Resistance and Propulsion

Chapter 12

Resistance and Propulsion Chapter Outline 12.1 Hydrostatically Supported Ships and Boats 12.1.1 Froude’s Analysis Procedure 12.1.2 Components of Calm Water Resistance 12.1.3 Methods of Resistance Evaluation 12.1.4 Propulsive Coefficients 12.1.5 The Influence of Rough Water 12.1.6 Restricted Water Effects 12.1.7 Air Resistance

314 315 316 326 339 341 343 344

Ships and boats are designed to possess a variety of characteristics and forms to suit the intended operating profile and function that they are required to perform. Indeed, the desired operating profile of an intended ship together with the required cargo or passenger capacity form two of the key design parameters of a ship and the ship type, hull design and the propulsion system must satisfy these fundamental requirements. Fig. 12.1 illustrates the more common ship types and forms that are adopted to suit different needs and requirements. As can be seen from Fig. 12.1, there are four principal support mechanisms for marine vessels. These are the hydrostatic, hydrodynamic, aerostatic, and aerodynamic support mechanisms. The majority of merchant ship forms belong to the hydrostatic support class of vessels where Archimedes principle applies: the weight of the displaced water equals the weight of the ship. While this is generally true, some relatively minor effects of the pressure distribution along the bottom of the hull due to the forward motion of the ship in deep water cause a slight modification to the draft derived from purely hydrostatic principles. This is also seen in a more exaggerated form when the ship enters shallow water and the phenomenon of squat manifests itself. Submarines also conform to the class of hydrostatic marine vehicles. In the case of multihull ships and craft, many operate in the domain of hydrostatic support vessels and others utilize, in part or wholly, the hydrodynamic physical characteristics of planing craft. The degree to which they tend to conform to either of the hydrostatic or hydrodynamic support classes of vessel is largely a function of design speed. Within the multihull class of marine vehicles, a number of different configurations Marine Propellers and Propulsion. https://doi.org/10.1016/B978-0-08-100366-4.00012-2 © 2019 Elsevier Ltd. All rights reserved.

12.1.8 Multihull Configurations 12.2 Hydrodynamic Support Propulsion 12.2.1 Planing Craft 12.2.2 Hydrofoil Craft 12.3 Aerostatically Supported Vehicle Propulsion 12.4 Aerodynamic Support Propulsion Bibliography

345 349 349 353 358 363 363

are found. The Small Waterplane Area (SWATH) are one class of ship while catamarans, trimarans, and pentamarans form another set of ship types having two, three, and five hulls, respectively. Hydrodynamic support vessels have two principal classes. The first and perhaps most widely deployed is the planing hull, while the second is the class of hydrofoil craft. Indeed, this latter class subdivides into two types: the totally submerged and surface piercing hydrofoil classes. Vessels and craft of this type frequently possess higher design speeds than those belonging to the hydrostatic classes; however, this is not always the case. Aerostatic support most commonly embraces the set of hovercraft vehicles, which rely for their support on an air cushion. These subdivide into those having a flexible skirt deployed around the perimeter of the hull and the side-wall types of which the former is an amphibious vehicle and can operate equally on land and at sea, albeit when at sea with a degree of shelter. The latter type is a marine vehicle in which the side longitudinal walls act as a barrier to air leakage along the sides of the craft while the air cushion is maintained forward and aft by a flexible skirt. Aerodynamic support craft are normally seen in the context of Ekranoplans, as was their original Russian name, or wing-in-ground effect (WIGs) craft. Although in many ways visually resembling an airplane, they rely on flying close to the sea surface in order to enhance the aerodynamic characteristics of their wing sections and, thereby, unlike a seaplane cannot maintain higher altitude flight. Such a craft when first observed was initially christened the Caspian Sea Monster due to the location where it was seen and perhaps a lack of appreciation of its characteristics and purpose. 313

314 Marine Propellers and Propulsion

Ship type

Hydrostatic support

Displacement

Multihulls

Hydrodynamic support

Planing hulls

Aerostatic support

Hydrofoils

Hovercraft

Aerodynamic support

Ekranoplans (WIGs)

Submarines Flexible skirt

Side-wall

SWATH Catamarans Trimarans Pentamarans FIG. 12.1 A classification of ship and hull types.

12.1 HYDROSTATICALLY SUPPORTED SHIPS AND BOATS Prior to the mid-19th century, comparatively little was known about the laws governing the resistance of ships and the power required to develop a particular speed. Brown (1983) gives an account of the problems of that time and depicts the role of William Froude, who can justly be considered to be the father of ship resistance studies. An extract from Brown’s account reads as follows: …In the late 1860s Froude was a member of a committee of the British Association set up to study the problems of estimating the power required for steamships. They concluded that model tests were unreliable and often misleading and that a long series of trials would be needed in which actual ships were towed and the drag force measured. Froude wrote a minority report pointing out the cost of such a series of trials and the fact that there could never be enough carried out to study all possible forms. He believed that he could make sense from the results of model tests and carried out a series of experiments in the River Dart to prove his point. By testing models of two different shapes and three different sizes he was able to show that there were two components of resistance, one due to friction and the other to wave-making and that these components obeyed different scaling laws. Froude was now sufficiently confident to write to Sir Edward Reed (Chief Constructor of the Navy) on 24 April 1868, proposing that an experiment tank be built and a two year programme of work be carried out. After due deliberation, in February 1870, Their Lordships approved the expenditure of £2000 to build the world’s first ship model experiment tank at Torquay and to run it for two years. The first experiment was run in March 1872 with a model of HMS Greyhound. Everything was new. The carriage was pulled along the tank at constant speed by a steam

engine controlled by a governor of Froude design. For this first tank he had to design his own resistance dynamometer and followed this in 1873 by his masterpiece, a propeller dynamometer to measure thrust, torque and rotational speed of model propellers. This dynamometer was made of wood, with brass wheels and driving bands made of leather boot laces. It continued to give invaluable service until 1939 when its active life came to an end with tests of propellers for the fast minelayers… …William Froude died in 1879, having established and developed a sound approach to hull form design, made a major contribution to the practical design of ships, developed new experiment techniques and trained men who were to spread the Froude tradition throughout the world. William was succeeded as Superintendent AEW by Edmund Froude, his son, whose first main task was to plan a new establishment since the Torquay site was too small and the temporary building was nearing the end of its life. Various sites were considered but the choice fell on Haslar, Gosport, next to the Gunboat Yard, where AEW, then known as the Admiralty Marine Technology Establishment (Haslar) or AMTE(H), remains to this day. A new ship tank, 400 ft long, was opened in 1877… ….Edmund was worried about the consistency of results being affected by the change to Haslar. He was a great believer in consistency, as witness a remark to Stanley Goodall, many years later, “In engineering, uniformity of error may be more desirable than absolute accuracy”. As Goodall said “That sounds a heresy, but think it over”. Froude took two measures to ensure consistent results; the first, a sentimental one, was to christen the Haslar tank with water from Torquay, a practice repeated in many other tanks throughout the world. The flask of Torquay water is not yet empty – though when Hoyt analysed it in 1978 it was full of minute animal life! The more practical

Resistance and Propulsion Chapter 12

precaution was to run a full series of tests on a model of HMS Iris at Torquay just before the closure and repeat them at Haslar. This led to the wise and periodical routine of testing a standard model, and the current model, built of brass in 1895, is still known as Iris, though very different in form from the ship of that name. Departures of the Iris model resistance from the standard value are applied to other models in the form of the Iris Correction. With modern water treatment the correction is very small but in the past departures of up to 14.5 per cent have been recorded, probably due to the formation of long chain molecules in the water reducing turbulence in the boundary layer. Another Froude tradition, followed until 1960, was to maintain water purity by keeping eels in the tanks. This was a satisfactory procedure, shown by the certification of the tank water as emergency drinking water in both World Wars, and was recognised by an official meat ration, six pence worth per week, for the eels in the Second World War!…

In contrast, the world’s first commercially run ship model experiment tank was built in 1881 by William Denny & Brothers of Dumbarton within their shipyard. This tank had a length of 73 m and was 6.7 m deep. Although closed in 1963, this tank forms a central part of the Scottish Maritime Museum located in Dumbarton, Scotland. So much then for the birth of the subject as we know it today and the start of the tradition of “christening” a new towing tank from the water of the first tank, sited at Froude’s home at Chelston Cross at Crockington near Torquay.

12.1.1

315

FIG. 12.2 Comparison of a ship and its model’s specific resistance curves.

flow and weather acting on the above-water surfaces. This second component was termed the frictional resistance RF. Froude’s major contribution to the ship resistance problem, which has remained useful to the present day, was his conclusion that the two sources of resistance might be separated and treated independently. In this approach, Froude suggested that the viscous resistance could be calculated from friction data while wave-making resistance RW could be deduced from the measured total resistance RT and the calculated frictional resistance RF as follows:

Froude’s Analysis Procedure

RW ¼ RT  RF

William Froude, Froude (1955), recognized that ship models of geometrically similar form would create similar wave systems, albeit at different speeds. Furthermore, he showed that the smaller models had to be run at slower speeds than the larger models to obtain the same wave pattern. His work showed that for a similarity of wave pattern between two geometrically similar models of different size, the ratio of the speeds of the models was governed by the relationship rffiffiffiffiffi V1 L1 ¼ (12.1) V2 L2

To provide data for calculating the value of the frictional component, Froude performed his famous experiments at the Admiralty-owned model tank at Torquay. These experiments entailed towing a series of planks ranging from 10 to 50 ft in length, having a series of surface finishes of shellac varnish, paraffin wax, tin foil, graduation of sand roughness and other textures. Each of the planks was 19 in. deep and 3 16 in. thick and was ballasted to float on its edge. Although the results of these experiments suffered from errors due to temperature differences, slight bending of the longer planks and laminar flow on some of the shorter planks, Froude was able to derive an empirical formula, which would act as a basis for the calculation of the frictional resistance component RF in Eq. (12.2). The relationship Froude derived took the form

By studying the comparison of the specific resistance curves of models and ships, Froude noted that they exhibited a similarity of form although the model curve was always greater than that for the ship (Fig. 12.2). This led Froude to the conclusion that two components of resistance were influencing the performance of the vessel and that one of these, the wave-making component RW, scaled with V/√ L and the other did not. This second component, which is due to viscous effects, derives principally from the flow of the water around the hull but also is influenced by the air

RF ¼ fSV n

(12.2)

(12.3)

in which the index n had the constant value of 1.825 for normal ship surfaces of the time and the coefficient f varied with both length and roughness, decreasing with length but increasing with roughness. In Eq. (12.3), S is the wetted surface area.

316 Marine Propellers and Propulsion

Because of this work, Froude’s basic procedure for calculating the resistance of a ship is as follows: 1. Measure the total resistance of the geometrically similar model RTM in the towing tank at a series of speeds embracing the design V/√ L of the full-size vessel. 2. From this measured total resistance, subtract the calculated frictional resistance values for the model RFM to derive the model wave-making resistance 3. Calculate the full-size frictional resistance RFS and add this to the full-size wave-making resistance RWS, scaled from the model value, to obtain the total full-size resistance RTS   DS RTS ¼ RWM (12.4) + RFS DM In Eq. (12.4), the suffixes M and S denote model- and full scale, respectively, and D is the displacement. The scaling law of the ratio of displacements derives from Froude’s observations that when models of various sizes, or a ship and its model, were run at corresponding speeds dictated by Eq. (12.1), their resistances would be proportional to the cubes of their linear dimensions or, alternatively, their displacements. This was, however, an extension of a law of comparison, which was known at that time. Froude’s law, Eq. (12.1), states that the wave-making resistance coefficients of two geometrically similar hulls of different lengths are the same when moving at the same V/√ L value, V being the ship or model speed and L being the waterline length. The ratio V/√ L is termed the speedlength ratio and is of course dimensional; however, the dimensionless Froude number can be derived from it to give Fn ¼

V √ ðgLÞ

(12.5)

in which g is the acceleration due to gravity (9.81 m/s2). Care needs to be exercised in converting between the speed length ratio and the Froude number: Fn ¼ 0:3193 Fn ¼ 0:1643

V √L V

√L

where V is in m=s; L is in meters: where V is in knots; L is in meters:

Froude’s work with his plank experiments was carried out prior to the formulation of the Reynolds number criteria and this undoubtedly led to errors in his results: for example, the laminar flow on the shorter planks. Using dimensional analysis, after the manner shown in Chapter 6, it can readily be shown today that the resistance of a body moving on the surface, or at an interface of a medium, can be given by ( ) R VLr V V s p0  pv , , , ¼f , (12.6) rV 2 L2 m √ ðgLÞ a grL2 rV 2

In this equation the left-hand side term is the resistance coefficient CR, while on the right-hand side of the equation: The The The The The

1st term is the Reynolds number Rn. 2nd term is the Froude number Fn (Eq. 12.5). 3rd term is the Mach number Ma. 4th term is the Weber number We. 5th term is the Cavitation number.

For the purposes of ship propulsion, the 3rd and 4th terms are not generally significant and can, therefore, be neglected. Hence, Eq. (12.6) reduces to the following for all practical ship purposes: CR ¼ f fRn , Fn , s0 g

(12.7)

In which, r is the density of the water m is the dynamic viscosity of the water p0 is the free stream undisturbed pressure pv is the water vapor pressure.

12.1.2 Components of Calm Water Resistance In the case of a vessel, which is undergoing steady motion at slow speeds, that is where the ship’s weight balances the displacement upthrust without the significant contribution of hydrodynamic lift forces, the components of calm water resistance can be broken down into the contributions shown in Fig. 12.3. From this figure, it is seen that the total resistance can be decomposed into two primary components: pressure and skin friction resistance. These can then be broken down further into more discrete components. In addition to these components, there is of course the added resistance due to rough weather and air resistance: these are, however, dealt with separately in Sections 12.1.5 and 12.1.7, respectively. Each of the components shown in Fig. 12.3 can be studied separately, if it is remembered that each will have an interaction on the others and, therefore, as far as the ship is concerned, need to be considered in an integrated way.

12.1.2.1 Wave-Making Resistance RW Lord Kelvin (1904a,b,c) studied the problem of the wave pattern caused by a moving pressure point. He showed that the resulting wave system comprises a divergent set of waves together with a transverse system, which are approximately normal to the direction of motion of the moving point. Fig. 12.4 shows the system of waves so formed. The pattern of waves is bounded by two straight lines, which

Resistance and Propulsion Chapter 12

317

FIG. 12.3 Components of ship resistance.

FIG. 12.4 Wave pattern induced by a moving-point pressure in calm water.

in deep water, are at an angle f to the direction of motion of the point, where f is given by   1 f ¼ sin 1 ¼ 19:471 degrees 3 The interference between the divergent and transverse systems gives the observed wave their characteristic shape and, since both systems move at the same speed, the speed of the vessel, the wavelength l between successive crests is l¼

2p 2 V g

wave system. In this model the bow pressure field will create a crest near the bow, observation showing that this occurs at about l/4 from the bow, while the suction field will introduce a wave trough at the stern: both wave systems have a wavelength l ¼ 2pV2/g. Fig. 12.5B shows a photograph of the comparable wave system generated by a twin-screw passenger ferry. The divergent component of the wave system derived from the bow and the stern generally does not exhibit any strong interference characteristics. This is not the case, however, with the transverse wave systems created by the vessel, since these can show a strong interference characteristic. Consequently, if the bow and stern wave systems interact such that they are in phase, a reinforcement of the transverse wave patterns occurs at the stern and large waves are formed in that region. For such a reinforcement to take place, Fig. 12.6A, the distance between the first crest at the bow and the stern must be an odd number of halfwavelengths as follows: l l L  ¼ k where k ¼ 1, 3, 5,…, ð2j + 1Þ 4 2

(12.8)

The height of the wave systems formed decreases fairly rapidly as they spread out laterally because the energy contained in the wave is constant and it has to be spread out over an increasingly greater length. More energy is absorbed by the transverse system than by the divergent system, and this disparity increases with increasing speed (Fig. 12.4). A real ship form, however, cannot be represented adequately by a single moving pressure point as originally analyzed by Kelvin. The simplest representation of a ship, Fig. 12.5A, is to place a moving pressure field near the bow in order to simulate the bow wave system together with a moving suction field near the stern to represent the stern

with j ¼ 0, 1,2, 3, … From which 4 l 2pV 2 ¼ ¼ ¼ 2pðFn Þ2 2k + 1 L gL that is,

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Fn ¼ pð2k + 1Þ

(12.9)

For the converse case, when the bow and stern wave systems cancel each other, and hence produce a minimum

318 Marine Propellers and Propulsion

FIG. 12.5 (A) Simple ship wave pattern representation by two pressure points. (B) Photograph of the wave pattern developed by a ferry. (Source: Unknown.)

FIG. 12.6 Wave reinforcement and cancellation at stern: (A) wave reinforcement at stern and (B) wave cancellation at stern.

Resistance and Propulsion Chapter 12

319

wave-making resistance condition, the distance L  l/4 must be an even number of half-wave lengths (Fig. 12.6B): l l L  ¼ k where k ¼ 2,4, 6, …, ð2jÞ 4 2 with j ¼ 0, 1,2, 3, … Hence,

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Fn ¼ pð2k + 1Þ

as before, but with k even in this case. Consequently, from Eq. (12.9), Table 12.1 can be derived, which for this particular model of wave action identifies the Froude numbers at which reinforcement (humps) and cancellation (hollows) occur in the wave-making resistance. Each of the conditions shown in Table 12.1 relates sequentially to maximum and minimum conditions in the wave-making resistance curves. The “humps” occur because the wave profiles and hence the wave-making resistance are at their greatest in these conditions while the converse is true in the case of the “hollows.” Fig. 12.7 shows the general form of the wave-making resistance curve together with the schematic wave profiles associated with the various values of k. The hump associated with k ¼ 1 is normally termed the main hump since this is the most pronounced hump and occurs at the highest speed. The second hump, k ¼ 3, is called the prismatic hump since it is influenced considerably by the prismatic coefficient of the hull form. The derivation of Fig. 12.7 and Table 12.1 relies on the assumptions made in their formulation: for example, a single pressure and suction field, bow wave crest at l/4; stern trough exactly at the stern, etc. Clearly, there is some latitude in all these assumptions and, therefore, the values of Fn at which the humps and hollows occur vary. In the case of warship, hull forms the distance between the first crest of the bow wave and the trough of the stern wave has been shown to approximate well to 0.9L and, therefore, this could be

FIG. 12.7 Form of wave-making resistance curve.

