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This fast, seaworthy 0.855 kg, 4.9W twin-screw no-frills speedboat based on the 74x18x7 LU City Lines hull is our fastest boat to date at 0.99 m/s.
About this creation
Please feel free to look over the images and skip the verbiage.

Nadine is a 0.855 kg, 4.9 W twin-screw monohull speedboat based on the long, blue 74x18x7 LU City Lines hull. A top speed of ≥0.99 m/s (Froude number ≥ 0.43) made her our fastest no-frills LEGOŽ speedboat for well over a year -- until Triton hit the water.

Nadine also has the distinction of being our most seaworthy speedboat. Great speed and seaworthiness aren't often found together in real boats.

On this page:

Designed for speed

Without doubt, the 4 most important factors behind Nadine's speed are

The CLH is one of only 2 LUHs with enough volume to support twin XLs in a monohull boat, but Nadine owes much of her speed to the CLH's unmatched length.

The rationale behind the twin inverted-V outdrives and third-party props is discussed here.

Nadine's current twin "XL/8.33/55" drivetrain (XL motors, 1:8.33 overdrives, and 55 mm props) came out of a methodical and rather tedious motor/gearing/prop (MGP) optimization process.

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Motor/gearing/prop (MGP) optimization

Our strategy for maximizing top speed relies on 3 key facts:
  • The total resistance encountered by a boat grows with the square to 4th power of her speed relative to the water.
  • Boat speed tops out when mounting total resistance finally consumes all of the mechanical power delivered to the water processed by the props. Further acceleration then becomes impossible.
  • LEGOŽ motors, like all DC electric motors, peak in mechanical power output at shaft speeds near 50% of their no-load speeds.
The trick, then, is to bring motor shaft speed to ~50% of no-load speed (NLS) just as the boat tops out.

When powered by the 7.4V output of a PF rechargeable battery, XL and L motors have no-load shaft speeds of ~180 and ~300 RPM, respectively. Their target shaft speeds for peak mechanical power are therefore ~90 and ~150 RPM, respectively. The peaks in their power-speed curves are broad enough that 40% NLS and 60% NLS are almost as good as 50% NLS WRT power output, but better to err on the high side if you have the choice.

In practice, MGP optimization takes
  • An inexpensive hand-held optical tachometer like the one in use above
  • Lots of fiddling with feasible motor, overdrive ratio, and prop size combinations
  • A reliable way to measure top speed as a final check
Optimization is further complicated by the fact that feasible overdrive ratios are limited by
  • Available LEGOŽ gears -- especially double-bevels
  • The need to maximize power transmission efficiency at every step from motor to prop
  • The need to minimize appendage drag by keeping the wetted volumes and wetted surface areas of prop supports to a minimum and streamline them as much as possible
We use double-bevel gears whenever possible, as they're more efficient and less prone to skipping under the high torques encountered in our powerboat drivetrains.

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Nadine's evolution from the prototype CLH monohull seen in the video below to the current version presented here shows why MGP optimization has to be done on a boat-by-boat, one-step-at-a-time basis.

The twin L/1:5/52 prototype, my 1st CLH-based boat, was (and still is) slated to become a replacement hull for this marine geology research vessel model -- hence, the heavy full-coverage decking. She was never weighed or formally clocked but must have displaced a good 0.15 kg more than Nadine does now.

This prototype had 2 major handicaps, both of which contributed to high total resistance: (i) Added displacement begat increased wetted surface area begat increased skin drag. (ii) Vertical outdrive struts begat increased appendage drag and reduced prop efficiency.

With 55 mm props, the prototype's L motors lugged along at well below target shaft speed under the added resistance. Of the 52 mm and 40 mm alternatives available at the time, the 52 mm props got her much closer to target shaft speeds. Feasible overdrive ratios were too far apart to help with any prop.

The twin L/1:5/52 prototype picked up a bit more top speed when fitted with our usual inverted-V outdrives, but resistance and prop efficiency didn't improve enough to allow 55 mm props. Having never had a hull capable of supporting twin XLs before, the option occurred to me only much later.

My 2nd CLH went to the original Nadine, a much faster no-frills speedboat with a twin L/1:5/55 drivetrain, inverted-V outdrives, a top speed of ~0.92 m/s, and a much-reduced displacement of 0.815 kg.

The original Nadine ran away from our then-reigning top-speed champ -- the original Trident seen in the pool trial video below. (Trident's now almost as fast as the current Nadine.)

The current Nadine (below) came with the realization that she'd still be seaworthy with twin XL motors onboard. The ensuing MGP re-optimization yielded the current twin XL/1:8.33/55 drivetrain, ≥0.99 m/s top speed, and 0.855 kg displacement.

The added displacement and low target speed of the XL motors might well have back-fired WRT total resistance and drivetrain efficiency, respectively, but the net result was positive -- this time.

The higher center of gravity reduced stability a bit, but current Nadine is more stable than the prototype and seaworthy enough for any swimming pool sea state I've ever seen.

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Guided trial and error

The section on MGP optimization underscores -- and Nadine's evolution illustrates -- an inescapable truth in boat design at all scales: Given the inherent unpredictability of the boat-water interaction,

The only way to know if a design change will help or hurt top speed or seaworthiness is to try it and see.

