Hull Speed
Hull speed is a phenomenon of displacement boats, and not of planing boats. Most sailing boats and all ships displace water—move it aside—as they plow through it. Planing craft, such as most motor boats, glide over the top like a surfboard. It takes more energy to push water aside than it does to slide over the top of it, and so displacement boats move at a more sedate pace than their lighter planing cousins. Some small sailing boats can be made to plane, but the general rule is that sailing boats are of the displacement type. Hull speed is usually an upper limit to the speed of displacement boats.* It is unsurprising that such a limit exists: we have seen how drag increases with speed, and so sooner or later drag will balance out the drive force and a sailboat will not be able to go faster. Yet there is a surprise in store for those of you who are not familiar with sailing: the hull speed of a given boat depends on its hull length at the waterline. It is not obvious from a simple consideration of drag why this should be so, but it is a wellattested fact, often quoted in the sailing literature, that the maximum natural speed of a displacement boat (in knots) is 4/3 the square root of waterline length in feet.
A key feature of the phenomenon, again well known to any sailor, is that hull speed has been reached when the bow wave of the boat lengthens to the waterline length. At lower speeds, there may be three or four complete waves seen to lap along the boat hull, but this number decreases as the boat picks up speed and reaches, pretty closely, one complete wave by the time the boat reaches her hull speed. It may be possible for her to go faster than hull speed, but this requires a disproportionate amount of effort. In other words, the hydrodynamic drag
* There is one trick by which a small displacement boat can exceed hull speed without expending enormous effort, and that is by surfing. Riding along the front of a wave is not the sole preserve of surfboards.
Figure 6.1. (a) Your hullspeed raft, viewed from above. Note the direction of motion. (b) When the bow wavelength is less than the distance between the long beams, drag is reduced compared to the case of (c). In (c) bow wavelength equals the distance between beams because the aft beam is more submerged. So hull speed is reached when hull length equals bow wavelength. Consequently, hull speed is limited by hull length.
Figure 6.1. (a) Your hullspeed raft, viewed from above. Note the direction of motion. (b) When the bow wavelength is less than the distance between the long beams, drag is reduced compared to the case of (c). In (c) bow wavelength equals the distance between beams because the aft beam is more submerged. So hull speed is reached when hull length equals bow wavelength. Consequently, hull speed is limited by hull length.
force that is acting to hold back the boat increases rapidly once hull speed is reached. My goal in this section is to explain to you, in simple physics terms, why these phenomena occur.
Which is why I have pressganged you into service onboard the undignified vessel illustrated in figure 6.1. She is a wooden raft with two long logs fore and aft that stretch way beyond her beam. These logs are not there to provide flotation, please note—we will suppose that the raft has enough buoyancy without them—but rather to illustrate hull speed. You set the primitive sail and drift off to the right. The forward log generates a bow wave which spreads out in the wake, as waves do. You notice something that you have seen many times before in other craft: the bow wave size (amplitude) increases as the vessel speed increases. This makes sense because the hull is pushing water aside, the displaced water has to go somewhere, and the faster you go, the more water is moved. So the wave size increases. Now you pick up speed, and so the wavelength of the wake, as observed alongside your hull, stretches out until exactly one wave lies between the two extended logs at bow and stern. The raft speed that gives rise to this condition is her top speed, you
Figure 6.2. Your hullspeed barge. Bow waves forward of the center of gravity, CG (open circle) exert a buoyancy force (vertical arrows) proportional to wave height that acts to rotate the barge hull counterclockwise. Similarly, waves aft of the CG act to rotate the hull clockwise. If we can assume that drag forces are proportional to counterclockwise torque (a dominant CCW torque means that the barge is climbing a hill created by its bow wave), we can show that hull speed occurs when bow wavelength equals hull length.
Figure 6.2. Your hullspeed barge. Bow waves forward of the center of gravity, CG (open circle) exert a buoyancy force (vertical arrows) proportional to wave height that acts to rotate the barge hull counterclockwise. Similarly, waves aft of the CG act to rotate the hull clockwise. If we can assume that drag forces are proportional to counterclockwise torque (a dominant CCW torque means that the barge is climbing a hill created by its bow wave), we can show that hull speed occurs when bow wavelength equals hull length.
find. It is clear why: the aft log is now submerged, and so experiences more drag than it did earlier, when there was no wave crest at the hull stern (see fig. 6.1). So, drag force peaks when bow wavelength equals hull length, in this simple example.
Now we are able to see where the old formula for hull speed comes from. The speed of a bow wave, or of any other surface water wave,1 is c where c2 = gk/2p. Here l is the water wavelength, and g is the constant acceleration due to gravity. Now the raft speed, v, equals the water wave speed, c, so that v = VgL/2p (since hull length, L, equals water wavelength at hull speed, as we just saw). Substitute numbers and we arrive at the old formula.