TABLE 12.2 Effect of Difference in Calculation Basis on Prediction of Hump and Hollow Froude Numbers k

1

2

3

4

5

L  l/4 basis

0.46

0.36

0.30

0.27

0.24

0.9L basis

0.54

0.38

0.31

0.27

0.24

TABLE 12.1 Froude Numbers Corresponding to Maxima and Minima in the Wave-Making Resistance Component K

Fn

Description

1

0.461

1st hump in Rw curve

2

0.357

1st hollow in Rw curve

3

0.301

2nd hump in Rw curve

4

0.266

2nd hollow in Rw curve

5

0.241

3rd hump in Rw curve

⋮

⋮

⋮

used to rederive Eq. (12.9). This would then derive slightly differing values of Froude numbers corresponding to the “humps” and “hollows.” Table 12.2 shows these differences, and it is clear that the greatest effect is formed at low values of k. Fig. 12.7, for this and the other reasons cited, is not unique but is shown here to provide awareness and guidance on wave-making resistance variations. A better approximation to the wave form of a vessel can be made by considering the ship as a solid body rather than two point sources. Wigley initially used a simple parallel body with two pointed ends and showed that the resulting

320 Marine Propellers and Propulsion

FIG. 12.8 Components of wave systems for a simple body.

wave pattern along the body could be approximated by the sum of five separate disturbances of the surface (Fig. 12.8). From this figure, it is seen that a symmetrical disturbance corresponds to the application of Bernoulli’s theorem with peaks at the bow and stern and a hollow, albeit with cusps at the start and finish of the parallel middle body, between them. Two wave forms starting with a crest are formed by the action of the bow and stern while a further two wave forms commencing with a trough originates from the shoulders of the parallel middle body. The sum of these five wave profiles is shown at the bottom of Fig. 12.8 and compared with a measured profile, which shows good general agreement. Since the wavelength l varies with speed and the points at which the waves originate are fixed, it is easy to understand that the whole profile of the resultant wave form will change with speed length ratio. This analysis was extended by Wigley for a more realistic hull form comprising a parallel middle body and two convex extremities. Fig. 12.9 shows the results in terms of the same five components and the agreement with the observed wave form. Considerations of this type lead to attempting to design a hull form to produce a minimum wave-making resistance using theoretical methods. The basis of these theories is developed from Kelvin’s work on a traveling pressure source; however, the mathematical boundary conditions are difficult to satisfy with any degree of

precision. Results of work based on these theories have been mixed in terms of their ability to represent the observed wave forms.

12.1.2.2 The Contribution of the Bulbous Bow Bulbous bows are today commonplace in the design of ships. Although their origins are to be found before the turn of the last century, the first application appears to have been in 1912 by the US Navy. The general use in merchant applications appears to have waited until the late 1950s and early 1960s. The basic theoretical work on their effectiveness was carried out by Wigley (1936) in which he showed that if the bulb was nearly spherical in form, then the acceleration of the flow over the surface induces a low-pressure region which can extend toward the water surface. This lowpressure region then reacts with the bow pressure wave to cancel or reduce the effect of the bow wave. The effect of the bulbous bow, therefore, is to cause a reduction, in the majority of cases, of the effective power required to propel the vessel, the effective power PE being defined as the product of the ship resistance and the ship speed at a particular condition in the absence of the propeller. Fig. 12.10 shows a typical example of the effect of a bulbous bow from which it can be seen that a bulb is, in general, beneficial above a certain speed and gives a penalty at low speeds. This is due to the balance between the bow pressure

Resistance and Propulsion Chapter 12

321

FIG. 12.9 Wave components for a body with convex ends and a parallel middle body.

2. The influence of the upper part of the bulb and its intersection with the hull to introduce a downward flow component near the bow. 3. An increase in the frictional resistance caused by the surface area of the bulb. 4. A change in the propulsion efficiency induced by the effect of the bulb on the global hull flow field. 5. The change induced in the wave-breaking resistance.

FIG. 12.10 Influence of a bulbous bow of the effective power requirement.

wave reduction effect and increase in frictional resistance caused by the presence of the bulb on the hull. The effects of the bulbous bow in changing the resistance and delivered power characteristics can be attributed to several causes. The principal of these are as follows: 1. The reduction of bow pressure wave due to the pressure field created by the bulb and the consequent reduction in wave-making resistance.

The shape of the bulb is particularly important in determining its beneficial effect. The optimum shape for a particular hull depends on the Froude number associated with its operating regime and bulbous bows tend to give good performance over a narrow range of ship speeds. Consequently, they are most commonly found on vessels, which operate at clearly defined speeds for much of their time. The actual bulb form, Fig. 12.11, is defined in relation to a series of form characteristics as follows: 1. length of projection beyond the forward perpendicular; 2. cross-sectional area at the forward perpendicular (ABT); 3. height of the centroid of cross-section (ABT) from the base line (hB); 4. bulb section form and profile; 5. transition of the bulb into the hull. With regard to section form, many bulbs today are designed with noncircular forms so as to minimize the effects of

322 Marine Propellers and Propulsion

FIG. 12.12 Flow around an immersed transom stern.

FIG. 12.11 Bulbous bow definition.

slamming in poor weather. There is, however, still work to be done in relating bulb form to power saving and many contemporary studies are proceeding. For current design purposes, reference can be made to the work of Inui (1962), Yim (1974), and Schneekluth (1987). In addition to its hydrodynamic behavior, the bulb also introduces a further complication into resistance calculations. Traditionally, the length along the waterline has formed the basis of many resistance calculation procedures because it is basically the fundamental hydrodynamic dimension of the vessel. The bulbous bow, however, normally projects forward of the forward point of the definition of the waterline length and since the bulb has a fundamental influence on some of the resistance components, there is a case for redefining the basic hydrodynamic length parameter for resistance calculations. Bulbous bows are only really effective over a limited range of draft conditions due to their interaction with the bow pressure wave. Consequently, when extreme changes in draft are required, such as with a tanker between loaded and ballast conditions, then cylindrical bow forms are contemplated: these being somewhat of a two-dimensional approximation to a conventional three-dimensional bulbous bow form.

12.1.2.3 Transom Immersion Resistance and Duck Tails Unlike several decades ago when cruiser and counter sterns were common place, in modern ships a transom stern is now normal practice. If at the design powering condition a portion of the transom is immersed, this leads to separation taking place as the flow from under the transom passes out astern of the hull as seen in Fig. 12.12. The resulting vorticity that takes place in the separated flow behind the transom leads to a pressure loss behind the hull, which needs to be taken into account in the analysis procedures.

The magnitude of this resistance is generally small and, of course, vanishes when the lower part of the transom is dry. Transom immersion resistance is largely a pressure resistance that is scale independent. Duck tails are a fixed appendage to the hull, which forms an integral part of the hull and which is a full width transom extension. Moreover, the duck tail differs from, for example, a flap, in that it is an extension to the buttock lines of the ship and frequently the duck tail lines flow upwards in contrast to a flap’s downward inclination. Hamalainen and van Heerd (1998) discuss the application of duck tails to fast monohull passenger ferries, which can effectively reduce the height of the stern wave and thereby, give a propulsion efficiency advantage.

12.1.2.4 Viscous Form Resistance The total drag on a body immersed in a fluid and traveling at a particular speed is the sum of the skin friction components, which is equal to the integral of the shear stresses taken over the surface of the body, and the form drag, which is the axial component of the integral of the normal forces acting on the body. In an inviscid fluid, the flow along any streamline is governed by Bernoulli’s equation and the flow around an arbitrary body is predictable in terms of the changes between pressure and velocity over the surface. In the case of Fig. 12.13A, this leads to the net axial force in the direction of motion being equal to zero since in the two-dimensional case shown in Fig. 12.13A, I p cos yds ¼ 0 (12.10) When moving in a real fluid, a boundary layer is created over the surface of the body, which, in the case of a ship, will be turbulent and is also likely to separate at some point in the afterbody. The presence of the boundary layer and its growth along the surface of the hull modifies the pressure distribution acting on the body from that of the potential or inviscid case. As a consequence, the left-hand side of

Resistance and Propulsion Chapter 12

FIG. 12.13 Viscous form resistance calculation: (A) inviscid flow case on an arbitrary body and (B) pressures acting on shell plate of a ship.

Eq. (12.10) can no longer equal zero and the viscous form drag RVF is defined for the three-dimensional case of a ship hull as n X RVF ¼ pk cos yk d Sk (12.11)

This equation, known as the Schoenherr line, was adopted by the American Towing Tank Conference (ATTC) in 1947 and to make the relationship applicable to the hull surfaces of new ships an additional allowance of 0.0004 was added to the smooth surface values of CF given by Eq. (12.12). By 1950 there was a variety of friction lines for smooth turbulent flows in existence and all, with the exception of Froude’s work, were based on Reynolds number. Phillips-Birt (1970) provides an interesting comparison of these friction formulations for a Reynolds number of 3.87  109, which is applicable to ships of the length of the former trans-Atlantic liner Queen Mary and is rather less than that for the large supertankers: in either case lying way beyond the range of direct experimental results. Indeed, the magnitude of the correlation allowance is striking between the two Schoenherr formulations: the allowance is some 30% of the basic value. In the general application of the Schoenherr line, some difficulty was experienced in the correlation of large and small model test data and wide disparities in the correlation factor CA were found to exist upon the introduction of all welded hulls. These shortcomings were recognized prior to the 1957 International Towing Tank Conference (ITTC) and a modified line was accepted. The ITTC-1957 line is expressed as CF ¼

k¼1

In Fig. 12.13B, the hull has been split into n elemental areas dSk and the contribution of each normal pressure pk acting on the area is summed in the direction of motion. Eq. (12.11) is a nontrivial equation to solve since it relies on the solution of the boundary layer over the vessel and this is a solution, which can only be approached using significant computational resources for comparatively simple hull forms. Consequently, for many practical purposes the viscous form resistance is normally accounted for using empirical or pseudoempirical methods.

12.1.2.5 Naked Hull Skin Friction Resistance

323

0:075 ð log 10 Rn  2:0Þ2

(12.13)

and this formulation, which is in use with most ship model basins, is shown together with the Schoenherr line in Fig. 12.14. It can be seen that the present ITTC line gives slightly higher values of CF at the lower Reynolds numbers than the Schoenherr line, while both lines merge toward the higher values of Rn. The frictional resistance RF derived from the use of either the ITTC or ATTC lines should be viewed as an instrument of the calculation process rather than producing a definitive magnitude of the skin friction associated with a particular ship. As a consequence, when using a Froude analysis

The original data upon which to calculate the skin friction component of resistance were that provided by Froude in his plank experiments conducted at Torquay. These data, as discussed previously, were subject to error and in 1932 Schoenherr reevaluated Froude’s original data in association with other work in the light of the Prandtl-von Karman theory. This analysis resulted in an expression for the friction coefficient CF as a function of Reynolds number Rn and the formulation of a skin friction line, applicable to smooth surfaces, of the following form: 0:242 √CF

¼ log ðRn  CF Þ

(12.12) FIG. 12.14 Comparison of ITTC (1957) and ATTC (1947) friction lines.

324 Marine Propellers and Propulsion

based on these, or indeed any friction line data, it is necessary to introduce a correlation allowance into the calculation procedure. This allowance is denoted by CA and is defined as CA ¼ CT ðmeasuredÞ  CT ðestimatedÞ

(12.14)

In this equation, as in the previous equation, the resistance coefficients CT, CF, CW, and CA are nondimensional forms of the total, frictional, wave-making, and correlation resistances, and are derived from the basic resistance summation RT ¼ R W + RV 1 1 by dividing this equation throughout by rVs2 S, rVs2 L2 , or 2 2 1 2 2=3 rV r according to convenience. 2 s

12.1.2.6 Appendage Skin Friction The appendages of a ship such as the rudder, bilge keels, stabilizers, sea chest openings, duct head-box arrangements, transverse thruster orifices, and so on introduce additions to the skin friction resistance above that of the naked hull resistance. At ship scale the flow over the appendages is turbulent, whereas at model scale it would normally be laminar unless artificially stimulated, which, in itself, may introduce a flow modeling problem. In addition, many of the hull appendages are working wholly within the boundary layer of the hull and since the model is run at Froude identity, not Reynolds identity, this again presents a problem. Consequently, the prediction of appendage resistance needs care if significant errors are to be avoided. The calculation of this aspect is further discussed in Section 12.1.3. In addition to the skin friction component of appendage resistance, if the appendages are located on the vessel close to the surface, then they will also contribute to the wavemaking component. This is because a lifting body close to a free surface, due to the pressure distribution around the body, will create a disturbance on the free surface. As a consequence, the total appendage resistance can be expressed as the sum of the skin friction and surface disturbance effects as follows: RAPP ¼ RAPPðFÞ + RAPPðW Þ

(12.15)

where RAPP(F) and RAPP(W) are the frictional and wavemaking components, respectively, of the appendages. In most cases of practical interest to the merchant marine RAPP(W) ’ 0 and can be neglected. This is not the case, however, for some naval applications, such as where submarine hydrofoils are operating just submerged near the surface.

12.1.2.7 Viscous Resistance Fig. 12.2 defines the viscous resistance as being principally the sum of the form resistance, the naked hull skin friction, and the appendage resistance. In the discussion on the viscous form resistance, it was said that its calculation by analytical means was an involved matter and for many hulls of a complex shape this is difficult to achieve with any degree of accuracy. Hughes (1954) attempted to provide a better empirical foundation for the viscous resistance calculation by devising an approach, which incorporated the viscous form resistance and the naked hull skin friction. To form a basis for this approach, Hughes undertook a series of resistance tests using planks and pontoons for a range of Reynolds numbers up to a value of 3  108. From the results of this experimental study, Hughes established that the frictional resistance coefficient CF could be expressed as a unique inverse function of aspect ratio AR and, moreover, that this function was independent of Reynolds number. The function derived from this study had the form: 

1 CF ¼ CF jAR¼∞  f AR



in which the term CF jAR¼∞ is the frictional coefficient relating to a two-dimensional surface; that is, one having an infinite aspect ratio. This function permitted Hughes to construct a twodimensional friction line defining the frictional resistance of turbulent flow over a plane smooth surface. This took the form CF jAR¼∞ ¼

0:066 ½ log 10 Rn  2:032

(12.16)

Eq. (12.16) quite naturally bears a close similarity to the ITTC-1957 line expressed by Eq. (12.13). The difference, however, is that the ITTC and ATTC lines contain some three-dimensional effects, whereas Eq. (12.16) is defined as a two-dimensional line. If it is plotted on the same curve as the ITTC line, it will be found that it lies just below the ITTC line for the full range of Rn and in the case of the ATTC line it also lies below it except for the very low Reynolds numbers. Hughes proposed the calculation of the total resistance of a ship using the basic relationship CT ¼ CV + CW in which CV ¼ CF jAR¼∞ + CFORM ,

Resistance and Propulsion Chapter 12

CT CV CF

Low Vs ~0 Cw -

325

CT

A Cw

B

Vis c

Cform

Bas

CV

line

CF

ou s

ic 2

res

is t a

nce l

Ds k in f

in e

rictio n

line

AR = ∞

C Reynolds number FIG. 12.15 Hughes model of ship resistance.

Rn

thereby giving the total resistance as CT ¼ CF jAR¼∞ + CFORM + CW

(12.17)

in which CFORM is a “form” resistance coefficient, which takes into account the viscous pressure resistance of the ship. In this approach, the basic skin friction resistance coefficient can be determined from Eq. (12.16). To determine the form resistance, the ship model can be run at a very slow speed when the wave-making component is very small and can be neglected. When this occurs, that is to the left of point A in Fig. 12.15, the resistance curve defines the sum of the skin friction and form resistance components. At the point A, when the wave making resistance is negligible, the ratio AC viscous resistance ¼ BC skin friction resistance skin friction resistance + viscous form resistance ¼ skin friction resistance viscous form resistance ¼1+ skin friction resistance viscous form resistance and if k ¼ skin friction resistance AC then ¼ ð1 + k Þ BC (12.18)

In Eq. (12.18), (1 + k) is termed the form factor and it is assumed constant for both the ship and its model. Indeed, the form factor is generally supposed to be independent of speed and scale in the resistance extrapolation method. In practical cases, the determination of (1 + k) is normally

FIG. 12.16 Determination of (1 + k) using Prohaska method.

carried out using a variant of the Prohashka method by a plot of CT against Fn 4 and extrapolating the curve to Fn ¼ 0 as indicated by Fig. 12.16. From this figure, the form factor (1 + k) is deduced from the relationship 

R 1 + k ¼ lim Fn!0 RF



This derivation of the form factor can be used in the resistance extrapolation only if scale-independent pressure resistance is absent; for example, there must be no immersion of the transom and slender appendages, which are oriented to the direction of flow. Although traditionally the form factor (1 + k) is treated as a constant with varying Froude number, the fundamental question remains as to whether it is valid to assume that the (1 + k) value, determined at vanishing Froude number, is valid at high speed. This is of particular concern at speeds beyond the main resistance hump where the flow configuration around the hull is likely to be very different from that when Fn ¼ 0, and, therefore, a Froude number dependency can be expected for (1 + k). In addition, a Reynolds dependency may also be expected since viscous effects are the basis of the (1 + k) formulation. The Froude and Reynolds effects are, however, likely to mostly affect the high-speed performance and have a lesser influence on general craft. The extrapolation from model- to full scale using Hughes’ method is shown in Fig. 12.17A. From this figure, it is seen that the two-dimensional skin friction line, Eq. (12.16), is used as a basis and the viscous resistance is estimated by scaling the basic friction line by the form factor (1 + k). This then acts as a basis for calculating the wave-making resistance from the measured total resistance on the model, which is then equated to the ship condition

326 Marine Propellers and Propulsion

FIG. 12.17 Comparison of extrapolation approaches: (A) extrapolation using Hughes approach and (B) extrapolation using Froude approach.

along with the recalculated viscous resistance for the ship Reynolds number. The Froude approach, Fig. 12.17B, is essentially the same, except that the frictional resistance is based on one of the Froude, ATTC, Eq. (12.12), or the ITTC, Eq. (12.13), friction lines without a (1 + k) factor. Clearly the magnitude of the calculated wave-making resistance, since it is measured as total resistance minus calculated frictional resistance, will vary according to the friction formulation used. This is also true of the correlation allowances as defined in Eq. (12.14) and, therefore, the magnitudes of these parameters should always be considered in the context of the approach and experimental facility used. In practice both the Froude and Hughes approaches are used in model testing; the latter, however, is most frequently used in association with the ITTC-1957 friction formulation rather than Eq. (12.16).