Bringing naval architure into LEGOŽ powerboat design doesn't change that, but it does help you figure out what to try next. Given the large number of variables involved and the counter-intuitive nature of many aspects of boat behavior, that's a great help indeed.

Real-world engineering always informs technical LEGOŽ building to my mind -- at the very least, by supplying the guidance in guided trial and error (GTE). Its influence is evident in the work of many of my favorite builders, and the harder one pushes the physical limits of what ABS plastic and LEGOŽ parts can reasonably do, the more valuable GTE becomes. If a non-LEGOŽ component has to be brought in to capture the functional essence of real-world counterparts now and then, so be it.

Coming up with fast and seaworthy LEGOŽ powerboats seems to push that envelope pretty hard, but that's half the fun. The other half, of course, is getting to play with the boats in open water without a sense of impending doom.

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The video above shows Nadine's oldest ancestor -- a slower, heavier CLH monohull prototype with an L/1:5/52 drivetrain and vertical outdrives -- running in moderate swimming pool chop. Nadine has even more freeboard now by virtue of lesser displacement, but the prototype had no trouble keeping her decks dry.

The current Nadine put in her fastest recorded top speed of ≥ 0.99 m/s (Froude number ≥ 0.43) during the recent pool trials above. Note the added speed and greater freeboard now, her excellent stability, and the frequent fits and starts and minor collisions -- the last due entirely to PF remote control failures.

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Wave wakes -- more than just a fascination

Also worth noticing in the videos are the wave wakes -- i.e., the complex wave patterns generated by forward boat motion. Note how the wave wakes move along with the boats, grow in amplitude (height) with speed, and become fixed in shape and amplitude at top speed. The wave wakes vary little in geometry from boat to boat, but they do vary in amplitude, and those amplitudes ultimately limit top speeds.

Wave wakes generally consist of 2 distinct types of waves: (i) Divergent wave sets angling outward from the boat's direction of travel, and (ii) simpler transverse waves with crests roughly perpendicular to the direction of travel. The latter are seen along the hull and behind the boat in the angle between the divergent waves formed on either side of the boat.

Nadine's videos show divergent waves coming off her bow, forward shoulders, and outdrive struts. Steeper faces, shorter wavelengths, and the lighting make them a lot more conspicuous than the transverse waves. The latter are higher at top speed but don't look it, because they're more spread out from crest to crest. The transverse waves in Trident's pool trial video below are hard to miss.

Wave wakes are fascinating to watch, but they have great practical significance as well: The power consumed in creating and maintaining wave wakes -- especially the transverse waves -- could have gone into top speed instead.

Above a certain length-dependent speed, wringing more top speed out of a boat largely becomes a matter of wave wake management.

The ongoing loss of power to wave-making is manifest as a force opposing the boat's forward motion. This force, known as wave-making resistance or wave drag, grows with the square of wave height and the square to fourth power of speed but diminishes with waterline length. Wave-making resistance limits our top speeds more than all other factors combined.

Nadine is all about the dependence on length. Trident nicely compensates for her shorter length by exploiting the dependence on wave height instead, as wave-making resistance can be suppressed to some extent by arranging for destructive interference among the various wave populations making up the wake wake.

The connection between waterline length and wave-making resistance -- and the implications for LEGOŽ powerboat design -- are explored below.

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PF remote control issues

Of course, speed isn't everything. Seaworthiness and play value are also essential in remote control boats, and reliable, long-range remote control is central to both.

Given that each and every control lapse captured in the videos -- and there are many -- came courtesy of the deeply flawed Power Functions infra-red remote control (IRRC) system system, it's quite clear that IRRC is neither long-range nor reliable.

We've had 2 near-sinkings due to IRRC failures. After the 2nd incident, Shawn decided to remove the radio-frequency remote control (RFRC) receiver from his 2002-vintage RC Race Buggy and substitute it for the IRRC receiver in a boat otherwise identical to Nadine.

This "RF-Nadine" ended up ~0.2 kg heavier, noticeably less stable (but still quite seaworthy), and 15-20% slower than "IR-Nadine". However, she delivered absolutely flawless remote control at ≥6 times the maximum IRRC range, and that put her in a play-value class by herself.

The photos above show Shawn's latest CLH-based RFRC boat, Radio Flyer. The new rudder simplifies the human interface, as the RC Race buggy transmitter control layout doesn't lend itself to differential-drive steering. The added weight of the RF reciever and rudder causes her to sit ~5 mm lower in the water than Nadine.

Thanks to the reliable control as much as anything else, this boat took 1st place in the boat drag races at BrickWorld 2015 using the legal 5.5L LEGOŽ props shown.

The bad news: The RC Race Buggy transmitter and receiver are as rare as they are expensive, and the CLH -- also rare and expensive -- is the only unitary hull capable of carrying the RFRC reciever in a monohull boat.

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More photos

At midships, Nadine's breadth, draft, and freeboard are 142 mm, 16 mm and 42 mm, respectively. She owes much of her stability to her breadth and huge breadth/draft ratio (8.9), both of which far exceed anything possible with any other unitary hull.