The ungainly raft has served her purpose, and you can now abandon her. The lesson learned is intuitive, and yet it gives us a basis for understanding quantitatively what hull speed is about. Now I can do another calculation, this time a little more realistic. The math is more involved (you need not wade though it), but the basic idea is again quite intuitive. Figure 6.2 shows the profile of a steepsided hull plowing through water and generating a bow wave, which oscillates along the line of the hull. This vessel is kept afloat by the buoyancy force, and we can see that the buoyancy force is going to be different at different points along the line of the hull because the wave height varies along the hull. Buoyancy that acts forward of the hull CG (shown in fig. 6.2) will create a counterclockwise torque that tends to twist the hull about the CG—trying to make it do a backflip. The buoyancy force aft of the CG produces a torque that acts in the clockwise sense. These two more or less cancel* but not quite. If the counterclockwise buoyancy torque is just a little bigger than the clockwise torque, the boat will tilt backwards, until her stern goes deep enough to generate a compensating torque. We would then be left with a boat that is going uphill, trying to reach the crest of her own bow wave.
Where am I going with all this? Roughly speaking, counterclockwise torque equates to uphill motion, and uphill motion leads to increased drag, for reasons that will soon be made clear. So, I am saying that increasing the unbalanced counterclockwise torque generated by a bow wave will increase drag. If this increase should suddenly take off at a certain speed, then we have found our hull speed. In fact, I can calculate the torque generated by the bow wave. You can see that as the bow wavelength changes, the torque will also change because the manner in which buoyancy force is distributed along the hull length changes with wavelength (fig. 6.2). The results of this calculation are plotted in figure 6.3. (For those interested, the math is provided in this endnote 2 in sufficient detail for you to reproduce the calculation.2) In figure 6.3 we see once again that drag force takes off for water wavelengths exceeding hull length, more or less.3
For simplicity, the hull of figure 6.2 was given vertical sides, but most boats don't have vertical sides, for a host of reasons. Recall that, in the Age of Sail, ships of the line were given a tumblehome cross section to deter boarders. Nowadays we are less likely to have to repel nefarious enemies swarming over our gunwales with cutlass in hand, casting a single bloodshot eye (the other being patched) in search of our gold doubloons. Hull sides are angled but the other way, with cross sections resembling a martini glass rather than a brandy glass. In plain language: more Vshaped. Here are some physics reasons for different hull cross sections.
——'Rounded hull bottoms are stronger than Vshaped hulls, but the latter will be deeper for the same displacement and so will better resist leeway.
*Just as well, because backflipping boats would be pretty uncomfortable.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Water wavelength / L
Figure 6.3. Hull speed is limited by drag. In the simple model described in the text, the drag increases with water wavelength, l, as shown (L is hull waterline length). Here, drag force is set arbitrarily to 1 at zero speed. If the bow wave is assumed to have constant amplitude, independent of speed, then drag changes with speed as shown. For a more realistic model, with bow wave amplitude increasing with speed, the curve looks similar. In this simple model, hull speed occurs at l « 1.2L because for longer waves (higher boat speed) the drag force becomes too strong.
•—A large deck area is desirable, but large hydrodynamic drag is not. For a hull of a given displacement, the choice of hull shape is constrained by the tradeoff between these two characteristics. •—'An angled hull—say one that is Vshaped—will have greater reserve buoyancy. That is, the righting moment will increase as the hull heels further and further. •—'During heeling, the waterline along an angled hull will not be symmetric about the longitudinal axis; the port side waterline length and shape will be different from that on the starboard side. This asymmetry can assist the boat to head up while heeling. Thus, even without aerodynamic assistance from her sails, a boat may automatically
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Water wavelength / L
Figure 6.3. Hull speed is limited by drag. In the simple model described in the text, the drag increases with water wavelength, l, as shown (L is hull waterline length). Here, drag force is set arbitrarily to 1 at zero speed. If the bow wave is assumed to have constant amplitude, independent of speed, then drag changes with speed as shown. For a more realistic model, with bow wave amplitude increasing with speed, the curve looks similar. In this simple model, hull speed occurs at l « 1.2L because for longer waves (higher boat speed) the drag force becomes too strong.
point to windward when heeling solely because of hydrodynamic forces acting on the hull. •—'Different angled hull shapes beneath the waterline assist with planing. For certain boats, such as racers, this is important because planing requires less displacement, less wetted area, and so less drag—and hence increased speed.
The physics of angled hull shapes casts an interesting light on the capabilities of some ancient ships. Certain ancient ships were built with a lot of overhang at the bow and stern, but this practice is usually thought to have been of little value for the old squareriggers because these ships were supposed to be nippy only when running or on a broad reach. Today, such hull shapes are utilized to increase hull speed while heeling because the waterline length is increased when the hull is heeled over. This lengthened waterline increases boat speed on a beam reach, for example. It seems plausible to suppose that ancient vessels with overlapping bows and sterns may have been capable of traveling across the wind at speed. Indeed, such a hull design offers no other advantage for these squarerigged vessels. (An overhanging bow and stern increases deck area, but for merchantmen—and in ancient times most of the sailing ships were merchant vessels because warships were oarpowered—deck area was not such a big deal. Volume of the hold was what mattered.) For a downwind point of sail, extended hull length above the waterline will increase pitching motion when traveling downwind; this is bad, and yet the overhanging bow and stern must have conferred some advantage or these ancient ships would not have been built this way.
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