12.1.3

Methods of Resistance Evaluation

To evaluate the resistance of a ship, the designer has several options available. These range, as shown in Fig. 12.18, from what may be termed the traditional methods through to

computational fluid dynamics (CFD) class of methods. The choice of method depends not only on the capability available but also on the accuracy desired, the funds available, and the degree to which the approach has been developed. Fig. 12.18 identifies four basic classes of approach to the problem: the traditional and standard series; the regression-based procedures; the direct model test; and the CFD approaches. Clearly these are somewhat artificial distinctions and, consequently, break down on close scrutiny. They are, however, convenient classes for discussion purposes. Unlike the CFD and direct model test approaches, the other methods are based on the traditional naval architectural parameters of hull form: for example, block coefficient, longitudinal center of buoyancy, prismatic coefficient, etc. These form parameters generally describe the hull shape and have served the industry well in the past for resistance calculation purposes. However, as requirements become more exacting and hull forms become more complex, these traditional parameters are less able to reflect the growth of the boundary layer and wave-making components. As a consequence, an amount of research has been expended in the development of form parameters, which

Resistance and Propulsion Chapter 12

327

FIG. 12.18 Ship resistance evaluation methods and examples.

will reflect the hull surface contours in a more equable way: in some methods, this has extended to around 30 or 40 geometric form parameters.

derived for each of the chosen V/√ L values from the relation EHPn ¼ ACT VS3

(12.19)

12.1.3.1 Traditional and Standard Series Analysis Methods

where A is the wetted surface area.

A comprehensive treatment of these methods would require a book in itself and also would also lie to one side of the main theme of this text. As a consequence, an outline of four of the traditional methods starting with that of Taylor and passing through Ayer’s analysis to the later methods of Auf’m Keller and Harvald is presented in outline to illustrate the development of this class of methods.

Ayre’s Method (1942) Ayre (1927, 1948) developed a method, again based on model test data, using a series of hull forms relating to colliers. His approach, which in former years achieved quite widespread use, centers on the calculation of a constant coefficient C2 which is defined by Eq. (12.20)

Taylor’s Method (1910–1943) Admiral Taylor in 1910 published the results of model tests on a series of hull forms. This work has since been extended, Taylor (1943), to embrace a range of V/√ L from 0.3 to 2.0. The series comprised some 80 models in which results are published for beam to draft ratios of 2.25, 3.0, and 3.75 with five displacement length ratios. Eight prismatic coefficients were used spanning the range 0.48–0.80, and this tends to make the series useful for the faster and less full vessels. The procedure is centered on the calculation of the residual resistance coefficients based on the data for each B/T value corresponding to the prismatic and V/√ L values of interest. The residual resistance component CR is found by interpolation from the three B/T values corresponding to the point of interest. The frictional resistance component is calculated on a basis of Reynolds number and wetted surface area together with a hull roughness allowance. The result of this calculation is added to the interpolated residuary resistance coefficient to form the total resistance coefficient CT from which the naked effective horsepower is

EHP ¼

D0:64 VS3 C2

(12.20)

This relationship implies that in the case of full-sized vessels of identical forms and proportions, the EHP at corresponding speeds varies as (D0.64V3S) and that C2 is a constant at given values of V/√ L. In this case, the use of D0.64 is intended to avoid the necessity to treat the frictional and residual resistances separately for vessels of around 30 m. The value of C2 is estimated for a standard block coefficient. Corrections are then made to adjust the standard block coefficient to the actual value and corrections applied to cater for variations in the beam-draft ratio, the position of the l.c.b., and variations in length from the standard value used in the method’s derivation. Auf’m Keller Method Auf’m Keller (1973) extended the earlier work of Lap (1954) to allow the derivation of resistance characteristics of large block coefficient, single-screw vessels. The method is based on the collated results from some 107 model test

328 Marine Propellers and Propulsion

The procedure adopted is shown in outline form by Fig. 12.21 in which the correction for zr and the ship model correlation CA are given by Eq. (12.21) and Table 12.3, respectively: h i (12.21) % change in zr ¼ 10:357 e1:129ð6:5L=BÞ  1 As in the case of the previous two methods the influence of the bulbous bow is not taken into account, but reasonable experience can be achieved with the method within its area of application. Harvald Method The method proposed by Harvald (1978) is essentially a preliminary power prediction method designed to obtain an estimate of the power required to propel a ship. The approach used is to define four principal parameters upon which to base the estimate; these parameters are: FIG. 12.19 Definition of ship class.

results for large single-screw vessels and the measurements were converted into five sets of residuary resistance values. Each of these sets is defined by a linear relationship between the longitudinal center of buoyancy and the prismatic coefficient. Fig. 12.19 defines these sets, denoted by the letters A to E, and Fig. 12.20 shows the residuary resistance coefficient for set A. As a consequence, it is possible to interpolate between the sets for a particular l.c.b. versus CP relationship.

1. 2. 3. 4.

the the the the

ship displacement (D), ship speed (Vs), block coefficient (Cb), length displacement ratio (L/r1/3).

By making such a choice, all the other parameters that may influence the resistance characteristics need to be standardized: such as hull form, B/T ratio, and l.c.b., propeller diameter. The method used by Harvald is to calculate the resistance of a standard form for a range of the four parameters cited here and then evaluate the shaft power using a Quasipropulsive Coefficient (QPC) based on the wake

FIG. 12.20 Diagram for determining the specific residuary resistance as a function of Vs/(CpL) and Cp. (Reproduced with permission from Auf’m Keller, W.H., 1973. Extended diagrams for determining the resistance and required power for single screw ships. ISP 20.)

Resistance and Propulsion Chapter 12

329

from 0.55 to 0.85 in 0.05 intervals as shown in Fig. 12.22. From these diagrams, an estimate of the required power under trial conditions can be derived readily with the minimum of effort. However, when using a method of this type, it is important to make allowances for deviations of the actual form from those upon which the diagrams are based. Standard Series Data

FIG. 12.21 Auf’m Keller resistance calculation.

TABLE 12.3 Values of CA Used in Auf’m Keller Method Length of Vessel (m)

Ship Model Correlation Allowance

50–150

0.0004 ! 0.00035

150–210

0.0002

210–260

0.0001

260–300

0

300–350

0.0001

350–450

0.00025

(Reprinted from Auf’m Keller, W.H., 1973. Extended diagrams for determining the resistance and required power for single screw ships. Int. Shipbuilding Progress 20 (225), 133–142. doi: 10.3233/ISP-1973-2022501. Copyright 1973 with permission from IOS Press.)

and thrust deduction method discussed in Chapter 5 and a propeller open water efficiency taken from the Wageningen B Series propellers. The result of this analysis led to the production of seven diagrams for a range of block coefficients

In addition to the more formalized methods of analysis, there is a wealth of data available to the designer and analyst in the form of model data and more particularly in model data relating to standard series hull forms. That is, those in which the geometric hull form variables have been varied in a systematic way. Most data have been collected over the years and Bowden (1970) gives a useful guide to the extent of the data available for single-screw ocean-going ships between the years 1900 and 1969. Some of the more recent and important series and data are given in Dawson (1953, 1954, 1956, 1959), Moor (1960a,b), DTMB Series 60 (n.d.), Van Manen et al. (1955, 1961), Moor et al. (1961), Moor (1965–1966), Lackenby and Parker (1966), Moor and Pattullo (1968), SSPA Standard Series (1969), Moor (1973), Moor (1974), and Pattullo (1974). Unfortunately, there is little uniformity of presentation in the work as the results have been derived over a long period of time in many countries of the world. Designers therefore must accept this state of affairs and account for this in their calculations. In addition, hull form design has progressed considerably in recent years and little of these changes is reflected in the data cited in these references. Consequently, unless extreme care is exercised in the application of these data, significant errors can be introduced into the resistance estimation procedure. In more recent times, the Propulsion Committee of the ITTC (1987) conducted a cooperative experimental program between tanks around the world. The data so far reported relate to the Wigley parabolic hull and the Series 60, Cb ¼ 0.60 hull forms.

12.1.3.2 Regression-Based Methods Ship resistance prediction based on statistical regression methods has been a subject of some interest for several years. Early work by Scott (1972, 1973) resulted in methods for predicting the trial performance of single- and twinscrew merchant ships. The theme of statistical prediction was then taken up in a series of papers by Holtrop (1977, 1978, 1988a,b) and Holtrop and Mennen (1978, 1982). These papers trace the development of a power prediction method based on the regression analysis of random model and full-scale test data together with, most recently, the published results of the Series 64 high-speed displacement hull terms. In this latest version, the regression analysis is now based on the results

330 Marine Propellers and Propulsion

FIG. 12.22 Harvald estimation diagram for ship power.

of some 334 model tests and the results are analyzed on the basis of the ship resistance equation: RT ¼ RF ð1 + k1 Þ + RAPP + RW + RB + RTR + RA

(12.22)

In this equation the frictional resistance RF is calculated according to the ITTC-1957 friction formulation, Eq. (12.13), and the hull form factor (1 + k1) is based on a regression equation. It is expressed as a function of afterbody form, breadth, draft, length along the waterline, length of run, displacement, prismatic coefficient: ð1 + k1 Þ ¼ 0:93 + 0:487118ð1 + 0:011Cstern Þ  ðB=LÞ1:06806 ðT=LÞ0:46106  0:36486  ðLWL =LR Þ0:121563 L3WL =r

(12.23)

 ð1  CP Þ0:604247 in which the length of run LR, if unknown, is defined by a separate relationship as follows:   0:06CP l:c:b: LR ¼ LWL 1  CP + 4Cp  1 The sternshape parameter Cstern in Eq. (12.23) is defined in relatively coarse steps for different hull forms, as shown in Table 12.4.

TABLE 12.4 Cstern Parameters According to Holtrop Afterbody Form

Cstern

Pram with gondola

25

V-shaped sections

10

Normal section ship

0

U-shaped sections with Hogner stern

10

The appendage resistance according to the Holtrop approach is evaluated from the equation X 1 (12.24) RAPP ¼ rVS2 CF ð1 + k2 Þequv SAPP + RBT 2 where the frictional coefficient CF of the ship is again determined by the ITTC-1957 line and SAPP is the wetted area of the appendages of the vessel. To determine the equivalent (1 + k2) value of the appendages, denoted by (1 + k2)equv, appeal is made to the relationship X ð1 + k2 ÞSAPP X ð1 + k2 Þequv ¼ (12.25) SAPP

Resistance and Propulsion Chapter 12

ð10Fn  4Þ RW ¼ RW jFn ¼0:4 + 1:5 h i  RW jFn¼0:55  RW jFn ¼0:4

TABLE 12.5 Tentative Appendage Form Factors (1 + k2) Appendage Type

(1 + k2)

Rudder behind skeg

1.5–2.0

Rudder behind stern

1.3–1.5

Twin-screw balanced rudders

2.8

Shaft brackets

3.0

Skeg

1.5–2.0

Strut bossings

3.0

Hull bossings

2.0

Shafts

2.0–4.0

Stabilizer fins

2.8

Dome

2.7

Bilge keels

1.4

The values of the appendage form factors are tentatively defined by Holtrop as shown in Table 12.5. In cases where bow thrusters are fitted to the vessel, their influence can be taken into account by the term RBT in Eq. (12.24) as follows: RBT ¼ prVS2 dT CBTO in which dT is the diameter of the bow thruster and the coefficient CBTO lies in the range 0.003–0.012. If the thruster is located in the cylindrical part of the bulbous bow, then CBTO ! 0.003. The prediction of the wave-making component of resistance has proved difficult and in the last version of Holtrop’s method, Holtrop (1988a), a three-banded approach is proposed to overcome the difficulty of finding a general regression formula. The ranges proposed are based on the Froude number Fn and are as follows: Range 1: Fn > 0.55 Range 2: Fn < 0.4 Range 3: 0.4 < Fn < 0.55 within which the general form of the regression equations for wave-making resistance in ranges 1 and 2 is    RW ¼ K1 K2 K3 rrg exp K4 FKn 6 + K5 cos K7 =F2n (12.26) The coefficients K1, K2, K3, K4, K5, K6, and K7 are defined by Holtrop (1988a) and it is of interest to note that the coefficient K2 determines the influence of the bulbous bow on the wave resistance. Furthermore, the difference in the coefficients of Eq. (12.26) between ranges 1 and 2 above lies in the coefficients K1 and K4. To accommodate the intermediate range, range 3, a more or less arbitrary interpolation formula is used of the form

331

(12.27)

The remaining terms in Eq. (12.22) relate to the additional pressure resistance of the bulbous bow near the surface RB and the immersed part of the transom RTR. These are defined by relatively simple regression formulae. With regard to the model-ship correlation resistance the most recent analysis has shown the formulation in Holtrop and Mennen (1982) to predict a value some 9%–10% high; however, for practical purposes that formulation is still recommended by Holtrop: 1 RA ¼ rVS2 SCA 2 where CA ¼ 0:006ðLWL + 100Þ0:16  0:005205 + 0:003√ ðLWL =7:5ÞC4B K2 ð0:04  c4 Þ (12.28) in which and

c4 ¼ TF/LWL c4 ¼ 0.04

when when

TF/LWL  0.04 TF/LWL > 0.04

where TF is the forward draft of the vessel and S is the wetted surface area of the vessel. The parameter K2, which also appears in Eq. (12.26) and determines the influence of the bulbous bow on the wave resistance, is given by h pffiffiffiffiffiffii K2 ¼ exp 1:89 C3 where c3 ¼

0:56ðABT Þ1:5

BT 0:31√ABT + TF  hB

in which ABT is the transverse area of the bulbous bow and hB is the position of the center of the transverse area ABT above the keel line with an upper limit of 0.6TF (see Fig. 12.11). Eq. (12.28) is based on a mean apparent amplitude hull roughness kS ¼ 150 mm. In cases where the roughness may be larger than this, use can be made of the ITTC-1978 formulation, which gives the effect of the increase in roughness DCA as

1=3 (12.29) DCA ¼ 0:105kS  0:005579 =L1=3 The Holtrop method provides a useful estimation tool for the designer. However, like many analysis procedures of this type, it relies to a very large extent on traditional naval architectural geometric parameters. As these parameters

332 Marine Propellers and Propulsion

cannot fully act as a basis for representing the hull curvature and its effect on the flow around the vessel, there is a natural limitation on the accuracy of the approach without using more complex hull definition parameters.

12.1.3.3 Direct Model Test Model testing of a ship in the design stage is an important part of the design process and one that, in a great many instances, is either not explored fully or undertaken. In the author’s view this is a false economy, bearing in mind the relatively small cost of model testing when compared to the cost of the ship and the potential costs that can be incurred in design modification to rectify a problem or the through-life costs of a poor performance optimization. General Procedure for Model Tests While the detailed procedures for model testing differ from one establishment to another, the underlying general principles are similar. Here the general concepts are discussed, but for a more detailed account reference can be made to Phillips-Birt (1970) or to the ongoing ITTC proceedings. With regard to resistance and propulsion testing, there are a limited number of experiments that are of interest: the resistance test, the open water propeller test, the propulsion test, and the flow visualization test. The measurement of the wake field was discussed in Chapter 5. A further test that may be of interest in certain cases is an added resistance test where the additional resistance due to sea conditions is estimated in a seakeeping basin for various weather and heading scenarios. Resistance Tests In the resistance test, the ship model is towed by the carriage in calm water and the total longitudinal force acting on the model is measured for various speeds (Fig. 12.23). The breadth and depth of the towing tank essentially govern the size of the model that can be used. Todd’s original criterion that the immersed cross-section of the vessel should not exceed 1% of the tank’s cross-sectional area was placed in doubt after the famous Lucy Ashton experiment. This showed that to avoid boundary interference from the tank walls and bottom this proportion should be reduced to the order of 0.4%. The model, constructed from paraffin wax, wood, or glass-reinforced plastic, must be manufactured to a high degree of finish and turbulence simulators placed at the bow of the model to stimulate the transition from a laminar into a turbulent boundary layer over the hull. The model is positioned under the carriage and towed in such a way that it is free to heave and pitch and ballasted to the required draft and trim. In general, there are two kinds of resistance tests: the naked hull and the appended resistance test. If appendages are present, local turbulence tripping is applied to prevent

the occurrence of uncontrolled laminar flow over the appendages. Furthermore, the propeller should be replaced by a streamlined cone to prevent flow separation in this area. The resistance extrapolation process follows Froude’s hypothesis and the similarity law is followed. As such the scaling of the residual, or wave-making component, follows the similarity law RWship ¼ RW mod el l3 ðrs =rM Þ provided that VS ¼ VM √ l, where l ¼ LS/LM. In general, the resistance is scaled according to the relationship   3 rS Rs ¼ ½RM  RFM ð1 + kÞl + RFS ð1 + kÞ + RA rM   r ¼ ½RM  FD l3 S rM (12.30) where 1 r 2 FD ¼ rM VM SM ð1 + kÞðCFM  CFS Þ  M RA =l3 2 rS that is, 1 2 FD ¼ rM VM SM ½ð1 + kÞðCFM  CFS Þ  CA  2

(12.31)

The term FD is known as both the scale effect correction on resistance and the friction correction force. The term RA in Eq. (12.30) is the resistance component, which is intended to allow for the following factors: hull roughness; appendages on the ship but not present during the model experiment; still air drag of the ship and any other additional resistance components acting on the ship but not on the model. As such, its nondimensional form CA is the incremental resistance coefficient for ship-model correlation. When (1 + k) in Eq. (12.30) is put to unity, the extrapolation process is referred to as a two-dimensional approach since the frictional resistance is then taken as that given by the appropriate line, Froude flat plate data, ATTC or ITTC1957, etc. The effective power (PE) is derived from the resistance test by the relationship PE ¼ Rs VS

(12.32)

Open Water Tests The open water test is carried out using either a stock propeller or an actual model of the propeller to be fitted to the ship to derive its open water characteristics in order to estimate the ship’s propulsion coefficients. In this context, a stock propeller is one which approximates, to a greater or lesser extent, the actual propeller to be used for the ship but comes from the model test establishment’s collection.

Resistance and Propulsion Chapter 12

333

FIG. 12.23 Ship model test facility.

The propeller model is fitted on a horizontal driveway shaft and is moved through the water at an immersion of the shaft axis, which is frequently equal to the diameter of the propeller as seen in Fig. 12.24. The loading of the propeller is normally carried out by adjusting the speed of advance and keeping the model revolutions constant. However, when limitations in the measuring range, such as a J-value close to zero or a high carriage speed needed for a high J-value, are reached the rate of revolutions is also varied. The measured thrust values are corrected for the resistance of the hub and streamlined cap, this correction being determined experimentally in a test using a hub only without the propeller.