Nadine's performance definitely benefits from the PF Li polymer rechargeable battery and V2 receiver -- especially during speed runs in long pools. The rechargeable battery delivers more current than AA and AAA battery boxes loaded with NiMH or alkaline cells, and does so without voltage sag.

Relative to the original V1 receiver, the V2 passes much more current to the motors, tolerates higher internal heat loads before its thermal protection trips, and is more efficient to boot.

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Comparison with other fast boats

The group shots below show my 3 of my 4 fastest boats together for comparison purposes. The monohull in the middle is Laverne, which share the 2nd fastest spot with Celine (not shown). Trident, with the 3rd fastest top speed, is the odd-looking trimaran opposite Nadine.

The top speeds represented here range from ~0.95 m/s for Trident to ≥0.99 m/s for Nadine. Trident has the shortest overall hull length, but Laverne has the shortest LWL.

Nadine's LWL of 0.540 m and top speed Umax of ≥0.99 m s-1 correspond to Froude number Fr ≥0.43. For Trident, LWL = 0.372 m at her center hull, Umax ≈ 0.95 m/s, and Fr ≈ 0.50, respectively. For Laverne, LWL = 0.345 m, Umax ≈ 0.97 m/s, and Fr ≈ 0.53, respectively.

Compared to Trident, Nadine is 47% longer at waterline, displaces 73% more water, and has ~67% more total wetted surface area S.

I'll explain the significance of the Froude numbers below.

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Length matters

As mentioned earlier, Nadine owes much of her speed and all of her seaworthiness to her 74x18x7 LU City Lines hull (CLH) -- the largest and most stable LUH ever offered.

The large and seaworthy part of that probably comes as no surprise (details here), but how does large and fast work??

I'll give the short answer now and explain its physical basis below. Though rather technical in places, the explanation introduces some valuable design concepts along the way.

The load waterline length (LWL) attainable with a CLH far exceeds that of any other hull. (Recall that LWL is just boat length taken at waterline at rest in calm water at operating displacement.) Overall length is irrelevant here.

First, 4 key facts:
  • LUHs are effectively displacement hulls in that they push through the water rather than plane (skim) over it at attainable speeds.

  • Wave-making and viscous resistance make up the lion's share of the total resistance encountered by well-designed powerboats.

  • Hydrodynamically, high speed means fast enough for wave-making resistance to dominate the total resistance budget. All of our boats top out in this high-speed regime.

  • Once in the high-speed regime, details of hull form have little impact on total resistance.
Now to put it all together.

Displacement hulls with greater LWL generally encounter less wave-making resistance but more viscous resistance -- the latter by virtue of the added wetted surface area (S) and load waterline breadth (BWL) that usually come with added length. The net effect on total resistance depends on speed.

In the high-speed regime of primary interest here, added length brings about a net reduction in total resistance -- often a substantial one. Hence, given 2 boats with reasonable hull forms and identical propulsion systems at high speed, the faster top speed will generally go to the one with the greater LWL.

The group shot above leaves no doubt that CLH-based boats are in a class by themselves when it comes to LWL. For CLH-based Nadine in the foreground, LWL = 0.540 m.

This shot doesn't include the next 2 LUHs down in overall (bounding box) length: The 52x12x6 Family Yacht hull (FYH) used by Lucille and the 52x12x6.33 Cargo Carrier hull (CCH) used by Celine. Under typical speedboat loads, the FYH and CCH have ~33% and ~26% waterlines than the CLH, respectively.

The long bow overhang on the next longest LUH overall, the 51x12x6 LU Police Boat hull, leaves Laverne (middle boat in the group shot) with an LWL of only 0.345 m.

Trident in the background gets an LWL of 0.374 m out of the next longest LUH overall, the 48x6x5 LU Speedboat hull.

The 3 speedboats gathered above top out in the 0.95 m/s to 0.99 m/s range -- a spread of <5%. Each gets there in her own way.

The reduced wave-making resistance afforded by Nadine's great length is enough to put her at the head of the pack despite major handicaps in displacement and viscous resistance. Her tanker-like hull form is largely irrelevant at these high speeds.

The other 2 boats, being much shorter, have to contend with much greater wave-making resistance. Hence, they have to work much harder to reach such speeds. Next-fastest Laverne gets there via a greatly increased power/displacement ratio and much reduced viscous resistance relative to Nadine.

Trident, the tricky trimaran, overcomes her shorter length with an extremely high power/displacement ratio, minimal viscous resistance, and interhull wave cancellation brought about by careful side hull positioning.

In the remaining sections, I'll try to make physical sense of all this. A number of helpful boat design concepts will come up along the way.

To simplify the discussion and maximize its relevance to LEGOŽ speedboats, the rest of this page will focus on propeller-driven monohull powerboats with displacement hulls moving through calm water at steady forward speeds in the absence of wind and current. Since air resistance is negligible at LEGOŽ scale in the absence of a stiff headwind, it will be ignored.

Total resistance and effective power

Total resistance (RT) is the resultant (vector sum) of all the forces opposing a boat's forward motion through the water. Many physical processes contribute to total resistance, but the net effect is single speed-dependent force antiparallel to the direction of travel.

Total resistance grows rapidly with speed U. The general trend is a U 2 dependence, but RT varies more like U 4 at certain high speeds.

