## Principles of Yacht Design

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exist: diverging waves moving sidewards and transverse waves at right angles to the direction of motion, moving with the ship.

Locally, the situation is quite different and the waves arc highly dependent 011 the shape of the hull. Within distances of a few hull lengths, waves from all points on the hull surface will in theory contribute to the wave system. Some of the points are, however, more important than others, since the disturbance is larger. For a sailing yacht the high pressure regions at the bow and stern are dominant," and it is usually assumed that only two wave systems exist (see Fig 5.14).

There is a very simple relation between wavelength and travelling speed for surface waves. As can be seen in Fig 5.15 the speed is equal to 1.25 times the square root of the length. For example, a 7 m long wave will have a speed of 3.3 m/s.

Since the wave system travels with the yacht, at the same speed in the longitudinal direction, the length of the generated waves will depend on the yacht speed. If, for instance, the speed is 1.25 times the square root of the waterline length, the length of the wave is the same as the waterline length. A yacht with an Lwl of 7 m will thus have one wave crest at the bow and the next one at the stern if the speed is 3.3 m/s.

The speed dependence of the waves gives rise to an important phenomenon: interference. An illustration of this is given in Fig 5.15. If the wave crests from the bow system coincide with those from the stern, large waves will be created. On the other hand, if the bow wave crests coincide with troughs in the stern waves, the result is an attenuated wave. The first case is illustrated in (a) and (c), where the wavelength is half and equal to the waterline length, respectively. In (b) the wavelength is 2A of Lwl, and the waves are attenuated. In the last figure (d) the wavelength is larger than the Lwl. The second wave crest then occurs aft of the stern, which, when the speed increases, will move into a trough, giving the hull a large trim angle.

Fig 5.14 Local bow and stem wave systems

Diverging waves

Transverse wave

Fig 5.15 Interference between the bow and stern waves

-Bow wave system

----Stern wave system

x = wavelength

In each of the cases (a)-(d) a quantity Fn is given. This is the so called Froude number, which plays a similar role for the wave resistance as the Reynolds number does for viscous resistance. The Froude number is a dimensionless speed, where the velocity in metres per second is divided by the square root of the waterline length times the acceleration of gravity (see Fig 5.15). It is the Froude number that determines how many waves there are along the hull. For instance, at Fn = 0.40 there is one wave, at 0.2cS there are two, etc. The properties of the wave resistance curve are highly dependent on the Froude number, as we will sec below. The Froude number is therefore a very important quantity and we use it extensively in the following discussion, rather than the velocity in knots or metres per second. Using the simple definition, the Froude number can always be converted easily into these dimensional quantities.

Since the wave resistance occurs because energy is transported away in the waves, the amplification and attenuation due to interference between the wave systems must have some effect on the wave resistance curve. Thus, at speeds where there is an amplification of the waves the resistance must be relatively large, while the opposite must be true at speeds where there is an attenuation. The wave resistance curve thus exhibits what is normally referred to as humps and hollows (see Fig 5.16). it may be assumed that wave resistance increases with speed to the sixth power, but in addition there are the fluctuations due to interference.

Fig 5.16 Humps and hollows on the wave resistance curve

Wave resistance

Humps and hollows may be more or less pronounced, depending on the hull shape. For many sailing yachts they are very small in the lower speed range, but the last hump is still important. The slope of the curve gets very large just below this speed and to get over the hump is difficult. If this can be achieved, however, the increase in resistance becomes more gradual, and the hull enters the semi-planing speed range. Catamarans and extremely light canoes and dinghies may accomplish this even beating to windward, while the lightest displacement hulls, like the America's Cup yachts enter the semi-planing range in the downwind legs. Most displacement hulls cannot, however, pass the barrier at the last hump.

According to the discussion above, the largest hump in the resistance curve should occur when the wavelength is equal to the wateiiine length, at Fn = 0.40, but in practice it occurs at a higher Froude number, ie at a higher speed. This is because the overhangs at the bow and stern cause the distance between the bow and stern waves to be larger than the nominal waterline length. The last hump thus occurs normally at a Froude number of about 0.5. Heavy displacement hulls cannot reach this value, except under special conditions, as when sailing in heavy following seas. Normally, it is difficult to reach higher Froude numbers than 0.45 for this kind of hull. The YD-40 has a waterline length of 10 m, so it is difficult to reach higher speeds than 0.45 *v!0 g = 4.5 m/s, corresponding to 8.7 knots. This is also apparent from the resistance curve of Fig 5.3. A hull twice as long would reach a speed of 0.45 V20 g = 6.3 m/s (12.2 knots). The speed has thus increased by a factor of V2.

It should be mentioned that in most literature 011 sailing theory an older quantity, the so-called 'speed length ratio" is used instead of the Froude number. This is defined as the speed in knots, divided by the square root of the waterline length in feet. In fact it differs only by a constant from the Froude number, but its disadvantages are that it is not dimensionless and that it is not based on metric quantities. Conversion between the two numbers can be made easily using the formula: Froude number = 0.30 • (speed length ratio).

### Influence of hull shape on wave resistance

Very extensive series of tests with models of sailing yachts have been carried out by Professor J Gerritsma and his co-workers at the Delft University of Technology in the Netherlands. The first series was run during the 1970s and comprised 22 models with a systematic variation of five different hull parameters: Lwl/Bwl, Bxvl/Tc, Cp, LCB and Lwl/Vc1/3. All hulls were derived from a Frans Maas designed parent model, a medium displacement, contemporary ocean racer. Its body-plan is shown in Fig 5.17(top). During the 1980s it became apparent, however, that an extension of the series to lighter displacements was

Fig 5.17 Body plans of the two Delft parent models

Parent model, medium to heavy displacement (No 1)

Parent model, light displacement (No 25)

0 0

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### Responses

• VAPPU PAASIO
What is hump and hollows relate to bow wave?
4 years ago