The measured torque and corrected thrust are expressed as nondimensional coefficients KQO and KTO in the normal way (see Chapter 6); the suffix O being used in this case to denote the open rather than the behind condition. The open water efficiency and the advance coefficient are then expressed as 0 ¼

J KTO 2p KQO

and Vc nD where Vc is the carriage speed. J¼

334 Marine Propellers and Propulsion

FIG. 12.24 Propeller open water test using towing tank carriage.

Unless explicitly stated, it should not be assumed that the propeller open water characteristics have been corrected for scale effects. The data from these tests are normally plotted on a conventional open water diagram together with a tabulation of the data.

Propulsion Tests In the propulsion test, the model is prepared in much the same way as for the resistance experiment and turbulence stimulation on the hull and appendages is again applied. For this test, however, the model is fitted with the propeller used in the open water test together with an appropriate drive motor and dynamometer. During the test, the model is free to heave and pitch as in the case of the resistance test. In the propulsion test the propeller thrust TM, the propeller torque QM, and the longitudinal towing force F acting on the model are recorded for each tested combination of model speed VM and propeller revolutions nM. Propulsion tests are carried out in two parts. The first comprises a load variation test at one or sometimes more than one constant speed while the other comprises a speed variation test at constant apparent advance coefficient or at the self-propulsion point of the ship. The ship selfpropulsion point is defined when the towing force (F) on the carriage is equal to the scale effect correction on viscous resistance (FD), Eq. (12.31).

The required thrust TS and self-propulsion point of the ship is determined from the model test using the equation:  ∂TM 3 rS (12.33) TS ¼ T M + ð F D  F Þ l ∂F rM In Eq. (12.33), the derivative ∂ TM/∂ F is determined from the load variation tests, which form the first part of the propulsion test. In a similar way, the local variation test can be interpolated to establish the required torque and propeller rotational speed at self-propulsion for the ship. In the extrapolation of the propulsion test to full-scale the scale effects on resistance (FD), on the wake field and on the propeller characteristics need to be considered. At some very high speeds, the effects of cavitation also need to be taken into account. This can be done by analysis or through the use of specialized facilities. Flow Visualization Tests Various methods exist to study the flow around the hull of a ship. One such method is to apply stripes of an especially formulated paint to the model surface; the stripes being applied vertical to the base line at different longitudinal locations. The model is then towed at Froude identity and the paint will smear into streaks along the hull surface in the direction of the flow lines. In cases where the wall shear stresses are insufficient, tufts can be used to visualize the flow over the hull. In

Resistance and Propulsion Chapter 12

general, woolen threads of about 5 cm in length will be fitted onto small needles driven into the hull surface. The tufts will be at between 1 and 2 cm from the hull surface and the observation made either visually or by using an underwater television camera. Interaction phenomenon between the propeller and ship’s hull can also be studied in this way by observing the behavior of the tufts with and without the running propeller.

335

Model Test Facilities Many model test facilities exist around the world, almost all of which possess a ship model towing tank. Some of the model facilities available are listed in Table 12.6; this, however, is by no means an exhaustive list of facilities and is included here to give an idea of the range of facilities available.

TABLE 12.6 Examples of Towing Tank Facilities Around the World Facilities

Length (m)

Width (m)

Depth (m)

Maximum Carriage Speed (m/s)

164

6.1

2.4

7.5

270

12.0

5.5

12.0

Experimental and Electronic Lab.

76,188

3.72.4

1.71.3

9.113.1

B.H.C. Cowes (UK)

197

4.6

1.7

15.2

MARIN Wageningen (NL)

100

24.5

2.5

4.5

216

15.7

1.25

5

220

4.0

4.0

15/30

252

10.5

5.5

9

MARIN Depressurized Facility, Ede (NL)

240

18.0

8.0

4

Danish Ship Research Laboratories

240

12.0

6.0

14

Ship Research

27

2.5

1.0

2.6

Institute of Norway (NSFI)

175

10.5

5.5

8.0

SSPA G€ oteborg, Sweden

260

10.0

5.0

14.0

Bassin d’Essais de Care`nes, Paris

155

8.0

2.0

5

220

13.0

4.0

10

120

8.0

1.1

4.2

250

8.0

4.8

20

30

6.0

1.2

0.0023–1.9

80

4.0

0.7

3.6

80

5.0

3.0

3.6

300

18.0

6.0

8.0

37.5

3.0

2.5

3

23

12.5

6.2

8

293

5.0

3.5

12

NSRDC Bethesda

845

15.6

6.7

10

USA

905

6.4

3.0–4.8

30

NRC, Marine

137

7.6

3.0

8

European facilities Qinetiq Haslar (UK)

VWS, Germany

H.S.V. Hamburg, Germany

B.I.Z. Yugoslavia

North American facilities

Continued

336 Marine Propellers and Propulsion

TABLE 12.6 Examples of Towing Tank Facilities Around the World—cont’d Facilities

Length (m)

Width (m)

Depth (m)

Maximum Carriage Speed (m/s)

Meguro model

98

3.5

2.25

7

Basin, Japan

235

12.5

7.25

10

340

6.0

3.0

20

Ship Research

20

8.0

0–1.5

2

Institute, Mitaka

50

8.0

4.5

2.5

Japan

140

7.5

0–3.5

6

375

18.0

8.5

15

KIMM—Korea

223

16.0

7.0

Hyundai—Korea

232

14.0

6.0

Dynamics and Ship Laboratory, Canada Far East facilities

Reproduced with permission from Clayton, B.R., Bishop, R.E.D., 1982. Mechanics of Marine Vehicles. Spon, London.

Two-Dimensional Extrapolation Method This, as discussed previously, is based on Froude’s original method without the use of a form factor. Hence the fullscale resistance is determined from   3 rS RS ¼ ðRM  FD Þl rM where 1 2 SM ðCFM  CFS  CA Þ FD ¼ rM VM 2 and when Froude’s friction data are used the value of CA is set to zero; however, this is not the case if the ATTC-1947 or ITTC-1957 line is used. When the results of the propulsion test are either interpolated for the condition when the towing force (F) is equal to FD or when FD is actually applied in the self-propulsion test, the corresponding model condition is termed the “selfpropulsion point of the ship.” The direct scaling of the model data at this condition gives the condition generally termed the “tank condition.” This is as follows:   9 3:5 rS > > PDS ¼ PDM l > > rM > > >   > > r > 3 S > > TS ¼ TM l > = rM (12.34) nS ¼ nM =√l > > > > > > VS ¼ VM =√l > >  > > r > > ; RS ¼ ðRM  FD Þl3 S > rM

The power and propeller revolutions determined from the tank condition as given by Eq. (12.34) require to be converted into a trial prediction for the vessel. In the case of the power trial prediction, this needs to be based on an allowance factor derived from the results of trials of comparable ships of the same size or, alternatively, on the results of statistical surveys. The power trial allowance factor is normally defined as the ratio of the shaft power measured on trial to the power delivered to the propeller in the tank condition. The full-scale propeller revolutions prediction is based on the relationship between the delivered power and the propeller revolutions derived from the tank condition. The power predicted for the trial condition is then used in this relationship to devise the corresponding propeller revolutions. This propeller speed is corrected for the over- or underloading effect and often corresponds to around a 0.5% decrease of rpm for a 10% increase of power. The final stage in the propeller revolutions prediction is to account for the scale effects in the wake and propeller blade friction. For the trial condition, these scale effects are of the order of 1 pffiffiffi l percent for single-screw vessels 2 1%–2% for twin-screw vessels The allowance for the service condition on rotational speed is of the order 1%. Three-Dimensional Extrapolation Method The three-dimensional extrapolation method is based on the form factor concept. Accordingly, the resistance is scaled

Resistance and Propulsion Chapter 12

under the assumption that the viscous resistance of the ship and its model is proportional to the frictional resistance of a flat plate of the same length and wetted surface area when towed at the same speed, the proportionality factor is (1 + k) as discussed in Section 12.1.2. In addition, it is assumed that the pressure resistance due to wave generation, stable separation, and induced drag from nonstreamlined or misaligned appendages follows the Froude similarity law. The form factor (1 + k) is determined for each hull from low-speed resistance or propulsion measurements when the wave resistance components are negligible. In the case of the resistance measurement of form factor, then this is based on the Prohaska derivation:   R ð1 + kÞ ¼ lim Fn !0 RF In the case of the propulsion test acting as a basis for the (1 + k) determination, then this relationship takes the form " # F  T=ð∂T=∂FÞ  ð1 + kÞ ¼ lim  Fn !0 FjT¼0 =R RF The low-speed measurement of the (1 + k) factor can only be validly accomplished if scale-independent pressure resistance is absent. This implies, for example, that there is no immersed transom. In this way, the form factor is maintained independent of speed and scale in the extrapolation method. In the three-dimensional method, the scale effect on the resistance is taken as 1 2 FD ¼ rM VM SM ½ð1 + kÞðCFM  CFS Þ  CA  2 in which the form factor is normally taken relative to the ITTC-1957 line and CA is the ship-model correlation coefficient. The value of CA is generally based on an empirical relationship and additional allowances are applied to this factor to account for extreme hull forms at partial draughts; appendages not present on the model; “contract” conditions; hull roughnesses different from the standard of 150 mm; extreme superstructures or specific experience with previous ships. In the three-dimensional procedure, the measured relationship between the thrust coefficient KT and the apparent advance coefficient is corrected for wake scale effects and for the scale effects on propeller blade friction. At model scale, the model thrust coefficient is defined as KTM ¼ f ðFn , J ÞM whereas at ship scale this is     1  wTS KTS ¼ f Fn , J + DKT 1  WTM

337

According to the ITTC-1987 version of the manual for the use of the 1978 performance reduction method, the relationship between the ship and model Taylor wake fractions can be defined as wTS ¼ ðt + 0:04Þ + ðwTM  t  0:04Þ 

ð1 + kÞCFS + DCF ð1 + kÞCFM

where the number 0.04 is included to take account of the rudder effect and DCF is the roughness allowance given by " #   ks 1=3  0:64  103 DCF ¼ 105 LWL The measured relationship between the thrust and torque coefficient is corrected for the effects of friction over the blades such that KTS ¼ KTM + DKT and KQS ¼ KQM + DKQ where the factors DKT and DKQ are determined from the ITTC procedure discussed in Chapter 6. The load of the full-scale propeller is obtained from the relationship KT S CTS ¼ 2 2 2D ð1  tÞð1  wTS Þ2 J and with KT/J 2 as the input value, the full-scale advance coefficient JTS, and torque coefficient KQTS are read off from the full-scale propeller characteristics and the following parameters calculated: ð1  wTS ÞVS JTS D KQTS PDS ¼ 2prD5 n3S  103 R KT 2 TS ¼ 2 JTS rD4 n2S J KQTS 5 2 QS ¼ rD nS R nS ¼

(12.35)

The required shaft power PS is found from the delivered power PDS using the shafting mechanical efficiency S as PS ¼ PDS =S

12.1.3.4 Computational Fluid Dynamics The analysis of ship forms to predict total resistance using the CFD approach is now an important and maturing subject with considerable research effort is being devoted to the topic.

338 Marine Propellers and Propulsion

Regarding the wave-making part of the total resistance, provided that the viscous effects are neglected, then the potential flow can be defined by the imposition of boundary conditions at the hull and free surface. The hull conditions are accounted for by placing a distribution of source panels over its surface. The problem comes in satisfying the free surface boundary conditions, which ought to be applied at the actual free surface and, which, of course, are unknown at the start of the calculation. A solution to this problem was developed by Dawson (1977) and is one method in the class of “slow-ship” theories. With this method, the exact free surface condition is replaced by an approximate one that can be applied at a fixed location such as the undisturbed water surface. In such a case, a suitable part of the undisturbed free surface is covered with source panels and the source strengths determined to satisfy the boundary conditions. Fig. 12.25 shows the wave pattern calculated using a variation of the Dawson approach for a Wigley hull at a Froude number of 0.40 (MARIN, 1986). Free surface models in the CFD process can pose problems for integrated solutions for the total resistance estimation. However, methods based on the transportation of species concentration show promise for an integrated

FIG. 12.25 Calculated wave profile for Wigley hull at Fn ¼ 0.4. (Courtesy: MARIN.)

FIG. 12.26 Zones for CFD analysis.

CFD solution. These transport models are then solved additionally to the Navier-Stokes equations within the computational code. A typical example of one such model is: ð ð ∂=∂t Ci du + Ci v  nds ¼ 0 v

with r ¼

X

s

ri Ci and m ¼

X

mi C i

and where Ci is the transport species concentration in a particular grid domain. In the case of the viscous resistance, the flow field it is often considered in terms of three distinct regions: a potential or, more correctly, nearly potential zone; a boundary layer zone for much of the forward part of the hull; and a thick boundary layer zone toward the stern of the ship as shown in Fig. 12.26. Analysis by CFD procedures has matured significantly in the last few years and in many cases yields good quantitative estimates of frictional resistance. It also enables the designer to gain valuable insights into the flow field around the ship, particularly in the afterbody region where unpleasant vorticity and separation effects may manifest themselves. In these computationally based analyses, turbulence modeling has been problematic and while reasonable estimates of the frictional resistance have been made for fine form ships using k-O and k-OSST models, deployment of the more computationally intensive Reynolds stress models has improved the accuracy of the prediction for the finer hull forms. Moreover, these more advanced models have extended the range of applicability in terms of quantitative estimates of resistance to full-form ships. Such developments, therefore, help to relieve concerns as to where the frictional resistance solution starts to diverge significantly from the true value for a given hull form. When considering the propulsion aspects of a ship’s design, the use of a combination of model testing and analysis centered on CFD coupled with sound design experience is advisable. Moreover, notwithstanding the advances that

Resistance and Propulsion Chapter 12

have been made with the mathematical modeling processes, they should not at present replace the conventional model testing procedures for which most correlation data exist: rather they should be used to complement the design approach by allowing the designer to gain insights into the flow dynamics and develop remedial measures before the hull is constructed.

12.1.4

Propulsive Coefficients

The propulsive coefficients of ship performance form the essential link between the effective power required to drive the vessel, obtained from the product of resistance and ship speed, and the power delivered from the engine to the propeller. The power absorbed by and delivered to the propeller PD to propel the ship at a given speed VS is PD ¼ 2pnQ

Q ¼ KQb rn D 2

5

PD ¼ 2pKQb rn3 D5

QPC ¼

KQo KQb

Recalling that the resistance of the vessel R can be expressed in terms of the propeller thrust T as R ¼ T (1  t), where t is the thrust deduction factor, which will be explained later. Also from Chapter 5 the ship speed Vs can be defined in terms of the mean speed of advance Va as Va ¼ Vs (1  wt), where wt is the mean Taylor wake fraction. Furthermore, since the open water thrust coefficient KTo is expressed as To ¼ KTo rn2D4, with To being the open water propeller thrust at the mean advance coefficient J, To ¼ rn2 D4 KTo and the QPC can be expressed from the foregoing as QPC ¼

To ð1  tÞVa KTo r ð1  wt Þ2pKQo nDT o

which reduces to

 QPC ¼

 1t   1  wt 0 r

since, from Eq. (6.8),

(12.39)

this relationship being the definition stated earlier in Chapter 6. Hence, Eq. (12.38) can then be expressed in terms of the relative rotative efficiency as KQo 3 5 rn D r

RV S r 2pKQo rn3 D5

(12.38)

If the propeller were operating in open water at the same mean advance coefficient J, the open water torque coefficient KQo would be found to vary slightly from that measured behind the ship model. As such the ratio KQo/KQb is known as the relative rotative efficiency r

PD ¼ 2p

which implies that:

(12.37)

where KQb is the torque coefficient of the propeller when working in the wake field behind the vessel at a mean advance coefficient J. By combining Eqs. (12.36), (12.37), the delivered power can be expressed as

r ¼

RV S ¼ PD QPC KQo 3 5 ¼ 2p rn D QPC r

(12.36)

where n and Q are the rotational speed and torque at the propeller, respectively. The torque required to drive the propeller Q can be expressed for a propeller working behind the vessel as

339

(12.40)

Now the effective power PE is defined as PE ¼ RV s ¼ PD QPC where the QPC is termed the quasipropulsive coefficient. Hence, from the foregoing and in association with Eq. (12.40),

0 ¼

J KTo 2p KQo

The quantity (1  t)/(1  wt) is termed the hull efficiency h and hence the QPC is defined as QPC ¼ h 0 r

(12.41)

or, in terms of the effective and delivered powers, PE ¼ PD QPC that is, PE ¼ PD  h  0  r

(12.42)

12.1.4.1 Relative Rotative Efficiency The relative rotative efficiency (r), as defined by Eq. (12.39), accounts for the differences in torque absorption characteristics of a propeller when operating at similar conditions in a mixed wake and open water flows. In many cases, the value of r lies close to unity and is generally within the range 0:96  r  1:04

340 Marine Propellers and Propulsion

In relatively few cases, it lies outside this range. Holtrop (1988a) gives the following statistical relationships for its estimation: 9 For conventional stern single-screw ships : > > > > > r ¼ 0:9922  0:05908 ðAE =AO Þ > > > = + 0:07424ðCP  0:0225l:c:b:Þ > For twin-screw ships : > > > > r ¼ 0:9737 + 0:111 ðCP  0:0225 l:c:b:Þ > > > ; 0:06325P=D

resistance augmentation factor can be derived from Eqs. (12.44), (12.45) as being t

ar ¼ 1t

If resistance and propulsion model tests are performed, then the relative rotative efficiency is determined at model scale from the measurements of thrust Tm and torque Qm with the propeller operating behind the model. Using the nondimensional thrust coefficient KTm as input data, the values of J and KQo are read off from the open water curve of the model propeller used in the propulsion test. The torque coefficient of the propeller working behind the model is derived from

in which TM and FD are defined previously and Rc is the resistance corrected for differences in temperature between the resistance and propulsion tests:

KQb ¼

QM rn2 D5

Hence, the relative rotative efficiency is calculated as r ¼

KQo KQb

The relative rotative efficiency is assumed to be scale independent.