The practical import of total resistance in the equation for the effective power (PE) needed from the motors to maintain a steady speed U:


From the speed dependence of RT already noted, it's clear that the PE required to reach U usually grows as U 3 to U 5.

Clearly, then, there must come a top speed (Umax) when the boat's given all the PE it has to give, and further acceleration becomes impossible.

Propulsive efficiency and installed power

The installed power PI needed to push a specific boat to target speed U will always exceed the effective power calculated from total resistance.

The reason is simple: Inefficiencies in power conversion and transmission between the boat's power supply (fuel or batteries) and the water processed by her props can often be reduced, but they can never be eliminated entirely.

When powered by a PF rechargeable 7.4V battery, Nadine's twin-XL powerplant puts out 4.9W of peak installed mechanical power at ~90 RPM, whereas the twin-L powerplants used by Laverne and Trident supply 4.3W at ~150 RPM.

Since all 3 boats optimized to the same 55 mm props, it fell to their overdrive ratios to get them as close to maximum-power shaft speed as possible when they top out.

Mathematically, installed and effective power are related by the total propulsive efficiency given by


Like all efficiencies, ηT is a dimensionless number between 0 and 1. It subsumes (i) all the electrical power losses associated with the batteries, RC receivers, and motors; (ii) all the mechanical losses associated with powertrain bearings, shafts, gears, and their mounts (including those inside the motors); and (iii) all hydrodynamic losses associated with the props.

Nadine's gearing is theoretically a bit less efficient for having three overdrive stages instead of two.

In the real world, ηT is generally on the order of 40-60% for well-designed displacement monohulls with efficient props. Our efficiencies are probably also in the 40-60% range when we run size-optimized third-party props like Nadine's.

However, I doubt that we'd get even 20% propulsive efficiency with the best LEGOŽ props (the left-handed 44 mm 2-blade "twisted" props, 4745), and all other LEGOŽ props are much, much worse in that regard. The huge hydrodynamic losses associated with LEGOŽ props can't be wished away.

If you stick with LEGOŽ electricals, as we've done so far, your leverage on electrical losses is limited to battery and IR receiver selection. The best battery option by far in nearly all cases is the lightweight 7.4V PF Li polymer rechargeable. The V2 receiver used by Nadine and most of my other speedboats is more efficient than the original (unmarked) V1, delivers nearly twice the current, and is much less prone to thermal protection shut-down.

Mechanical losses, on the other hand, are very much under the LEGOŽ naval architect's control. Stiff motor mounts, solid axle support, short or reinforced shafts, and few gears are the keys to efficient mechanical power transmission. These strategies are evident throughout Nadine's powertrain, including her 3rd generation outdrives.

The bottom-line equation below expresses target speed U in terms of installed power PI, propulsive efficiency ηT, and total resistance RT:

U = ηT PI / RT

This important equation is worth remembering, as it clearly identifies installed power, propulsive efficiency, and total resistance as limiting factors of potentially equal importance when it comes to maximizing either (i) top speed or (ii) battery life at a lower speeds. You'd need some way of measuring or estimating ηT and RT at speed to use it quantitatively, but that's not the point.

Seaworthiness, always the top design priority, isn't mentioned in this equation but ultimately constrains every design decision involving installed power, efficiency, and resistance. Juggling the many difficult trade-offs involved just so is the name of the game in powerboat design at all scales. See my tutorial on powerboat seaworthiness for details.

Playing the trade-offs surrounding RT well requires some understanding of the main components of total resistance and their interactions. The next few sections tease apart total resistance in a practical way.

Viscous resistance

For our purposes, the most useful breakdown of total resistance is

RT = RV + RW + RA,

where RV and RW are respectively the viscous resistance and wave-making resistance due to the hull. The viscous and wave-making resistances due to appendages like props and outdrive struts are lumped together under the RA term.

We're mainly interested in the wave-making term here, as RW ultimately limits top speed in fast LEGOŽ powerboats. However, there are several important things to know about the viscous term:
  • Viscous resistance is further broken down as RV = RF + RVP, where RF is frictional resistance, and RVP is viscous pressure resistance.

  • As the name implies, frictional resistance, aka skin drag, is due to friction between the hull and the water flowing past it and depends mostly on the roughness and extent of the hull's wetted surface.

  • Viscous pressure resistance, aka form drag, is ultimately due to separation of flow from the hull at changes in hull curvature along the waterline. Most of the form drag in a well-designed hull occurs at the stern, where all flow along the hull finally separates to form the boat's eddy-filled turbulent wake. Forward and aft shoulders also contribute. Slender hull forms with high length-breadth (L/B) ratios at waterline (e.g., Trident's center hull) generate less form drag.

  • Overall, RV is proportional to the hull's wetted surface area (S) in square meters (m2) and grows as U 2 at all speeds.

  • Minimizing RV runs up against some tough trade-offs. Once breadth has been reduced to the minimum required for stability, further gains in L/B can only be had by adding length at the cost of increased wetted surface area.

  • By definition, RV dominates total resistance in the low-speed regime.
RW progressively overshadows RV in the high-speed regime -- at some speeds, overwhelmingly so. We'll have a precise boundary between these regimes in just a moment.

