10 year old thread!

but the relevant article is:

Anchor Testing and Rode Loads - Practical Sailor

If you want to extend your analysis I might commend you to have a look at the app SCraMP which uses the accelerometer in a tablet, or phone. I confess to have only recently been introduced to the concept, and there are other apps offering the same recordings. I have a parallel (and well know interest) in the forces at play during anchoring and SCraMP might allow some of the unknowns to be defined (and if you find it useful I'd value some advise!

Jonathan
 
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Birdseye

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On my boat at the moment without access to books and a decent Internet source but I seem to remember that it’s possible to get a reasonably accurate number for the efficiency of a propeller interning talk into forward thrust. In that case it should be possible with the engine working at a given power output and the boat travelTravelling at a steady speed in flat water to Derive a figure for thrust.
Failing that a spring balance between the boat and a Bullard with the engine at certain revs That you know translates into a given speed in flat water should give you a drag figure
 

Laminar Flow

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On my boat at the moment without access to books and a decent Internet source but I seem to remember that it’s possible to get a reasonably accurate number for the efficiency of a propeller interning talk into forward thrust. In that case it should be possible with the engine working at a given power output and the boat travelTravelling at a steady speed in flat water to Derive a figure for thrust.
Failing that a spring balance between the boat and a Bullard with the engine at certain revs That you know translates into a given speed in flat water should give you a drag figure
I am sure it's not that easy. Prop efficiency is actually surprisingly poor and subject to many variables. The (assumed) efficiency of 60% in JDC's post is already quite high and very much dependent on hull and prop arrangement. Significantly higher values are only possible in planing craft.

I doubt that a bollard pull, without progress through the water or the effect of "dead" water carried along with the hull (slip) and the consequential reduction in RPM, will provide a realistic picture of the power transmitted to the water when in motion at various speeds.

I was always more interested in the effects of drag on sailing speeds, but the logical cruising speed for a power driven displacement vessel is at or just short of the point where the curves for form drag and frictional drag diverge.

Picking the right prop is not even something the pros always get right. When we re-engined I did quite a bit of "calculatory fretting" and came to the conclusion that I was not getting optimal performance from our original 40 year old prop, even though RPM, HP and transmission hadn't changed. I had the prop pitched up one and a half inches (for small change) and gained another half a knot at 1500 RPM to let us cruise at 5.5 kts. At this speed we use 1.8 ltrs/hour, including the Webasto, so, realistically, about 1.6 ltrs/hour.

In some ways it is more reliable to calculate power input from sails to equate a projected speed.
Without adding any particular factors for sail type such as jib, main, mizzen, full battens etc. the powers generated by a sail on a reach are:
F3 0.015 hp/sqft
F4 0.020 hp/sqft
F5 0.040 hp/sqft
F6 0.070 hp/sqft
This way I was able to calculate that, with our increased SA, we should reach "hull speed", 7.1 kts, in a F4.
This has, subsequently, proven to be correct.
 

TernVI

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.....
Without adding any particular factors for sail type such as jib, main, mizzen, full battens etc. the powers generated by a sail on a reach are:
F3 0.015 hp/sqft
F4 0.020 hp/sqft
F5 0.040 hp/sqft
F6 0.070 hp/sqft
This way I was able to calculate that, with our increased SA, we should reach "hull speed", 7.1 kts, in a F4.
This has, subsequently, proven to be correct.
These numbers seem uite low to me, what is their provenance please?
 

Laminar Flow

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These numbers seem uite low to me, what is their provenance please?
Dave Gerr, "Nature of Boats".

Edit:
Also, to calculate force generated by a sail:
W = K x A x v2 (squared) x PL/2

Where W is the force generated, K is a factor of 0.5 for propulsive force and 1.32 for side force, A is SA, v is the wind speed in m/sec, PL is the specific gravity of air 1.204kg/cube meter.
 
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jdc

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Thank you to for resurrecting a rather old thread of mine!

Waves are hard to estimate! I did make some estimates based on Prof Gerritsma's work, and added it to my memo here: https://awelina.com/hull_drag/fuel_consumption2.pdf It shows a substantial reduction in effective speed with waves - that part is more or less ok - but it's estimating the period and amplitude of the waves which is nigh impossible and I made a few rather brutal assumptions. The shape seems ok 'tho.
 

mark_tolly61

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Thanks I loved your drag calculations.
Way too much drag for semi-displacement hulls trying to go beyond hull speed.
Playing around with fits I got a better match with this:

1636369095424.png

I think it'll still work well for lower F...

Any further insights on this? I get, and think I've read of, a bit of improvement on dV/dRPM as F grows > 0.4 so I'm looking to match that, preferably with something more than mere data fits...
 

Wing Mark

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Thank you to for resurrecting a rather old thread of mine!

Waves are hard to estimate! I did make some estimates based on Prof Gerritsma's work, and added it to my memo here: https://awelina.com/hull_drag/fuel_consumption2.pdf It shows a substantial reduction in effective speed with waves - that part is more or less ok - but it's estimating the period and amplitude of the waves which is nigh impossible and I made a few rather brutal assumptions. The shape seems ok 'tho.
As a dinghy sailor, it seems obvious to me that waves are very variable.
Maybe waves are easier to model out further from the shore in deeper water than close to the shore where dinghies sail, but evn sailing in the same place, they are never the same from one day to another.
With a light boat, you can feel that some waves stop you far more than bigger waves of a 'better' shape.
Sometimes the boat feels very slow on one tack compared to the other ,so the angle into the waves makes a big difference.
Also how you steer through the waves is a big factor.

There is a limit to what you can reduce to a few equations.
What is the aim of the exercise? To build a faster boat? To have a better estimate of fuel consumption for a given boat?
 

ianat182

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I'm not a maths person but have read a book by JOHN TEALE -"How to Design A Boat" reprinted 1995. It contains various formulae for calculating boat speeds for various types of hull design ranging from round hull to veed and flat hulls, and the HP required to attain these speeds. Surprising to me was the difference a canoe sterned hull or flat transom will also make to velocity. The book also contains a Table to calculate the HP for these various types.
As a dinghy sailor I was very aware that we went faster than displaceme-though someone has in the model yacht world.
By the way he also mentions wind resistance for sailing yachts is to be considered.

ianat182
 
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mark_tolly61

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As a dinghy sailor, it seems obvious to me that waves are very variable.
Maybe waves are easier to model out further from the shore in deeper water than close to the shore where dinghies sail, but evn sailing in the same place, they are never the same from one day to another.
With a light boat, you can feel that some waves stop you far more than bigger waves of a 'better' shape.
Sometimes the boat feels very slow on one tack compared to the other ,so the angle into the waves makes a big difference.
Also how you steer through the waves is a big factor.

There is a limit to what you can reduce to a few equations.
What is the aim of the exercise? To build a faster boat? To have a better estimate of fuel consumption for a given boat?


I'm interested in fuel consumption for sure, also just general.
I'm not talking about waves in particular but meant by "wave drag" that part of resistance that comes from climbing up your own bow wave. When you start planing, it's no longer accurate, right? So if I can get one equation that can match my dinghy and my big boat both, up to, say F# 7, I'll call it good. I've noticed vicprop in Canada do seem tto get the right anwers, but they don't show their work! : )
 
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