12.1.4.2 Thrust Deduction Factor When water flows around the hull of a ship, which is being towed and does not have a propeller installed, a certain pressure field is set up, which is dependent on the hull form. If the same ship is now fitted with a propeller and is propelled at the same speed, the pressure field around the hull changes due to the action of the propeller. The propeller increases the velocities of the flow over the hull surface and hence reduces the local pressure field over the after part of the hull surface. This has the effect of increasing, or augmenting, the resistance of the vessel from that which was measured in the towed resistance case and this change can be expressed as T ¼ Rð 1 + ar Þ

t¼

Rc ¼

TM + FD  Rc TM

ð1 + kÞCFMC + CR RTM ð1 + kÞCFM + CR

where CFMC is the frictional resistance coefficient at the temperature of the self-propulsion test. In the absence of model tests, an estimate of the thrust deduction factor can also be obtained from the work of Holtrop (1988a) and Harvald (1978). In the Holtrop approach, the following regression-based formulas are given: 9 For single-screw ships : > >

0:2624 > > > 0:28956 > > 0:25014ðB=LÞ √ ðB=T Þ=D > > > = t¼ 0:01762 ð1  CP + 0:0225 l:c:b:Þ (12.46) > > +0:0015Cstern > > > > > For twin-screw ships : > > > ; t ¼ 0:325CB  0:1888D=√ ðBT Þ In Eq. (12.46), the value of the parameter Cstern is found from Table 12.4. The alternative to this approach is that of Harvald for the calculation of the thrust deduction factor. This assumes that it comprises three separate components as follows: t ¼ t1 + t2 + t3

(12.47)

in which t1, t2, and t3 are basic values derived from hull form parameters, a hull form correction, and a propeller diameter correction, respectively. The values of these parameters for single-screw ships are reproduced in Fig. 12.27.

(12.44)

where T is the required propeller thrust and ar is the resistance augmentation factor. An alternative way of expressing Eq. (12.44) is to consider the deduction in propeller effective thrust, which is caused by the change in pressure field around the hull. In this case, the relationship R ¼ T ð1  tÞ

If a resistance and propulsion model test has been performed, then the thrust deduction factor can be readily calculated from the relationship defined in the ITTC-1987 proceedings

(12.45)

applies, in which t is the thrust deduction factor. The correspondence between the thrust deduction factor and the

12.1.4.3 Hull Efficiency The hull efficiency can readily be determined once the thrust deduction and mean wake fraction are known. However, because of the pronounced scale effect of the wake fraction, there is a difference between the full-scale ship and model values. In general, because the ship wake fraction is smaller than the corresponding model value, due to Reynolds effects, the full-scale efficiency will also be smaller.

Resistance and Propulsion Chapter 12

12.1.5

341

The Influence of Rough Water

The discussion so far has centered on the resistance and propulsion of vessels in calm water or ideal conditions. Clearly the effect of bad weather is either to slow the vessel down for a given power absorption or, conversely, an additional input of power to the propeller to maintain the same ship speed. To gain some general idea of the effect of weather on ship performance, appeal can be made to the NSMB Trial Allowances (1976) and de Jong and Fransen (1976). These allowances were based on the trial results of 378 vessels and formed an extension to the 1965 and 1969 diagrams. Fig. 12.28 shows the allowances for ships with a trial displacement between 1000 and 320,000 tons based on the Froude extrapolation method and coefficients. Analysis of the data upon which this diagram was based showed that the most significant variables were the displacement, Beaufort wind force, model scale, and the length between perpendiculars. Consequently, a regression formula was suggested as follows: trial allowance ¼ 5:75  0:793D1=3 + 12:3Bn + ð0:0129LPP  1:864Bn Þl1=3

FIG. 12.27 Thrust deduction estimation of Harvald for singlescrew ships. (Reproduced with permission from Harvald, Sv.Aa., 1978. Estimation of power of ships. ISP 25(283).)

12.1.4.4 Quasipropulsive Coefficient It can be deduced from Eq. (12.41) that the value of the QPC is dependent upon the ship speed, pressure field around the hull, the wake field presented to the propeller, and the intimate details of the propeller design, such as diameter, rate of rotation, radial load distribution, amount of cavitation on the blade surfaces, etc. Consequently, the QPC should be calculated from the three component efficiencies given in Eq. (12.41) and not globally estimated. Of interest when considering general trends is the effect that propeller diameter can have on the QPC. As the diameter increases, assuming the rotational speed is permitted to fall to its optimum value, the propeller efficiency will increase and hence for a given hull form the QPC will tend to rise. In this instance, the effect of propeller efficiency dominates over the hull and relative rotative efficiency effects.

(12.48)

where Bn and l are the Beaufort number and the model scale, respectively. Apart from global indicators and correction factors such as Fig. 12.27 or Eq. (12.48), considerable work has been undertaken in recent years to establish methods by which the added resistance due to weather can be calculated for a particular hull form. Latterly, attention has been paid to the effects of diffraction in short waves, which is a particularly difficult area to consider analytically. In general, estimation methods range from those which work on databases for standard series hull forms whose main parameters have been systematically varied, to those where the calculation is approached from fundamental considerations. In its most simplified form, the added resistance calculation is of the form RTW ¼ RTC ð1 + DR Þ

(12.49)

where RTW and RTC are the resistances of the vessel in waves and calm water, respectively, and DR is the added resistance coefficient based on the ship form parameters, speed, and irregular sea state. Typical of results of calculation procedures of this type are the results shown in Fig. 12.28 for a container ship operating in different significant wave heights (HS) and a range of heading angles from directly ahead (y ¼ 0 degree) to directly astern (y ¼ 180 degrees). Shintani and Inoue (1984) have established charts for estimating the added resistance in waves of ships based on a study of the Series 60 models. These data take into account various values of CB, B/T, L/B, and l.c.b. position and allows interpolation to the required value for a

342 Marine Propellers and Propulsion

FIG. 12.28 NSMB 1976 trial allowances. (Reproduced with permission from MARIN, 1986. MARIN Report No. 26. MARIN, Wageningen.)

particular design. In this work, the compiled results have been empirically corrected by comparison with model test data to enhance the prediction process. In the context of added resistance, numerical computations have suggested that the form of the bow above the calm water surface can have a significant influence on the added resistance in waves. Such findings have also been confirmed experimentally and have shown that a bluntbow ship could have its added resistance reduced by as much as 20%–30% while having minimal influence on the calm water resistance. To make estimations of the added resistance in waves, several semiempirical methods have been developed. While some of these become quite involved during application, the method developed for the speed loss due to added resistance in wind and waves by Townsin and Kwon (1983) has been adapted for use as a simplified estimation procedure by Kwon (2008). This gives the percentage speed loss as: % Speed Loss ¼ am

DV 100% V

where DV/V is the speed loss in head weather given by the following equations: For Cb in the range 0.75 < Cb < 0.85 in the loaded DV Bn 6:5 100% ¼ 0:5Bn + condition: V 2:7r2=3

For Cb in the range 0.75 < Cb < 0.85 in the ballast DV Bn 6:5 100% ¼ 0:7Bn + condition: V 2:7r2=3 For Cb in the range 0.55 < Cb < 0.70 in the normal DV Bn 6:5 100% ¼ 0:7Bn + condition: V 2:2r2=3 in which, Bn is the Beaufort number and r is the volume displacement in m3. The values of the coefficient a for a range of block coefficients and load conditions are given by Table 12.7. Table 12.8 gives the equations to derive the coefficient m for bow quartering, beam, and stern seas. In general, the majority of the practical estimation methods are based in some way on model test data: either for deriving regression equations or empirical correction factors. In the case of using theoretical methods to estimate the added resistance and power requirements in waves, methods based on linear potential theory tend to underpredict the added resistance when compared to equivalent model tests. In recent years, some nonlinear analysis methods have appeared, which indicate that if the water surface due to the complete nonlinear flow is used as the steady wave surface profile then the accuracy of the added resistance calculation can be improved significantly: for

Resistance and Propulsion Chapter 12

TABLE 12.7 Values of the Coefficient a Cb and Load Condition

Α

0.55 (normal)

1:7  1:4Fn  7:4Fn 2

0.60 (normal)

2:2  2:5Fn  9:7Fn 2

0.65 (normal)

2:6  3:7Fn  11:6Fn 2

0.70 (normal)

3:1  5:3Fn  12:4Fn 2

0.75 (loaded or normal)

2:4  10:6Fn  9:5Fn 2

0.75 (ballast)

2:6  12:5Fn  13:5Fn 2

0.80 (loaded or normal)

2:6  13:1Fn  15:1Fn 2

0.80 (ballast)

3:0  16:3Fn  21:6Fn 2

0.85 (loaded or normal)

3:1  18:7Fn  28Fn 2

0.85 (ballast)

3:4  20:9Fn + 31:8Fn 2

TABLE 12.8 Values of Coefficient m Relative Heading

Μ

Relative Angle (degrees)

Bow quartering seas

0.5 [1.7  0.03(Bn  4)2]

30–60

Beam seas

0.5 [0.9  0.06(Bn  6)2]

60–150

Following seas

0.5 [0.4  0.03(Bn  8)2]

150–180

example Raven (1996) and Hermans (2004). Although CFD analyses are relatively limited, those published so far show encouraging results when compared to measured results, for example, Orihara and Miyata (2003). In the context of added resistance, numerical computations have suggested that the form of the bow above the calm water surface can have a significant influence on the added resistance in waves. Such findings have also been confirmed experimentally and have shown that a bluntbow ship could have its added resistance reduced by as much as 20%–30% while having minimal influence on the calm water resistance.

12.1.6

Restricted Water Effects

Restricted water effects derive essentially from two sources. First, a limited amount of water under the keel and second, limitations in the width of water on each side of the vessel, which may or may not be in association with a depth restriction.

343

To assess the effects of restricted water operation, these being particularly complex to define mathematically, the ITTC have cited typical influencing parameters: ITTC (1987). These are as follows: 1. An influence exists on the wave resistance for values of the Froude depth number Fnh in excess of 0.7. The Froude depth number is given by Fnh ¼

V √ ðghÞ

where h is the water depth of the channel. 2. The flow around the hull is influenced by the channel boundaries if the water depth to draft ratio (h/T) is <4. This effect is independent of the Froude depth number effect. 3. There is an influence of the bow wave reflection from the lateral boundary on the stern flow if either the water width to beam ratio (W/B) is <4 or the water width to length ratio (W/L) is less than unity. 4. If the ratio of the area of the channel cross-section to that of the midship section (Ac/AM) is <15, then a general restriction of the waterway will start to occur. In the case of the last ratio, it is necessary to specify at least two of the following parameters: width of water, water depth or the shape of the canal section, because a single parameter cannot identify unconditionally a restriction on the water flow. The most obvious sign of a ship entering into shallow water is an increase in the height of the wave system, Fig. 12.5B, in addition to a change in the ship’s vibration characteristics. As a consequence of the increase in the height of the wave system, the assumption of small wave height, and consequently small wave slopes, cannot be used for restricted water analysis. This, therefore, implies a limitation to the use of linearized wave theory for this purpose; as a consequence, higher-order theoretical methods need to be sought. Currently several researchers are working in this field and endeavoring to enhance the correlation between theory and experiment (Fig. 12.29). Barrass (1979) suggests the depth-draft ratio at which shallow water just begins to have an effect is given by the equation h=T ¼ 4:96 + 52:68ð1  Cw Þ2 in which the Cw is the water-plane coefficient. Alternatively, Schneekluth (1987) provides a set of curves based on Lackenby’s work and shown in Fig. 12.30 to enable the estimation of the speed loss of a vessel from deep to shallow water. The curves are plotted on a basis of the square of Froude depth number to the ratio √ AM/h. Beyond data of this type, there are little else available with which to

344 Marine Propellers and Propulsion

and model tests with block coefficients in the range 0.5–0.9 for both open water and restricted channel conditions. In his analysis, the restricted channel conditions were defined in terms of h/T ratios in the range 1.1–1.5. For the conditions of unrestricted water in the lateral direction such that the effective width of the waterway in which the ship is traveling must be greater than [7.7 + 45(1  Cw)2]B, the squat is given by

Smax ¼ Cb ðAM =AC Þ2=3 Vs2:08 =30 for Fnh  0:7 More recently, Barrass has extended his studies and developed further relationships for squat and bank interaction (Barrass, 2009). More recently, Gilardoni and Presedo (2017) have compiled a compendium of information on navigation in shallow waters.

12.1.7 FIG. 12.29 Estimated power increase to maintain ship speed in different sea states for a container ship.

Air Resistance

The prediction of the air resistance of a ship can be evaluated in a variety of ways ranging from the extremely simple to undertaking a complex series of model tests in a wind tunnel or in using CFD methods. At its simplest the still air resistance can be estimated as proposed by Holtrop (1988b) who followed the simple approach incorporated in the ITCC-1978 method as follows: 1 RAIR ¼ ra VS2 AT Cair 2

FIG. 12.30 Loss of speed in transfer from deep to shallow water. (Reproduced from Schneekluth, H., 1987. Ship Design for Efficiency and Economy. Butterworths, London.)

readily estimate the added resistance in shallow water beyond recourse to numerical methods. One further effect of shallow water is the phenomenon of ship squat. This is caused by a venturi effect between the bottom of the vessel and the bed of the seaway, which causes a reduction of pressure to occur. This reduction of pressure then induces the ship to increase its draft to maintain static equilibrium. Barras developed a relationship for ship squat by analyzing the results from different ships

(12.50)

in which VS is the ship speed, AT is the transverse area of the ship, and Cair is the air resistance coefficient, taken as 0.8 for normal ships and superstructures. The density of air (ra) is normally taken as 1.23 kg/m3. For more advanced analytical studies, appeal can be made to the works of van Berlekom (1981) and Gould (1982). The approach favored by Gould is to determine the natural wind profile on a power law basis and select a reference height for the wind speed. The yawing moment center is then defined relative to the bow and the lateral and frontal elevations of the hull and superstructure are subdivided into so-called universal elements. Additionally, the effective wind speed and directions are determined from which the Cartesian forces together with the yawing moment can then be evaluated. The determination of the air resistance from wind tunnel measurement would generally only be undertaken in exceptional cases and would most probably be associated with flow visualization studies; for example, in the design of suitable locations for helicopter platforms, ensuring exhaust gases and particulate matter do not fall on passenger deck areas as well as to check that abnormal wind flows and vortex patterns do not occur in unwanted locations. When such tests are contemplated, then most commonly they would be undertaken in open-jet wind tunnels and using

Resistance and Propulsion Chapter 12

345

FIG. 12.31 CFD model of the flow around a ship and emissions from the funnel.

smoke for flow visualization purposes. However, for most commercial applications, the cost of undertaking wind tunnel tests cannot be justified since air resistance is by far the smallest of the resistance components. An alternative to wind tunnel testing is through the use of CFD methods. These computations, given the possession of an adequate model mesh generation capability, are relatively easy and quick to undertake; albeit they use a significant computation resource to model the ship to a required level of accuracy. Fig. 12.31 shows the results of one such computation, which was addressing a perceived exhaust gas problem.

12.1.8

Multihull Configurations

12.1.8.1 Catamarans In recent years, there has been a growth in the use of multihull vessels of varying sizes and configurations. These include catamarans, trimarans, and pentamarans and for general ferry, passenger, and Ro/Ro services the catamaran has become popular. One reason for this is the greater deck

FIG. 12.32 Comparison of two catamaran ferries.

area per ton of displacement when compared to monohull forms. Fig. 12.32 shows two catamaran ferries from which an appreciation of the range in sizes of these vessels can be made: the one on the left being solely for passenger transport, while that on the right is a Ro/Pax craft. The resistance of a catamaran form is strongly dependent upon the mutual wave interference between the two hulls. The prediction of the resistance and propulsion properties can be achieved in a number of ways and one such method is to estimate the resistance of each hull separately, by for example a Holtrop and Mennen type of procedure based on the ITTC 1957 line, and then estimate the wave making resistance by either: i. Superimposing the waves generated by each hull and calculated from potential theory as if they were individual hulls without the presence of each other. ii. Or calculate the wave-making resistance for a single hull based on a regression function of the type: Cw ¼ exp ðaÞðL=BÞb1 ðB=T Þb2 Cb b3 ðs=LÞ

346 Marine Propellers and Propulsion

TABLE 12.9 Range of the Southampton Series Hull Parameters

TABLE 12.10 Regression Coefficients a and b Fn

a

n

R2

0.4

152

1.76

0.946

1.5, 2.0, 2.5

0.5

2225

3.00

0.993

Longitudinal center of buoyancy (l.c.b.)