Appendage drag

Resistance due to submerged appendages like rudders and prop shafts and supports can easily account for >25% of RT at speed, especially in twin-screw monohulls like Nadine. Many if not most of the LEGOŽ powerboats found on YouTube appear to have been designed without RA in mind.

Since appendage drag is primarily viscous in nature at all speeds, it always pays (i) to minimize wetted appendage surface area -- i.e., to minimize the amount of plastic in the water beyond the the hull itself -- and (ii) to streamline appendages as much as possible to minimize their form drag. Nadine's outdrives reflect that effort.

Appendages also make waves, as seen from 0:21 on in the video below. The associated wave-making resistance can also be significant. Judicious positioning of well-streamlined appendages helps.

We now turn our attention to the dominant component of total resistance in the high-speed regime -- i.e., the wave-making resistance generated by the hull itself.

Froude number and the scaling of length and speed

The most important parameters in boat hydrodynamics are displacement (Δ), speed (U), load waterline length (LWL), load waterline breadth (BWL), draft (T), and wetted surface area (S). Displacement, breadth, and draft are especially important at low speeds, as they have major impacts on viscous resistance via S and slenderness (or lack thereof).

The effect of LWL on RT via RV via wetted surface area and slenderness applies at all speeds, but its greatest impact on RT lies in the high-speed regime, where wave-making resistance RW eclipses RV. The boundary between the low- to high-speed regimes lies at Fr ≈ 0.30, where

FrU / √(g LWL),

and g is the local acceleration of gravity (~9.81 m s-2).

The dimensionless quantity Fr defined above is a length-adjusted measure of speed known as the Froude number. Its great power lies in the fact that monohull boats running at the same Froude number encounter (i) similar relative growth rates in RT with increasing speed or Fr, and (ii) similar RW / RV ratios -- regardless of size or hull type! Hence, resistance-speed relationships worked out from scale models can be applied to full-scale ships and vice-versa.

Because LWL is such an important parameter, naval architects scale model boats and their full-size counterparts by their LWL ratio, here denoted as α. Hence, if a model and the corresponding full-size ship have waterline lengths of Lm and Ls, respectively, then

αLs / Lm

Boats are said to possess geometric similarity if all angles are identical and all linear dimensions differ by same scaling factor α. They possess Froude similarity if they operate at the same Froude number.

Froude similarity is the most meaningful basis for scaling speed, as it insures hydrodynamic similarity WRT wave-making. From Frs = Frm, one easily obtains the speed scaling formula

Us / Um = √(Ls / Lm) = √(α)

I used this formula to calculate a scale speed of 12 knots for my 1:120 scale marine geology research vessel R/V Stormin' Norma, which tops out at 0.56 m/s.

Nadine vs. US Navy destroyer

The two rather different high-speed boats shown below illustrate the significance and utility of the Froude number. The focus is on their hulls and installed power, not their superstructures and outfit.

The USS Howard (DDG 83) in the US Navy photo above represents the current batch of Arleigh Burke-class twin-screw guided missile destroyers (Flight IIA, DDG 79 on). In many ways, these boats epitomize the modern fast monohull displacement ship.

The USS Howard's basic specs: Δ = 9.6 x 106 kg at full load, PI = 8.8 x 106 W, PI / Δ = 0.91 W/kg, LWL = 154.5 m, BWL = 19.4 m, LWL / BWL = 8.0, T = 10.2 m, BWL / T = 1.9, Umax = 31 kt = 15.9 m s-1, and Frmax = 0.41.

Nadine's basic specs: Δ = 0.855 kg, PI = 4.9 W at 7.4V, PI / Δ = 5.7 W/kg, LWL = 0.540 m, BWL = 0.142 m, LWL / BWL = 3.8, T = 0.016 m, BWL / T = 8.9, Umax = 0.99 m s-1, and Frmax = 0.43.

For the USS Howard and Nadine, the LWL ratio α = 282. Their hulls clearly don't possess geometric similarity, but they're reasonably close to Froude similarity in the sense that they top out at nearly identical Froude numbers.

As we'll see below, Froude similarity in the 0.40 to 0.54 range means that Nadine and Arleigh Burke-class destroyers are bucking the same resistance mix (RW >> RV) at top speed despite the vast difference in scale.

Hence, substantial gains WRT top speed (or fuel consumption or battery life) would require all-out war on RW instead of RV. Efforts to improve top speed via reductions in RV would still be valuable but lower in potential yeild.

The most straightforward way to reduce RW is to increase LWL. That's not an option for Nadine, as she already has the longest LUH available. However, a new batch of Arleigh Burke-class destroyers lengthened at midships like stretch limos has been proposed. The existing powerplants (four 22,000 kW gas turbines each) would be retained.

The brute-force approach -- i.e., upping installed power in the existing hull to achieve a comparable gain in top speed -- might have some merit in Nadine's case if non-LEGOŽ motors and batteries were on the table, but it wouldn't work in the destroyers' case. The current Arleigh Burke-class hull couldn't accommodate the huge increase in powerplant size and weight required, fuel consumption would skyrocket, and tactical range would fall well below that mandated by the destroyer's job description.