6.4% aft

0.6

1702

2.96

0.991

Separation of demihull centerlines to length ratio (Sc/L)

0.2, 0.3, 0.4, 0.5

0.7

896

2.76

0.982

0.8

533

2.58

0.982

Block coefficient of the demihulls

0.4

0.9

273

2.31

0.970

1.0

122

1.96

0.950

Length to volumetric displacement ration (L/r1/3)

6.3–9.5

Breadth to draft ratio (B/T)

and then combine this with the frictional resistance as follows: CT ¼ 2ð1 + bkÞCF + τCw Alternatively use can be made of model test series data; for example, the Southampton Series (Insel and Molland, 1992; Molland et al., 1996; Molland and Lee, 1997) or the VWS catamaran series (M€ uller-Graf, 1993; Zips, 1995; M€ uller-Graf and Radojcic, 2002). In the former test series the hull forms are of the round bilge configuration, while the latter series is based on hard chine forms. The Southampton series embraces the range of hull form and configuration parameters shown in Table 12.9. In the Southampton series, the value of the friction coefficient (CF) for the hulls is derived from the ITTC 1957 correlation line and (Insel and Molland, 1992) define the total resistance of a catamaran as, CT ¼ ð1 + fkÞsCF + τCW in which (1 + k) is the form factor of a demihull and where f is introduced as a modification for the change in pressure field around the hull. s is a velocity augmentation term between the hulls. τ is the wave resistance interference factor between the hulls. CW is the wave resistance for a demihull in isolation. By combining f and s into a viscous interference factor b, the total resistance can then be written as: CT ¼ ð1 + bkÞCF + τCW There is some debate concerning appropriate values of 1 + k with the ITTC recommending a value of unity for highspeed vessels. However, from analysis of the data, it is claimed that a satisfactory representation of the (1 + bk) term can be achieved from the relationship,   L 0:4 ð1 + bkÞ ¼ 3:03 r1=3

With regards to the residuary resistance CR use can be made directly from the data given in Molland et al. (1996) or from a regression based representation of the NPL monohull series corrected for wave interference by τ, the wave interference factor. For a monohull in isolation, the NPL series can be represented by   L n 1000CR ¼ a r1=3 where the coefficients a and n vary with Froude number (Fn). This variation is shown in Table 12.10. Then to convert this the introduction of the residuary resistance interference factor (τr) is introduced and this is for the most part dependent on Froude number, the L/r1/3 ratio, and the lateral separation of the hulls. In an analogous way to the estimation of the residual resistance coefficient CR a value of τr can be derived from   L q τr ¼ p r1=3 where the values of p and q may be found from Table 12.11. Based on the work done at Southampton, it is interesting to note that the values of τr did not vary significantly with hull form when, for example, comparing the NPL series with the results for the Series 64 hull form catamaran. This suggests that the interference factors could be applied to a wider range of monohull forms. Consequently, to estimate the total resistance coefficient for the full size catamaran CT, this is derived from the following relationship noting that the model length for the NPL extended series was 1.6 m.  CT ¼ CF + τr CR  bk CFðmodelÞ  CFðshipÞ from which the total hydrodynamic resistance of the catamaran is given by where S is the total wetted surface area of the hulls. 1 RT ¼ rCT SV 2 2

Resistance and Propulsion Chapter 12

347

TABLE 12.11 Regression Coefficients p and q Sc/L 5 0.2

Sc/L 5 0.3

Sc/L 5 0.4

Sc/L 5 0.5

Fn

p

q

p

q

p

q

p

q

0.4

1.862

0.15

0.941

0.17

0.730

0.28

0.645

0.32

0.5

1.489

0.04

1.598

0.05

0.856

0.20

0.485

0.45

0.6

2.987

0.34

1.042

0.09

0.599

0.34

0.555

0.36

0.7

0.559

0.40

0.545

0.39

0.456

0.47

0.518

0.41

0.8

0.244

0.76

0.338

0.61

0.368

0.57

0.426

0.51

0.9

0.183

0.89

0.300

0.67

0.352

0.60

0.414

0.52

1.0

0.180

0.90

0.393

0.55

0.541

0.40

0.533

0.39

To this total resistance an air resistance should also be estimated where the air drag coefficient is likely to lie within the range 0.5 < CD < 0.6 for first estimation purposes, and consequently the air resistance will be given by, 1 Rair ¼ rair CD AT V 2 2 where AT is the above water transverse projected area of the catamaran. Furthermore, if the catamaran is to be driven by a conventional propeller system, then an appendage drag Rapp must also be taken into account such that the total resistance becomes RTotal ¼ RT + Rair + Rapp and hence the required effective power becomes PE ¼ RTotalVS. The VWS test series of catamarans comprises a set of data relating to hard chine hull forms. This test series embraced the range of hull and speed parameters given in Table 12.12 where the value of the describing parameters is based on a demihull. TABLE 12.12 Characteristic Parameters of the VWS Catamaran Series Volumetric Froude Number (V/√(gr1/3)

1.0–3.5

Lwl/b

7.55–13.55

Lwl/r1/3

6.25–9.67

Midship deadrise angle (bM)

16–38 degrees

Longitudinal center of buoyancy (l.c.b.)

0.38 Lwl at bM ¼ 16 degrees–0.42 Lwl at bM ¼ 38 degrees

Transom flap angle (dW) (wedge)

0–12 degrees

In this series, the clearance between the demihulls is constant and specified as G/Lwl ¼ 0.167. This leads to separation between the demihull centerlines ratio (SC/Lwl) of between 0.24 and Lwl/b of 13.55 up to 0.30 for an Lwl/b ¼ 7.55. A regression analysis (Zips, 1995) was carried out upon the resistance data derived for the series. In this study the residuary resistance, derived using the ITTC 1957 line, was nondimensionalized with respect to displacement (RR/D). Additionally, three independent hull form descriptors were transformed to the normalized parameters as follows:   Lwl  10:55 =3 X1 ¼ b   X2 ¼ bM  27° =11° X3 ¼ dw =12° The length to displacement ratio for the demihull can be expressed as follows:

Lwl ¼ 7:651877 + 1:694413X1 + 0:282139X12 r1=3  0:052496X12 X2 and the wetted surface area coefficient (S/r2/3) for a demihull is given as: . X S

¼10 r2=3

CSi  XSi

where the values of the constituent components of the equation are given by Table 12.13. The residual resistance ratio (RR/D) is given by the relationship: X ðXRi  CRi Þ=100 RR =D ¼ where the values of XRi and Cri can be found in Table 12.14.

348 Marine Propellers and Propulsion

TABLE 12.13 Wetted Surface Area Coefficients CSi

XSi

+1.103767

1

+0.151489

X1

+0.00983

X22

0.009085

X12

+0.008195

X12X2

0.029385

X1X22

+0.041762

X13X22

FIG. 12.33 False bow fitted to a catamaran.

When operating in a seaway there will also be an added resistance acting on the hulls. This will generally comprise two parts: the first, due to the perturbed water through which the catamaran hulls are passing and, secondly, due to the seakeeping characteristics of catamarans. Regarding this latter characteristic, to give added buoyancy to bow region of the craft, catamarans are often fitted with a false, or dummy, bow situated forward between the two hulls,

Fig. 12.33. In a seaway, this center or dummy bow will frequently become partially or fully immersed and, therefore, a time-varying additional drag force will be incurred. Wave piercing catamarans are a subclass of catamarans, which are designed with a fine bow form. The fineness of the bow has a reduced buoyancy, which when a wave is encountered tends to pierce through the wave rather than trying to ride over it. This, in turn, reduces the crafts resistance in a seaway.

TABLE 12.14 Values of XRi and Cri CRi XRi

Fr— 5 1.0

X0 ¼ 1

1.25

1.50

1.75

2.0

2.5

3.0

3.5

2.348312

4.629531

5.635988

5.627470

5.690865

6.209794

7.243674

7.555179

X1 ¼ f(L/b)

0.706875

2.708625

2.371713

2.266895

2.500808

2.900769

3.246017

2.647421

X2 ¼ f(bM)

0.272668

0.447266

0.328047

0.428999

0.422339

0.391296

0

0.453125

X3 ¼ f(dW)

0.558673

0

0

0

0.288437

0.447655

0

0

X4 ¼ X1

2

0.256967

0.701719

0.349687

0.416250

0.571875

0.832031

0.554213

0.332042

X5 ¼ X1

3

0

0

0.165938

0.342187

0.496875

0.658719

1.196250

1.884844

X6 ¼ X2

2

0

0.148359

0

0

0

0

0

X7 ¼ X3

1/2

0

0

0

0

0

0

X8 ¼ X3

1/3

X9 ¼ X3

1/4

0.152163

X10 ¼ X1X2

0

0.251026

0.429128

0.450245

0.866917

0

0

0

0

0

0

0

0.149062

0.090188

0.089625

0.076125

0

0.135563

0.194250

0.190125

X11 ¼ X1X6

0.151312

0.090188

X12 ¼ X4X6

0.0592

0.322734

0

0.225938

1.87743 0 0 0.332250 0.211125

0.276875 0 0 1.036289 0.767250 0

0

0

0

0

0

X13 ¼ X4X2

0

0.148359

0.096328

0

0

0

0

0

X14 ¼ X1X3

0

0.409500

0.484800

0

0.817200

1.189350

1.007700

0

X15 ¼ X4X7

0

0

0

0

0

0

0.588758

0.683505

0

0

0

0

0

0

0

0

0

0

0.704463

0

0

0

0

0

0

0.120516

0.137585

0.257507

0

0

X16 ¼ X1X9 X17 ¼ X1X7 X18 ¼ X2X8

3

0.083789

0.241426

Resistance and Propulsion Chapter 12

12.1.8.2 Trimarans and Pentamarans Trimarans can offer advantages in some aspects of marine technology over conventional designs. Depending upon their design, the trimaran concept can facilitate speeds that may be difficult to contemplate in a conventionally designed hull form. Moreover, they can have improved behavior over the characteristics of a catamaran vessel in relation to seakeeping behavior and operational limits. Additionally, a trimaran ship’s seakeeping and performance is likely to degrade more gracefully than that of an equivalent catamaran. Trimaran ships and, for that matter, pentamarans offer significantly greater degrees of freedom during the design process. These freedoms and the best way in which to exploit them to advantage can, however, be confounded by a lack of insight into the particular nature of the hydromechanics of outrigger supported central hulls. Moreover, the induced structural loads and by implication, the stresses, both steady and dynamic, upon which the structural integrity depends need to be carefully considered. The design of trimaran craft is particularly weight sensitive and ship layout options may place constraints on the breadth of the main hull and, consequently, can have a significant influence on the resistance of the ship. With these craft, the initial location of the longitudinal center of buoyancy needs careful selection to achieve an equitable balance between the seakeeping, propulsion, and passenger comfort characteristics. In addition, the hydrodynamic interactions between the outriggers and the central hull need to be fully taken into consideration such that longitudinal position and spacing relative to the central hull are optimized. If this is not achieved, with respect to the Froude numbers being considered for the operational profile of the ship, then undue hydrodynamic interactions may build up and, in this way, disappointing speed performance can result. Moreover, test results have shown (Ackers et al., 1997) that outrigger symmetry and position are the most important factors influencing trimaran resistance. (Capizanno et al., 2003) have observed a strong relationship between speed and outrigger configurations and suggest that for increasing Froude number outrigger locations toward the bow and stern are more beneficial. However, while clearance effects between the main hull and the outriggers are less easy to interpret, the greater the clearance seems to offer the least interference. Ackers et al. (1997) suggest that the addition of outrigger angles of attack or increasing the side hull displacement is unlikely to enhance performance. Furthermore, the depth of immersion of the outriggers has a significant influence on the ship’s resistance at high speed. Consequently, efforts should be made to lift the outriggers relative to the ship’s waterline in these highspeed conditions. At high speed, the horizontal velocity of water along the vessel’s hull dominates the flow encountered during oper-

349

TABLE 12.15 Comparison of Two Trimaran Vessels Triton

Triumphant

Length overall

98 m

Length overall

54.5 m

Beam overall

22.5 m

Beam (Mld)

15.3 m

Beam (main hull)

6m

Depth

5.5 m

Outrigger length

34 m

Power installed

2  2320 kW

Outrigger beam

1.45 m

Service speed

40 kts

Installed power

2  2000 kW

Maximum speed

44.5 kts

Propulsion

Propeller

Propulsion

3 Waterjets

Range

3000 nm @ 12 kts

Passenger capacity

484

ation in waves. Additionally, model tests should be deployed to quantify the air gap required under the deck bridging the central hull to the outriggers to avoid choking of the flow or slamming, which might induce whipping, which could be significant due to the lightness of the structure. Indeed, it can be difficult to define an air gap between the main hull and the outriggers that will satisfy all ship operating conditions in a seaway. Several trimaran vessels have been designed for operational service either as demonstrators of the technology or as design studies. Typical of these are the experimental ship Triton, which was built in 2000 and the operational passenger ferry Triumphant. The outline dimensions of these ships are given in Table 12.15. Pentamarans have evolved from the trimaran concept in that two sets of outriggers are accommodated instead of a single set in the trimaran. The pentamarans are claimed to be able to maintain high speeds due to enhanced seakeeping qualities. As with trimarans, there is no general solution the optimum location for the outriggers relative to the hull: it is dependent upon the desired operational profile of the ship and the anticipated duty.

12.2 HYDRODYNAMIC SUPPORT PROPULSION 12.2.1

Planing Craft

When at rest or at low speed, a planing craft is little different from a displacement vessel in that Archimedes principle applies: that is, the upthrust caused by the amount of water displaced by the vessel is equal to its weight. However, as

350 Marine Propellers and Propulsion

R

TABLE 12.16 Published Data for Displacement and Planing Craft FH

Standard Series Data Displacement Data

Planing Data

Norstrom Series (1936)

Series 50 (1949)

de Groot Series (1955)

Series 62 (1963)

Marwood and Silverleaf (1960)

Series 65 (1974)

Series 63 (1963) Series 64 (1965) SSPA Series (1968) NPL Series (1984) NSMB Series (1984) Robson Naval Combatants (1988)

the speed of the craft increases, the under sides of the vessel act as a lifting body until the vehicle becomes supported by the lifting forces generated by the action of the lower surfaces of the hull. Additionally, as the craft gains speed, the hull lifts both vertically and also changes trim. A considerable amount of data is available by which an estimate of the resistance and propulsion characteristics can be made. Table 12.16 identifies some of the data published in the open literature for this purpose. In addition to basic test data of this type, various regression-based analyses are available to help the designer in predicting the resistance characteristics of these craft; for example, van Oortmerssen (1971) and Mercier and Savitsky (1973). Additionally, Savitsky and Ward-Brown (1976) offer procedures for the rough water evaluation of planing hulls. Although an oversimplification of the physics of the process, it is instructive to first consider the action of an

Plate

length

ed Wett

(L)

length

a

V

S

FIG. 12.34 Free jet of water impacting on a flat plate.

inclined plate subject to an inviscid, horizontal incident jet of fluid having a velocity V (Fig. 12.34). In this case, the stagnation streamline forms a curved path as seen in the figure and finishes up normal to the plate at the point S. Since the streamline, unlike the case where the plate is normal to the axis of the jet, is neither coincident with the jet axis nor straight, the location of the resultant reaction force (R) does not pass through the stagnation point. It acts somewhat aft of the stagnation point, at the center of pressure of the resulting pressure distribution acting on the plate caused by the impingement of the jet of water of the under surface of the plate. This force may then be resolved into two components: one vertically FV and the other in the direction of the water jet FH. In this latter case, the horizontal force may be thought of as a resistance or drag force of the plate and the vertical component of force opposing the weight of the plate. In addition to the system of forces acting on the plate, it should be noted that the water jet divides in the vicinity of the plate with the larger portion of the flow being deflected downwards while a smaller amount is thrown forward. A more realistic representation of the planing craft situation is a two-dimensional plate of length (L), which moves over the surface of the water at a small angle of incidence (a) with a relative horizontal velocity V; Fig. 12.35. The body of water is assumed to be stationary and without wave disturbances.

tive Plate velocity V rela l city Velo plate V re e th to city V abs te velo Absolu

(L w)

S te

FV

Spray root

d

Cp

te FIG. 12.35 Velocity and pressure distributions on a wetted planing surface.

S

Resistance and Propulsion Chapter 12

Clearly, there is a similarity between Figs. 12.34 and 12.35; however, the wetted length (lw) of the plate is defined as that length between the trailing edge of the plate and the root of the spray. When developing a model of the hydrodynamic action of a planing craft, the water thrown forward from the root of the spray is assumed to extend forward to infinity; however, in reality, the water breaks up into spray relatively quickly. Also shown in Fig. 12.35 is the pressure distribution on the wet side of plate generated by the planing motion from which it is seen that the peak pressure and, of course velocity, is generated at the stagnation point on the plate. From these maxima, the distributions of pressure and velocity fall away quickly either side of the stagnation point. In developing an analysis for the 2-dimensional model shown in Fig. 12.35, it is assumed that the water is at rest, consequently, ignoring deformations to the surface of the water and neglecting viscosity, the only energy imparted to the water is that required to throw the water forward from the stagnation point. Applying Bernoulli’s equation to the free surface streamline, it is seen that the velocity (V) is constant since the pressure remains sensibly constant for small angles of incidence; consequently, the water thrown forward also has a uniform velocity V relative to the inclined plate. To determine the absolute velocity of the water thrown forward and given the plate velocity is V horizontally, then from Fig. 12.35 it can be seen that: Absolute velocity of the water thrown forward ¼ 2V cos ða=2Þ Therefore, noting that the depth of the stagnation streamline relative to the free surface is (d) the mass of water thrown forward is rwVd, where rw is the density of the water. Consequently, kinetic energy associated with the water thrown forward per unit width is given by: 1 E ¼ rw Vd ½2V cos ða=2Þ2 2 Now by assuming R0 is the resultant force per unit width of plate, then the rate at which work is done on the water by this element of the plate is, 1 R0 V sin a ¼ rw Vd½2V cos ða=2Þ2 2 Given that the ambient pressure has been assumed to be constant, we may rearrange the foregoing equation and with a little manipulation using the trigonometric addition theorems show that: R0 ¼ rw V 2 d cot ða=2Þ This relationship suggests that a useful reaction force and hence lifting force on the hull can only be developed if water is thrown forward and, consequently, a sprayless planing craft is not possible: as such in the model developed here d > 0.

351

d FIG. 12.36 Definition of deadrise angle (d).

So far in the discussion the bottom of the hull has been approximated to a flat plate of infinite spanwise dimension. Clearly this is impractical, and planing hulls take up round bilge or hard chine, V bottom forms; however, it has to be recognized that some vessels exhibit characteristics of both these hull types. To describe practical planing hull forms, it is necessary to introduce the concept of aspect ratio, which is defined in this context as the ratio of the mean wetted beam squared of the vessel to plan area of the wetted surface of the hull. Additionally, the concept of deadrise angle of the hull (d) has to be introduced as seen in Fig. 12.36. Given that consideration now turns toward a threedimensional hull instead of the two-dimensional concept previously considered, the assumptions about the water thrown forward and spray produced no longer hold true. While the velocity of the flow will still be maintained however, its direction will tend to take up a transverse component relative to the hull and in some cases develop an astern component. When this happens the absolute velocity of the flow reduces, as does its kinetic energy, and this results in a decrease in the hydrodynamic forces on the hull. The trajectory of the spray emerging from under the vessel depends on the deadrise angle of the hull, its curvature, and the running trim of the craft of the hull. With low deadrise of the hull, the spray will be directed toward the sides of the craft, whereas when higher hull deadrise angles are used the spray will be directed well aft. To detach the sheet of spray from the hull, chines or spray rails are used with the aim of keeping the emerging spray to a low trajectory. However, problems with spray control may arise if the spray rails are poorly located; are of an inappropriate thickness; or are of the wrong shape. The effectiveness of the spray rails is strongly governed by their shape: the undersides should be horizontal or transversely tapered by a few degrees so that the outboard extremity is lower than the inboard attachment to the hull and the outboard edges must be sharp. The hull deadrise angle also influences the magnitude of the hull surface pressures acting on the hull, with smaller angles giving rise to higher pressures: both of a steady and dynamic nature. To consider the forces acting on a fully planing craft, the vessel can be approximated as a rectangular box having a wetted beam and length, of bw and lw, respectively, and operating at a running trim a. For most practical cases, excluding certain classes of power boat, the running trim seldom exceeds 4–5 degrees. Additionally, in these cases,

352 Marine Propellers and Propulsion

a

z

E FIG. 12.37 Planing hull at incidence.

Hence, integrating,  ðz  rw gbw 1 zdz ¼ rw gbw lw 2 sin a F¼ 2 sin a 0 This resultant force, which lies through a point on the hull, will therefore be capable of resolving into a vertical hydrostatic force and horizontal components, FV and FH, respectively, as F cos(a) and F sin(a). The system of forces acting on a planing hull when undergoing steady motion and including the effects of viscosity comprises the following: i. The hydrostatic force (Fh), which acts at the center of pressure on the hull. ii. A reaction force (Fp), which derives from an integration of the hydrodynamically induced pressure field over the wetted part of the hull and this acts at a point astern of the stagnation streamline on the hull—see Fig. 12.34. iii. The net force (Fs) arising from the skin friction acting on the wetted hull. iv. The all-up weight of the vessel (W) acting through the center of gravity. v. The propulsor thrust (T) acting along the shaft line, which may be different from the line of the craft’s keel. vi. The vertical bearing force (Pb) resulting from the propeller operation. vii. The hull surface pressure forces (Pz) arising from the passage of the propeller blades beneath the hull and the growth and collapse of cavitation developed by the propeller. viii. Normal forces from the rudder and appendages, (Nr) and (Na), respectively. ix. The total weight of the craft (W), which acts through the center of gravity. For steady motion of the craft, the resultant forces and moments arising from these loadings must be zero about

FIG. 12.38 Force system acting on a planning craft in steady motion.

Resistance (R/W)

the transom can be assumed to be unwetted. Referring to Fig. 12.37, at some location, E, on the hull at a depth (z) below the free surface, the static pressure on the wetted hull is p ¼ rwgz and this will give rise to a force (dF) acting on an elemental area of the hull surface, bw(dz/ sin a), given by   r gb dF ¼ w w zdz sina

Total

Speed

FIG. 12.39 Typical planing craft.

performance

characteristics

of

hard-chine

the longitudinal center of gravity of the craft. Some variation in the lines of action of the resultant hydrostatic and hydrodynamic forces does occur due to the curvature of the hull and the vessel speed being considered and these variations must be taken into account (Fig. 12.38). In essence, the total resistance (RT) can be expressed as the sum of three components. These are the induced resistance, which derives from the inclination of the net hydrodynamic forces; the wave-making and pressure resistance; and the skin friction resistance. Fig. 12.39 shows, in outline, an example of the resistance characteristics of a hard chine planning craft. Performance studies on planing craft, both analytical and from sea trials, have shown that the application of stern wedges changes the running trim angle. Indeed, du Cane (1947) showed how the application of wedges and trim tabs reduced the squat from planing craft and Monk (1958) gave guidance on suitable sizes for wedges for small high-speed craft. Savitsky and Brown (1976) noted that while a wedge was helpful in minimizing resistance, they did not offer the flexibility of trim tabs for optimizing the planing resistance when the speed and loading change.

Resistance and Propulsion Chapter 12

12.2.1.1 Stern Wedges, Flaps, and Interceptors Stern wedges are not a new invention in that they date back to the mid-1930s where they were fitted to the Type 34 Destroyers of the German Navy. However, the hydrodynamic mechanisms that apply to large ships and small craft are different. All wedges and stern flaps create a vertical lift force and modify the pressure distribution over the after part of the hull. In the case of large ships, these changes do not generally change the ship’s trim significantly, but the greatest benefit at appropriate higher speeds is in the modification the pressure distribution over the hull after-body and the modification to the flow field and any consequent reduction in wave-making resistance. This is in contrast to the application to small high-speed craft where the influence of the stern wedge or flap modifies the running trim of the vessel, sometimes to a significant effect since these craft can be very sensitive to these devices. As such, a wedge or flap can be used to optimize the hull trim angle with a view to minimizing resistance and gives rise to the concept of variable trim flaps so that differing circumstances across the operational spectrum can be accommodated. The design of a stern wedge or flap should be carried out in conjunction with model tests and computational fluid dynamic (CFD) methods. Scale effects in model testing need to be considered with care because scale effects can be a function of Reynolds number, the boundary layer thickness, the model scale ratio, and frictional resistance coefficient. Karafiath et al. (1999) discuss experience gained through a long programme of research with naval ships of the cruise, destroyer, and frigate types from which several conclusions were drawn. In summary, these were as follows: i. Stern flaps modify the flow around the transom so that energy lost due to eddy making and turbulence can be attenuated. ii. The near-field wake can be reduced in width and height by the introduction of a stern flap, similarly with the wave breaking directly behind the transom and at the edges of the inner stern wave system. iii. A stern flap has the potential to reduce the far-field wave energy of the ship and hence the wave-making resistance of the ship. iv. The addition of a stern flap increases the effective waterline length of the ship, which, in turn, by this physical lengthening of the ship may enhance the flap’s resistance reduction. v. In terms of propulsion interaction, there were a number of findings from this quite extensive study, which were: a. The primary influence of a stern flap is to slow the speed of the water flow under the stern of vessel and, thereby, increase pressure in this region.

353

b. From model tests with stern flaps, it was found that on average the delivered power reductions were a few percent greater than the resistance reductions. c. The increase in pressure under the hull developed by the introduction of a stern flap tended from visual full scale observations to suppress propeller cavitation and reduce thrust breakdown in the higherspeed range. d. The suppression of cavitation leads to the possibility of developing a higher cavitation inception speed. e. An increase in pressure combined with a reduced propeller loading may lead to an enhanced propeller efficiency due to a reduction in cavitation activity. vi. The stern flap, due to the lift forces and increased pressure under the stern region of the hull, induces a trim by the bow. This effect for displacement ships is small when compared to the trim changes experienced by planing craft. As a general guide, stern flaps are relatively ineffective for Froude numbers less than about 0.2; however, they do seem to significantly adversely affect low-speed ship operation. Interceptors, which can be either fixed or adjustable, are essentially a plate positioned as an extension to the transom and extending downwards beyond the shell plating. Their function is similar to the wedge or flap in controlling the dynamic trim of the craft to which they are fitted. In operation, they lie typically inside the boundary layer of the craft as seen in Fig. 12.40 where a schematic illustration of the operation of an interceptor of height h. Clearly, the interceptor introduces a drag force on the craft but increases the pressure beneath the hull. This, in turn, influences the trim of the craft. At sufficiently high Froude numbers, the flow separates from the interceptor plate and in doing so creates a hollow behind the transom of the craft. To achieve the same effect as a flap at speed, the interceptor plate can be much smaller than the trailing edge height of the wedge: typically, of the order of 50% depending on the circumstances prevailing. This leads to an advantage of the interceptor in that its weight is likely to be less than that for a wedge and weight is important for small high-speed craft. Clearly, these smaller dimensions can be an advantage at low speed due to the smaller proportion of immersed transom area (Fig. 12.40).

12.2.2

Hydrofoil Craft

Unlike conventional planing craft, at high speed the hull of a hydrofoil supported vehicle is designed to be clear of the water and its weight supported on struts attached to hydrofoils as seen in Fig. 12.41. The underlying principles of hydrofoils were considered in Chapter 7; consequently, here only the application to the hydrofoil supported craft will be considered.

354 Marine Propellers and Propulsion

FIG. 12.40 Outline of an interceptor and its operation.

Cp

Interceptor

Hull

FIG. 12.41 A passenger hydrofoil craft. (Courtesy: Hellenic Seaways.)

The hydrofoils of these craft are configured from either submerged or surface piercing foils. These foils are normally arranged with one foil forward which is used for both support and steering while the aft supporting foil combines a supporting role and the inlet flow ducting for the waterjet propulsion system if, indeed, that mode of propulsion is selected. An alternative propulsion method is to use propellers, normally in a twin screw configuration as would normally be the case with waterjets. In terms of definition, if >65% of the craft’s weight is supported by the forward or aft foil arrangements at maximum speed, the configuration is termed Conventional or Canard. These are the most common types of hydrofoil craft in operation and within these definitions the foils may either be designed as comprising a single continuous foil forward and aft or as having a split configuration at each or one of the longitudinal stations. In many modern craft, the hydrofoils are fitted with flaps for the purposes of controlling heel, trim, and stabilization in a seaway. Fig. 12.42 outlines a general sketch of the hydrofoil craft arrangements. When hydrofoils are operating at speed and close to the free surface of the water, a number of additional factors require consideration beyond those when they are deeply submerged. These additional considerations include:

i. The hydrofoils, upon which the craft is supported, must be sufficiently far away from the craft’s hull so as to avoid contact with the waves at cruising speed. ii. The scantlings and profile of the hydrofoils and struts need to be consistent with the required strength to accommodate the steady and transient loadings safely, while at the same time minimizing the hydrodynamic drag of the struts. iii. The hydrofoils and their supporting structure need to be located, constrained, or protected from either grounding or fouling the quay when docking or lying alongside. iv. The proximity to the surface of the hydrofoil will introduce a surface wave component on the surface of the otherwise undisturbed free surface of the water. v. The proximity to the surface of the hydrofoils introduces the possibility of significant cavitation occurring at high speeds. vi. The design of the struts must minimize the possibility of air drawing, or ventilation, down the strut to the lowpressure regions on the surfaces of the hydrofoils.

12.2.2.1 The Free Surface When considering free surface disturbance in relation to hydrofoil craft hydrodynamics, the developed pressure

Resistance and Propulsion Chapter 12

355

FIG. 12.42 Hydrofoil configurations. (From Johnston, R.J., 1985. Hydrofoils. Naval Eng. J. 97 (2), 142–199. Reproduced with permission from the American Society of Naval Engineers.)

distributions around the hydrofoil together with rigid body motion of the foil in proximity to the free surface introduce a wave resistance component. This has long been an issue confronting hydrofoil craft designers, and Keldysh and Lavrent’ve (1936) and Kochin (1936) developed a relationship for the wave drag coefficient (Cw) as follows: 2gd=V 2 Cw  ¼ gc=2V 2 2 CL where CL is the lift coefficient V is the hydrofoil speed c is the chord length of the hydrofoil g is the acceleration due to gravity d is the submergence of the hydrofoil. While such a relationship suggests that the contribution of the wave to the lift-drag characteristics of the hydrofoil is small, the presence of the free surface alters the 2dimensional hydrofoil characteristics as well as introducing a wave drag component. Hough and Moran (1969) considered this problem by representing thin hydrofoils with a bound vortex distribution with the potential representing the vortex satisfying the free surface boundary condition for small waves; that is, V2ux + gv ¼ 0. Such a modeling of the flow situation leads to a Froude number dependent distribution of doublets trailing downstream from the vortex and under the linearizing assumptions leads to an integral equation formulation of the problem—the solution of which yields a relationship as shown in Fig. 12.43 where it is seen that a chordally based Froude number of around 10 is required before the gravitational effects become negligible.

FIG. 12.43 Influence of chordal Froude number on a hydrofoil lifting properties (Hough and Moran, 1969).

Real hydrofoil situations are constrained by finite aspect ratios and an early consideration of the problem was by Wu (1954). More recently within this context Breslin (1961) showed that the nonviscous drag comprised three components: i. The induced drag due of the foil in an unbounded fluid. ii. The induced drag due to the foil’s induced drag in its biplane image in the free surface. iii. An additive wave resistance, which decays asymptotically with the chordal Froude number squared. Nishiyama (1965) in a series of papers developed an optimum circulation distribution for dihedral and surface piercing hydrofoils for operation in surface waves and a constant induced velocity distribution. The resulting circulation distribution tends to be rather fuller than the classical

356 Marine Propellers and Propulsion

elliptical distribution. In other papers, he has considered the effects of flaps and has also looked at the comparisons between the lifting line and lifting surface approaches where it was estimated that the lifting line approach overestimated the lift slope by around 25% at an aspect ratio of 4. Wang and Rispin (1971) showed that for aspect ratios exceeding unity reduces the gravity effect decreases lift, whereas below about unity the lift is increased.

12.2.2.2 Hydrofoil Cavitation The foils of a hydrofoil craft are never far away from the free surface of the water and, consequently, the static head over the foil tends to be low. This coupled with the dynamic head, which is a function of ship speed increases the risk of cavitation. When cavitation occurs, depending upon its type and severity, this may either result in a modification of the lifting capabilities of the foils due to its influence on the pressure distributions generated around the foil or in some cases to the destruction of the foil material by cavitation erosion. Figs. 9.21 and 9.22 outline the general influence that cavitation can have pressure distributions and lifting properties of hydrofoils. In general, hydrofoils can be designed to operate at speeds of up to around 50–60 knots without incurring the significant effects of cavitation, although some will be present. Beyond these speeds, increasing amounts of cavitation should be expected until a supercavitating state is eventually reached. Because the foils of hydrofoil craft are complex structures when combined with their supporting struts, the cavitation is usually very three dimensional: this applies to both the leading and trailing sets of foils. Systems of three-dimensional cavitation do have a tendency to become unstable from which significant fluctuating loads can be created. Cavitation buckets, of the type seen in Fig. 9.23, define the likelihood and the type of cavitation that may occur for a given set of design situations. While in many applications some limited amounts of leading edge cavitation are likely to be encountered, and is perhaps unavoidable, this is FIG. 12.44 Typical foil lift characteristics in relation to flap angle. (From Johnston, R.J., 1985. Hydrofoils. Naval Eng. J. 97 (2), 142–199. Reproduced with permission from the American Society of Naval Engineers.)

unlikely to have a marked effect on the hydrodynamic properties of the foils and has the possibility of reducing the foil’s response to wave-induced variations in the section angle of attack. More serious, however, is the occurrence of midchord cavitation and this needs to be avoided for both its adverse hydrodynamic and erosion potential. To control hydrofoil craft, the use of flaps is commonly used. Their modes of operation are not dissimilar to the use of control surfaces fitted to airplanes in that the lifting characteristics of the foil are modified by the chosen angle of flap, Abbott and von Doenhoff (1959). In the case of hydrofoils the effects of cavitation and, if it occurs, ventilation has to be considered. With regard to cavitation, its influence will be dependent upon its type and extent. Up to a certain point in the case of leading edge cavitation, the effect will be marginal; however, as cavitation progresses toward supercavitation, the effect becomes progressively more pronounced as is the case with propeller blade sections. Indeed, the use of flaps will modify the pressure distribution around the foil, so cavitation may transform from one form to another. Fig. 12.44 outlines the effects of flap angles at different foil angles of attack on the lift for cavitating and ventilated aerofoils. The systems of aerofoils such as are found in these craft to support the weight of the vessel foils are placed in forward and aft locations. Clearly the forward foils system will influence the aft located foils since cavitation inception and development is to a very large extent dependent upon the angle of attack of the flow incident on the hydrofoil. The forward foil, because it is of finite dimensions, generates a downwash due to the shed vorticity as discussed in Section 7.9. This variable downwash along its span alters the flow incident upon the after set of hydrofoils and, consequently, the cavitation characteristics of that foil. This interaction effect can be most pronounced if the span of the forward system of hydrofoils is greater than that of the aft foil system. A further cause for concern is the design of the intersections between the supporting struts and the hydrofoil. In much the same way as with propeller blade roots, trailing

Resistance and Propulsion Chapter 12

12.2.2.3 Ventilation Ventilation from atmospheric air into the foil systems can be a problem for hydrofoil craft. Ventilation is the entrainment of air along a hydrofoil, which will induce a loss of lift for the foil. The paths ventilation may take can be quite complex; for example, down the supporting struts of the craft and on to the hydrofoils. This phenomenon occurs due to the low pressures generated on the hydrofoil surfaces and, which, if a suitable differential with atmospheric pressure is created, an air flow is induced to travel along the foil. In many cases, the situation can be attenuated by providing suitable fences along the hydrofoils in the chordal direction. See also Section 6.9.

12.2.2.4 Hydrofoil Propulsion When at rest or operating at very low speed, the propulsion of a hydrofoil is broadly similar to a displacement hull but with the addition of the system of drag forces associated with the hydrofoil system. However, when foil-borne the resistance components comprise: 1. The viscous resistance of the foils and struts. 2. The induced drag developed by the trailing vortex sheet created by the foils. 3. The wave resistance of the foils and struts, which arises from the lift, or pressure distribution, and thickness of the foils in the lightly submerged condition. 4. The spray resistance acting on the struts, which is caused by both the potential and viscous flows. For vertical equilibrium, the lift forces generated by the forward and after foil system, (Lfwd foil) and (Laft foil), respectively, together with the vertical component of the aerodynamic force acting on the hull and superstructure (Fv aerodynamic) must balance the weight of the craft (W), its cargo and passengers, stores, fuel, lubricating oils, and other contained fluids. i:e:, Lfwd foil + Laft foil + Fv aerodynamic ¼ W Similarly, for equilibrium in the horizontal direction, the drag forces (D) generated by the foils, the strut spray resistance, and the horizontal component of the aerodynamic drag of the hull and superstructure (Fh aerodynamic) must balance the propulsor thrusts (T) for steady motion. Consequently, assuming there are n propulsors,

Dfwd foil + Daft foil + Rspray + Fh aerodynamic ¼

n X

Ti

i¼1

Recognizing the longitudinal distribution of these various forces and their disposition relative to reference frame, the resultant moments must also balance to enable the craft to achieve its desired operating trim to maintain the necessary lift forces from the hydrofoil sections and their control flaps. Furthermore, it should also be borne in mind that all of these forces will have both a steady and fluctuating component, which will vary depending upon the weather and its direction relative to the craft. Indeed, when in operation, hydrofoil craft may have to comply with operating envelopes with respect to the weather. If the sea becomes lumpy or wave heights become high and come from an adverse relative direction, shock loading through a slamming type of action can become severe. This, in addition, to the natural rolling tendency of these craft may become uncomfortable for passengers and in extreme cases induce structural damage. Fig. 12.45 illustrates a typical resistance curve for a hydrofoil craft where it is seen that as the speed of the craft is increased a resistance hump occurs at some point between the maximum speed and rest. At the resistance hump, the hull is still in the water and the resistance comprises both that of the hull and the drag of the foils and supporting systems. As speed increases, the resistance falls away as the hull comes out of the water before rising again under the influence of the drag associated with the foil systems. Unless sufficient power is provided within the machinery system to get over the resistance hump then, as with other high-speed craft types, the vessel will not reach its full speed. During the acceleration process, the trim angle of the hull will vary, typically rising to a maximum in the region of the resistance hump the falling off somewhat at high speed.

Resistance and thrust (R/W & T/W)

vortices are generated because both the foil and strut are lifting surfaces. Consequently, without due care over the detail of these component parts drag in its various forms and cavitation development can arise. Nevertheless, apart from the hydrodynamic consideration, strength considerations must prevail.