As discussed in the previous section, Nadine's slightly higher Frmax means that she tops out at a slightly higher scale speed than the USS Howard. That can only be due to her substantial advantage in installed power per unit displacement (5.7 vs 0.91 W/kg), as everything else here favors the destroyer WRT total, viscous, and wave-making resistance per unit displacement.

For starters, DDG-51 hulls have much finer entries (narrower bow angles), lower wetted surface area per unit displacement, much greater slenderness (L/B of 8.0 vs. 3.9), and much better streamlining with no significant bottom roughness. They surely have more efficient screws if nothing else. The relevant scale effects (jargon for hydrodynamic effects that vary significantly with scale despite Froude similarity) all work in the USS Howard's favor as well.

Critical speed

Physically, Fr relates waterline length to the wavelength λU of a free-standing deep-water wave traveling at boat speed U. The transverse waves generated separately at a moving boat's bow and stern are just such waves. Since they consume most of the power lost to wave-making, we'll focus on them hereafter.

The transverse waves generated at the bow and stern differ in only 2 ways: (i) The bow wave first appears as a crest just aft of the bow end of the waterline, whereas the stern wave first appears as a trough just ahead of the stern end of the waterline. (ii) Bow wave crests are slighter higher than stern wave troughs are deep.

The distance between the first bow wave crest and first stern wave trough is the effective waterline length given by

Leff = β LWL,

where β ≡ Leff / LWL ≈ 0.90 for most boats.

The bow and stern transverse wave systems have identical speed-dependent wavelengths given by

λU = 2 π U 2 / g = 2 π Fr 2 LWL

Rearranging yields an expression for Fr more in keeping with its physical significance:

Fr = Frcrit √(λU / LWL),

where Frcrit ≡ √(1 / 2 π) = 0.40 is the critical Froude number.

Clearly, Fr = Frcrit when λU = LWL.

The critical speed corresponding to Fr = Frcrit is given by

UcritFrcrit √(g LWL)

Ucrit marks the onset of dramatic growth in wave-making and total resistance -- a major event in the evolution of resistance with speed for a displacement monohull of waterline length LWL.

The notion of critical speed illustrates the value of thinking in terms of Froude number rather than speed. Critical speed applies to a single waterline length, whereas displacement monohulls of all lengths experience the same dramatic increase in wave-making and total resistance at Fr = Frcrit = 0.40.

Speeds in excess of Ucrit and Froude numbers in excess of Frcrit are said to be supercritical. All of our speedboats top out at supercritical speeds against rapidly mounting wave-making and total resistance.

Wave making at supercritical speeds

When transverse wavelength exactly matches effective length -- i.e., λU = Leff, a boat finds her 2nd bow wave crest superimposed on her initial stern wave trough.

The resulting destructive interference largely eliminates the power-sapping transverse wave field that would otherwise have formed behind the boat. The Froude number corresponding to λU = Leff is ~0.38 in the usual case of Leff / LWL ≈ 0.90.

In effect, the bow is laying down transverse waves of speed U and wavelength Leff as it goes, and the stern is (almost) erasing them exactly one wavelength behind. Divergent waves produced by monohulls are immune to this kind of interference because divergent waves emitted at different points along the hull never meet. Multihulls like Trident are a different story, but I'll save that for another page.

Take another look at the videos. Nadine's transverse waves are hard to see because some transverse wave cancellation is still in effect when she tops out at Fr = 0.43. The transverse waves behind the much slower CLH prototype in the 1st video were easier to spot because the prototype topped out well below its critical speed under the triple-whammy of lesser installed power, greater wetted surface area, and greater appendage drag.

Trident leaves a trail of very prominent transverse waves at top speed because she's then well beyond critical speed at Fr ≈ 0.50. At this Froude number, λU / Leff ≈ 1.4. That puts the 1st bow wave trough partially on top of the 1st stern wave trough. This constructive interference causes the transverse waves behind her to grow to particularly high amplitudes.

Perfect constructive interference occurs when the first bow wave trough coincides with the first stern wave trough -- i.e., when Leff = λU / 2, or equivalently, when Fr ≈ 0.54. At that point, the transverse waves behind a boat are at maximum amplitude, and RW is at its greatest.

Humps, hollows, and the wave wall

Wave-making resistance is proportional to the square of transverse wave amplitude. As we saw in the last section, this amplitude is least when the bow and stern transverse wave systems reach maximum destructive interference at Fr ≈ 0.38, and greatest when they reach maximum constructive interference at Fr ≈ 0.54.

Since total resistance RT is dominated by wave-making resistance RW at Fr > 0.30, and overwhelmingly so by Fr > 0.40, these fluctuations in RW result in large swings in RT.

Naval architects refer to Froude number intervals of lower than usual total resistance as hollows for the corresponding dips seen total resistance vs. Froude number (RT vs. Fr) curves. All hollows are due to destructive transverse wave interference.

The biggest hollow reaches its lowest point at Fr ≈ 0.38 and is gone by Fr ≈ 0.40, the critical Froude number. Progressively smaller hollows bottom out at Fr ≈ 0.27, 0.22, and 0.19.

Heavy ship models with battery life or access issues like Stormin' Norma enjoy extended battery life when operated in hollows. Likewise, long-haul merchant vessels are optimized to transit entire oceans in a single well-chosen hollow in the low-speed regime (Fr < 0.30) so as to minimize RW and fuel consumption per unit displacement.