357

T/W

Maximum speed

R/W

Take-off speed

Speed FIG. 12.45 Typical variations in resistance and thrust with speed.

358 Marine Propellers and Propulsion

FIG. 12.46 Flexible skirt air cushion vehicles.

12.3 AEROSTATICALLY SUPPORTED VEHICLE PROPULSION Aerostatically supported vehicles take on a variety forms and for the purposes of this chapter craft supported on an air cushion will be considered. Air Cushion Vehicles (ACVs) have been developed with either flexible skirts or having fixed side walls with flexible skirts located at the bow and stern of the craft. They have been produced in a wide range of sizes as shown in Fig. 12.46 for several fully flexible skirt hovercraft. Furthermore, this figure vividly contrasts the size difference between the SRN 4 car and passenger hovercraft and much smaller passenger carrying craft. In the case of side walled hovercraft, these are normally propelled by marine propellers, which is in contrast to the air propellers fitted to hovercraft having all-round flexible skirts. Furthermore, in the case of side wall hovercraft, some contribution to buoyancy is obtained from the light immersion of the side walls, which are aligned in a fore and aft direction. This is not the case for the air cushion vehicle having flexible skirts where the craft relies solely on building and maintaining an air cushion to support itself since the craft is designed with a nominal clearance between the bottom of the skirt and the ground and, in this way, these vehicles can be deemed to be amphibious in that they may be operated over land or water. When operating over water, the pressure under the hovercraft distorts the water surface since that pressure differs from the ambient conditions and this distorted surface moves with the hovercraft. Of primary importance to the design is maintaining the leakage of air to a minimum to avoid energy being wasted from the prime mover generating the air cushion. However, leakage of some magnitude is inevitable if the craft is to hover above the surface upon which it is traveling and this leakage will be a variable with time depending upon the roughness of the land or sea surface. The control of this leakage from the flexible skirt is clearly of importance to enhance the overall efficiency of the craft. There are two basic forms that a flexible skirt hovercraft can take for delivery of the floatation air. The first is where air is pumped into a large plenum chamber from the top of the craft and then exhausts back into the atmosphere

between the base of the skirt and the ground: whether this be land or water. The size of the plenum in these versions may be equivalent to the size of the craft. The alternative, and the one that will be pursued further here, is for the incoming air to be channeled around the internal periphery of the hovercraft and then exhausted through a system of nozzles back into the atmosphere between the base of the skirt and the ground as outlined in Fig. 12.47. To develop a simple theoretical approach to hovercraft propulsion, it is necessary to make several assumptions about the development of the air cushion. These are as follows: i. The pressure in the air cushion is constant. ii. The effects of viscosity are negligible. iii. The air is stationary in the air cushion and in the surrounding atmosphere. iv. The flow may be treated in a two-dimensional sense. v. The density and atmospheric pressure of the air is constant. Referring to Fig. 12.47, which shows a two-dimensional slice along a simple hovercraft, we see a configuration comprising a nozzle discharging into the area of height (h) under the hovercraft and above the ground. Due to the pressure (pc) within the cushion area, the air is deflected outwards along the ground to atmosphere where the ambient pressure (pa) is considered constant. Now, consider the flow of the jet in which the pressure is taken as being constant (pt) and by ignoring the small changes in height, the static pressure across the jet must rise from pa on the outside to a higher value pc at the inside streamline adjacent to the cushion. As a consequence, this velocity v0 along the streamline S0 must, therefore, be greater than the velocity vc along the streamline Sc. Furthermore, since the velocity of the fluid in the nozzle vn is constant, since the pressure is constant, then a discontinuity must exist in the velocity and pressure at the emergence of the nozzle. By appealing to Bernoulli’s theorem and ignoring small changes in heights and where r is the density of the air: 1 1 1 1 pt ¼ p + rV 2 ¼ pn + rVn2 ¼ pa + rVn2 ¼ pa + rV02 2 2 2 2 ¼ Constant (12.51)

Resistance and Propulsion Chapter 12

359

FIG. 12.47 Simplified flow regime from the jets of an ACV vehicle. (After Harting, A., 1969. A literature survey of the aerodynamics of air cushion vehicles (AGARD Rep., No. 565).)

This implies that nozzle velocity and the velocity along to the streamline S0 are equal; that is Vn ¼V0. By now considering the equilibrium of any fluid element in the jet as seen in Fig. 12.47, it can be shown for the element of flow highlighted in the figure that for equilibrium of the flow system rotating about the point O at constant radii: Pressure Forces of the Element ¼ Centrifugal Forces on the Element i:e:, ½ðp + dpÞ  prdf ¼ rrdfdr dp rV 2 ¼ Hence, dr r

V2 r

that is, p  pt ¼

1 Kr 2

Consequently, by using the boundary conditions r ¼ r0 and p ¼ pa, an expression for K can be derived and then the local pressure in the jet (p) can be expressed as, r 2 0 p ¼ pt  ð pt  pa Þ (12.53) r Returning to Eq. (12.51) and transforming we find that

(12.52)

in which r is the local radius of curvature of the streamline along which the element moves and, moreover, since in steady flow the velocity V along the streamline is constant it follows that dp/dr is also constant. Clearly, further refinements to such a relationship can be introduced by, for example, assuming the streamlines form an elliptical motion. The width of the jet emanating from the nozzle (a) remains constant under this model. Therefore, since from Eq. (12.51) rV 2 ¼ 2ðpt  pÞ we have by substituting into Eq. (12.52) the following relationship dp dr ¼2 pt  p r which upon solution gives,

 ln ðp  pt Þ ¼ 2 ln ðr Þ + constant

1 pt  pa ¼ rVn2 2 And then substituting into Eq. (12.53) we have,  r 2 1 0 p ¼ pa + rVn2 1  2 r and consequently, the air cushion pressure (pc), which is identical to that of the pressure of the streamline Sc is given by, "  2 # 1 2 r0 pc ¼ pa + rVn 1  2 rc An air cushion pressure coefficient can then be defined as: pc  pa Cpc ¼ 1 2 rV 2 n

360 Marine Propellers and Propulsion

FIG. 12.48 Complexity of the flow field beneath a simple jet model of an ACV. (After Harting, A., 1969. A literature survey of the aerodynamics of air cushion vehicles (AGARD Rep., No. 565).)

In the derivation of the relationship for air cushion pressure (pc), the major simplification that has been made that the air remains stationary in the air cushion. However, Harting (1969) showed the complexity that exists in the flow field beneath a simple peripheral jet ACV model that is hovering; Fig. 12.48. From this figure, it is seen that two large vortices are created and that a stagnation line is formed between the jet efflux and the ring vortex system. This induces the flow near the ground to divide with one part flowing inward while the other flows outwards away from the hovercraft. Clearly, if the vehicle is circular in planform then the vortex formed will comprise principally of a toroidal ring vortex, however, if different planforms are created, then the vortex will assume very different forms. The size of the internal vortices is dependent, to an extent, on the height at which the craft is hovering. However, there comes a point where the flow character changes completely to a focused jet format (Harting 1969). On the outside of the jet formed by the nozzle, air is also set in motion as seen in the figure. If the air cushion vehicle is hovering over water, then a modified regime ensues. The cushion pressure gives rise to a depression in the water surface, which gives the appearance of the craft settling into the water and, consequently, reduces the clearance of the craft relative to the still water. As such, due to the difference between the cushion and atmospheric pressures, water in the form of spray is forced out and to a lesser extent back into the cushion region. Consequently, while the principle of land and sea hovering is similar, there are certain differences, which need to be taken into account in the design of the system. In the case of flight over land, with a solid surface, the external flow over the hovercraft is strongly dependent on the forward speed of the vehicle and can be thought of as comprising six phases: i. At low speeds, the air flow is not dissimilar to that for hovering flight although there is an interaction between

ii.

iii.

iv.

v.

vi.

the ambient air and the efflux from the forward jets. This interaction takes the form of forcing the jet air upwards from the ground with eddies being blown over the upper works of the craft. When the speed is increased, then the eddies over the upper works of the hovercraft tend to die away and the main flow over the superstructure starts to assert itself. Nevertheless, when the first critical speed is attained, the jet flow from the forward jets is carried aft over the upper works in a thin layer. Increasing the speed further, the flow then develops into a stable flow regime in the transition region where some of the forward jet flow enters the air cushion while the main part of this flow is deflected upwards to enclose a bubble, which has been produced by the attachment of the jet to the forward structure. Further increases in craft speed promote the situation where more of the jet is deflected aft and the bubble decreases in size and starts to rotate with increasing intensity. As the transition flow develops, the Poisson-Quinton critical condition is reached. This is where the bubble pressure has increased to that of the air cushion pressure, which then gives rise to the forward jet having little curvature. Nevertheless, some of the jet close to the ground is still deflected forward. As the speed increases further, increasing proportions of the air move aft in more of a streamlined fashion. The jet flow then starts to move aft and a second critical speed is identified when no further flow is projected forward. At supercritical speed, the jet flow moves aft and some of the incident flow on the hovercraft is swept under the craft with streamlined flow establishing itself over the upper surfaces, albeit with some separation occurring in places.

Clearly, the three-dimensional profile of the hovercraft influences the flow patterns considerably and makes for

Resistance and Propulsion Chapter 12

difficulty in predicting precisely the flow regimes. Wind tunnel tests and computational fluid dynamics procedures can be helpful in resolving the true three-dimensional flow behavior over the craft in aligned air streams as well as for air flows incident at nonaligned angles to the direction of motion of the craft. When operating over water, due to the depression of the water under the influence of the air cushion pressure, the regimes although show some resemblance to those over land are, nevertheless, somewhat different. In relation to the six phases discussed, the first two show a distinctive behavior when over water: i. In phase (i) spray in large amounts over water are thrown over the craft. ii. Over water in phase (ii) the spray created at the front of the craft disappears. Under hovering flight over water, the depression in the water for a simple craft is almost symmetric. However, as the speed increases the depression remains nearly symmetrical and parallel to the base of the craft, but the depression and craft take up a small angle of incidence by the bow in the sense of a nose-up trim. Further increases in this subcritical speed region lead the water depression and craft to take-up a modified form: the water depression, trim angle of the hovercraft, and its clearance height all increase until the critical speed is reached. This is due to the piling up of water ahead of the craft. Further increase of forward speed in the supercritical speed range produce longer and shallower depressions in the water surface, which have increased area and reduced slope because of the inertial lag of the water particles in response to the fast moving pressure field created by the hovercraft. In this condition, the craft’s trim angle reduces; however, the clearance height tends to remain unchanged. The forces on an amphibious air cushion vehicle center upon the concept that the total lift force acting on the vehicle comprises the sum of four components: i. ii. iii. iv.

The The The The

aerodynamic lift. cushion lift. jet lift. pressure lift.

If the vehicle is traveling over water, it will assume a positive angle of trim (a) due to the deformation of the water surface under the action of the air flow and at high speed it can be assumed, with little error, that the craft and the water surface are parallel to each other. Consequently, the total lift will be given by:   L ¼ Laero + Lcushion + Ljet + Lpressure cos a In general, the angle of trim (a) is small aero and hence cosa ! 1. Furthermore, the aerodynamic lift is normally small; therefore, the total lift to a first approximation can be written as,

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L ffi Lcushion + Ljet + Lpressure With regard to the propulsion of the craft, the drag forces require consideration. When operating over land, the total drag force may written as D ¼ Dprofile + Dmomentum + Dinduced The profile drag comprises two components: i. The skin friction drag on the craft ii. The viscous pressure drag on the vehicle’s superstructure. Air cushion vehicle superstructures, except for a relatively few, are seldom streamlined bodies and, therefore, the concept of drag coefficient CD is introduced such that the drag of the craft’s superstructure is estimated as being, 1 Dprofile ¼ ra CD SV 2 2 where ra is the density of air. S is the frontal area of the craft normal to the direction of V. V is the relative velocity of the craft. While the minimization of drag coefficient is sought in design, noting that in some cases this has risen to 0.7, some typical examples are shown in Table 12.17 for some wellknown designs. To estimate the momentum drag, it is assumed that relative to craft the momentum of the air, which enters the fan intakes in the horizontal direction is destroyed. Consequently, by writing m_ f as the total mass flow rate entering the fans, equal in the absence of leakage to the jet air flow mass flow rate, this drag component of drag can be approximated as, Dmomentum ¼ m_ f V In the case of the induced, or attitude drag, this can be written in the form   Dinduced ¼ Lcushion + Ljet + Lpressure sin a and if the trim angle a is small, then sina ! a and Lcushion + Ljet + Lpressure ¼ W where W is the weight of the craft, then Dinduced ¼ Wa

TABLE 12.17 Typical Drag Coefficients (CD) SRN-2

0.25

SRN-4

0.40

SRN-6

0.38

362 Marine Propellers and Propulsion

Consequently,    1 D ¼ ra CD SV 2 + m_ f V + Lcushion + Ljet + Lpressure sin a sin a 2 or if a can be considered small, then  1 2 D ¼ ra CD SV + m_ f V + Wa a 2 For steady flight at a speed V, the total drag must balance the thrust produced by the propulsors. When considering operation over water, the wavemaking resistance must also be introduced. The transmission of this component of resistance is via the air cushion and this occurs whether the vehicle is in contact with the water or not. To derive an expression for this component, the wave-making resistance (DWave) can be considered to be the component of the pressure forces acting on the water depression in the opposite direction to the velocity of the craft. As such, ð Dwave ¼ pw sin fdSD SD

In which SD is the surface area of the depression in the water and f is the inclination of the depression. Moreover, to preserve equilibrium at the water surface, the air cushion pressure (pc) must equal the adjacent water pressure (pw) assuming they are uniformly distributed under the craft, then the following expression for the wave drag can be written ð Dwave ¼ pc sin fdSD SD

Now by assuming that the weight of the craft is supported by the air cushion, thereby neglecting the effects of the jet, then we may also write for the weight of the craft, ð W ¼ pc cos fdSD SD

Consequently, further assuming that the base of the craft is parallel to the water surface, then Dwave ¼ tan f W And for small trim and water depression angles Dwave ¼b¼a W The interaction of two-dimensional disturbances of constant pressure on initially plane liquid surface was considered by Lamb (1879). If a pressure disturbance pc having a length l moves over an initially plane water surface, he found that for a two-dimensional system the ratio 0 of the wave making drag Dwave to the applied force per unit 0 width W is given by,

  D0wave 2pc  1  cos Fn 2 ¼ W0 rgl 0

where Fn is the Froude number and W ¼ lpc. Now, clearly the maximum of the ratio D0wave rgl 2W 0 pc 2 will occur when cos(F2 n ) ¼  1. That is, when Fn ¼ (2n +1)p where n ¼ 0, 1, 2, … and for these conditions the maximum value of the two-dimensional wave-making drag 0 Dwavemax is:

4W 0 pc rgl When n ¼ 0 the primary hump will occur Fn ¼ 0.56. Barratt (1965) extended the study to three-dimensional air cushion vehicles when traveling over deep water and from this found that the primary hump occurred between 0.5 < Fn < 1.0 depending upon the planform of the craft. However, in shallow water, this hump occurs at a generally lower Froude number. Within the discussion so far, it has been assumed that the craft is traveling over initially undisturbed water. When proceeding over unlevel ground or in waves, there will be intermittent disturbances arising from the occasions when parts of the vehicle make contact with the surface over which it is traveling. As such, on these occasions, there will be transient loadings imparted to the craft. In the context of wetting drag, this can be a significant component when compared to the other drag contributions. It is difficult to estimate because it comprises a number of components: for example, i. Spray impact on the hovercraft. ii. Those parts of the craft, which transiently become submerged. iii. The increased weight of water thrown up on to the craft in various locations. Consequently, the total drag experienced by an air cushion vehicle is given by: DTotal ¼ Dprofile + Dmomentum + Dinduced + Dwave + Dwet + Dover-wave in which Dwet is the wetting drag and Dover-wave accounts for wave impacts when the craft is operating in very poor weather. These latter terms are often estimated with the aid of model testing. In the case of nonamphibious side walled hovercraft, these are usually fitted with side walls. While these have an advantage in being able to support relatively conventional marine shafting systems and marine propellers, the immersed part of the hull will incur a further drag penalty.

Resistance and Propulsion Chapter 12

This can at high speed be a significant force due principally to the effects of skin friction between the hulls and the water.

12.4 AERODYNAMIC SUPPORT PROPULSION The Ground Effect on airplanes when they fly close to the ground distorts the flow field generated by the wings in free flight. While not changing the magnitude of air in the flow field, the ground effect effectively squeezes and redistributes the flow field such that the effective, rather than actual, span of the wing becomes larger: Fig. 12.49. This, in turn, increases the effective aspect ratio Æg from its design value Æ due to the outward displacement of the tip vortices, see for comparison Fig. 7.30. Furthermore, this reduces the effective angle of incidence of the wing section and in so doing, it reduces the lift-dependent induced drag. Typically, the ground effect manifests itself at heights less than half the wing span of the airplane; consequently, ground-effect airplanes with lower positioned wings experience a greater ground effect than do those with wings located more highly. Aerodynamic support vehicles such as Wing in Groundeffect (WIG) airplanes or Ekranoplans to use their Russian name rely on their wings to generate lift and, as such, require a forward speed to achieve the required lift. However, when operated sufficiently close to a nominally flat surface, the lift-drag characteristics of the wings are enhanced. As outlined here at these conditions, the downwash angle, angle of attack, lift slope, and the effective aspect ratio are all affected. L The ideal lift slope is dC dai ¼ 2p. However, the larger the aspect ratio of the wing, the more closely the lift slope approaches the ideal, such that for an infinite aspect ratio the lift slope is 2p. For lesser aspect ratios, the lift slope of a wing decreases by a factor Æ/(Æ + 2) and, as such, the lift slope becomes

FIG. 12.49 Distortion of the flow field around the wings caused by the proximity of the ground.

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dCL ¼ 2p=ð + 2Þ da Defining the lift slope ratio (g) as being the lift slope when affected by the ground effect to that which is unaffected,     dCL dCL viz g ¼ da G da Consequently, the aspect ratio influenced by ground effect conditions is given by, g ¼

2g 2 + ð1  gÞ

where Æ is the aspect ratio of the wing in free unconstrained flight. The suffix G refers to the value of a parameter close to the ground. A similar reduction in downwash is found when birds fly in Vee formation. The leading bird in the formation experiences a reduction in downwash, which implies a lower drag than if the wing birds were absent. Consequently, changing the leading position quite regularly on long flights, such as migration, allows each bird to have a rest periodically.

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