In the same spirit, Froude number intervals of greater than usual total resistance are called humps for the corresponding rises seen in RT vs. Fr curves. All humps are due to constructive transverse wave interference.

The biggest hump of all, the main hump, starts at FrFrcrit = 0.40, peaks at Fr ≈ 0.54 and is gone by Fr ≈ 0.59. The main hump dwarfs the progressively smaller humps peaking at Fr ≈ 0.31, 0.24, and 0.20. The 2nd most prominent hump, the one at Fr ≈ 0.31, is called the prismatic hump.

Practically speaking, the main hump is the single most important feature on the RT vs. Fr curve. High-speed displacement craft typically top out on the main hump, and our speedboats are no exception.

Acceleration past the last hollow at Fr ≈ 0.38 is met with rapidly mounting constructive transverse wave interference at the stern and beyond. The transverse wave field behind the boat then roars back to life as never before, RW skyrockets, and RT follows suit. The constructive interference peaks at Fr ≈ 0.54 and vanishes at Fr > 0.59.

No further humps appear out to at least Fr ≈ 7, as transverse waves aren't generated at Fr > 0.59. However, divergent wave-making continues to sap power at levels comparable to the power lost to viscous resistance out to Fr ≈ 1, the onset of the planing regime.

Away from humps and hollows, RT generally grows with the square of speed. However, on the very steep ascending limb of the main hump between Fr ≈ 0.40 and Fr ≈ 0.54, RT grows more like U 4.

This very steep rise in RT with speed is sometimes referred to as the wave wall in recognition of its root cause -- explosive growth in RW. Since the installed power PI needed to climb the wave wall grows roughly as U 5, the wave wall represents a formidable practical barrier in boat design and operation.

Given that powerplant mass tends to grow with PI, powering through the wave wall generally won't be a viable option for boats with displacement hulls. Boats with planing and semi-planing hulls, on the other hand, are made to do just that. For such hulls, hydrodynamic lift starts to kick in at Fr ≈ 0.60, and full planing commences at Fr ≈ 1.0.

The fastest Class 1 offshore powerboats (13-16 m at static waterline), arguably the ultimate planing craft short of hydroplanes, top out at Fr > 6. Getting there takes a very special planing catamaran hull and 2 or more big-block V-8s or V-12s with a combined installed power of ~2.2 kW (~3,000 HP)!

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Dimensions and hull form coefficients
All measurements taken at rest in fresh water (density 1,000 kg m-3).

Overall dimensions:592 x 106 x 58 mm (LxWxH, excluding outdrives)
Displacement:0.855 kg
Displacement volume:8.55 x 10 -4 m3
Depth:68, 58 mm (bow, midships)
Waterline length:540 mm
Waterline breadth:142 mm
Draft at keel:16 mm (midships)
Freeboard:42 mm (midships)
Midship section area:~2.3 x 10 -3 m2
Waterplane area:~6.9 x 10 -2 m2
Block coefficient:0.70
Prismatic coefficient:0.70
Wetted surface area:~8.0 x 10 -2 m2
Midship coefficient:~0.99
Waterplane area coefficient:0.90
Length-breadth ratio:3.8
Breadth-draft ratio:8.9
Length-displacement ratio:5.7
Form factor:0.64

Performance measures

Installed power:4.9W at 7.4V
Installed power to displacement ratio:5.7 W/kg
Critical speed:0.92 m/s
Top speed:≥0.99 m/s
Froude number at top speed:≥0.43
Reynolds number at top speed:≥5.8 x 105
High-speed index:≥0.87

Design features

Construction:Aside from outdrives, mostly studded
Hull:74x18x7 LU City Lines hull (Set 7994)
Propulsion:Twin inverted-V outdrives
Motors:2, 1 XL on each prop
Propellers:55 mm 3-blade counter-rotating pair (non-LEGOŽ)
Gearing:3-stage 1:8.33 overdrive
Prop separation:194 mm on center
Steering:Differential power to props (no rudder)
Electrical power supply:7.4V PF rechargeable Li polymer battery
IR receiver:V2
IR receiver connections:2, 1 for each motor
Modified LEGOŽ parts:Prop hubs
Non-LEGOŽ parts:Props and electrician's tape (bottom fairing)
Credits:Entirely original MOC

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Most of the titles below are free online for the digging.

Anonymous, 2011, Basic Principles of Ship Propulsion, MAN Diesel & Turbo, Copenhagen, Denmark

Barrass, C.B., 2004, Ship Design and Performance for Masters and Mates, Elsevier Butterworth-Heinemann

Barrass, C.B., and Derrett, D.R., 2006, Ship Stability for Masters and Mates, 6th ed., Butterworth-Heinemann

Bertram, V., 2000, Practical Ship Hydrodynamics, Butterworth-Heinemann

Biran, A.B., 2003, Ship Hydrostatics and Stability, 1st ed., Butterworth-Heinemann

Blount, D.L., 2014, Performance by Design (self-published book)

Carlton, J.S., 2007, Marine Propellers and Propulsion, 2nd ed., Butterworth-Heinemann

Faltinsen, O.M., 2005, Hydrodynamics of High-speed Vehicles, Cambridge University Press

Hlavin, J., 2010, Hydrostatic and Hydrodynamic Analysis of a Lengthened DDG-51 Destroyer Modified Repeat, unpublished master's thesis, Naval Postgraduate School, Monterey, CA

Moisy, F., and Rabaud, M., 2014, Mach-like capillary-gravity wakes, Physical Review E, v.90, 023009, p.1-12

Moisy, F., and Rabaud, M., 2014, Scaling of far-field wake angle of non-axisymmetric pressure disturbance, arXiv: 1404.2049v2 [physics.flu-dyn] 6 Jun 2014

Molland, A.F., Turnock, S.R., and Hudson, D.A., 2011, Ship Resistance and Propulsion: Practical Estimation of Ship Propulsive Power, Cambridge University Press

Noblesse, F., He, J., Zhu, Y., et al., 2014, Why can ship wakes appear narrower than Kelvin’s angle? European Journal of Mechanics B/Fluids, v.46, p.164–171

Rawson, K.J., and Tupper, E.C., 2001, Basic Ship Theory, vol. 2: Ship Dynamics and Design, 5th ed., Butterworth-Heinemann

Schneekluth, H., and Bertram, V., 1998, Ship Design for Efficiency and Economy, 2nd ed., Butterworth-Heinemann

Tupper, E.C., 1996, Introduction to Naval Architecture, 3rd ed., Butterworth-Heinemann

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 I made it 
  January 5, 2015
Quoting Walter Lee I really love your design and your explanations. BTW: Lego Motors do output more power if the input voltage is increased and there is a way to double the voltage/power output of Lego battery Packs by putting them in series... but I'm not sure if the Lego Radio Control or the PF IR remote controller can handle the higher voltage. If I had access to a pool to run my lego boats - I'd be there every week!
Walter, Thank you very much. Can I hook up PF batteries in series? If so, I'm all ears! Sounds like you have as much fun with your boats as we do with ours. (Our land-based MOCs see far less play time.) During the summer, we can walk our boats to a nice 25 m pool. Having to drive to a rec center pool at other times of year really adds to the time it takes a new design to mature due to the lengthy testing involved, as you're well aware. Amazing how many folks have no idea that working LEGO boats are even possible!
 I like it 
  January 3, 2015
I really love your design and your explanations. BTW: Lego Motors do output more power if the input voltage is increased and there is a way to double the voltage/power output of Lego battery Packs by putting them in series... but I'm not sure if the Lego Radio Control or the PF IR remote controller can handle the higher voltage. If I had access to a pool to run my lego boats - I'd be there every week!
 I made it 
  November 19, 2014
Quoting matt rowntRee All three of those look like they are way too much fun! Always wanted an R/C boat until I watched my friend finally get his and immediately sink it in a lake. I'm sure it's still at the bottom along with his desire to do that again. XD I'm surprised at how fast that monohull can travel, figured it's displacement would counter any power thrown at it. Wonderfully pleasing result. Excellent!
Matt, I would have been surprised a year ago, too. Turns out, that total water resistance Rt = Rv + Rw, where Rv is viscous resistance and Rw is wave-making resistance. At "low" speeds, Rt is mostly Rv, which depends mainly on the hull's wetted surface area Aw. This hull clearly has no shortage of Aw. However, at "high" speeds, Rv becomes small potatoes compared to Rw, and a longer boat generates less Rw than a shorter one. Granted, Aw grows with length, but the reduction in Rw usually more than offsets the gain in Rv. Hence, it can be easier to push a longer boat to a given "high" speed than a shorter one. The divide between "low" and "high" speed also depends on length. All 3 of these boats top out at "high" speeds, mainly due to mounting Rw. If interested, I should have that section of the page done late tomorrow.
Jeremy McCreary
 I like it 
matt rowntRee
  November 18, 2014
All three of those look like they are way too much fun! Always wanted an R/C boat until I watched my friend finally get his and immediately sink it in a lake. I'm sure it's still at the bottom along with his desire to do that again. XD I'm surprised at how fast that monohull can travel, figured it's displacement would counter any power thrown at it. Wonderfully pleasing result. Excellent!
 I made it 
  November 18, 2014
Quoting Dr. Monster Awesome. You threw some heavy duty science at that boat! Killer MOC and write up.
Dr. Monster, Appreciate the comment and like. Being a geek who loves boats, I've had a lot of fun using real-world engineering to turn something that looked at first like a very bad idea (i.e., putting expensive electricals into a LEGO boat) into a boat I actually get to play with in open water.
 I made it 
  November 18, 2014
Quoting Yann (XY EZ) Impressive! I just hope it won't turn over ... 5/5
Yann, Many thanks for the comment and like. Not to worry. She's proven herself in water =much= rougher than that in the 1st video. Barring a torpedo hole in her side, it would take the swimming pool equivalent of a rogue wave to turn her over, as the hull itself is very stable, and she has a low center of gravity and plenty of freeboard to keep her decks dry.
 I like it 
  November 18, 2014
Awesome. You threw some heavy duty science at that boat! Killer MOC and write up.
  November 18, 2014
Impressive! I just hope it won't turn over ... 5/5
By Jeremy McCreary
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