The total resistance of a ship is due to several causes, and the phenomena involved are extremely complicated. This has been discussed by many investigators as stated in Saunders (1957), Comstock (1967), Lewis (1989b), Schneekluth and Bertram (1998) and oth- ers. It is, therefore, usual to simplify the problem by regarding the total resistance to be composed of several components independent of each other and to disregard the possible interaction between the different components. For a ship moving at the surface of water, the total resistance is composed of the resistance of the above-water part of the ship (air and wind resistance or aerodynamic resistance) and the resistance of the underwater part of the ship (hydrodynamic resistance). The hydrodynamic resistance, which constitutes the largest component of total drag, is mainly the bare hull resistance which increases due to various appendages such as rudders, bilge keels, stabilizer fins, sonar domes, etc.
The total resistance (hydrodynamic, ignoring aerodynamic drag) of the bare hull mov- ing at a constant speed in calm water can be divided into two main components. When the
hull moves in water, the motion is resisted by the viscosity of water and thus experiences viscous resistance RV. The motion of the hull at or near the surface also generates waves
at the surface, and this gives rise to wave (or wave-making) resistance RW. Thus, the total
resistance RT is given by the sum of these two components as
RT=RV+RW.
Looking at the drag in a different way, the total resistance to forward motion is the sum of the longitudinal components of stresses tangential to the surface (friction), RF, and normal
to the surface (pressure), RP:
RT=RF+RP.
The viscosity of water also alters the pressure distribution around the hull and thereby causes an increase in the pressure resistance. The part of the pressure resistance due to the viscosity around a 3D form is called the viscous pressure resistance RVPgiven by
RVP=RV-RF=RP-RW
Then,
RT=RV+RW=(RF+RVP)+RW=RF+(RVP+RW)=RF+RP
The viscous pressure resistance is usually a small component of the total resistance. However, if the hull is excessively curved at the stern and there are large waterline or but- tock line slopes or discontinues, the flow separates from the hull surface and gives rise to eddies or vortices. This results in a significant increase in the viscous pressure resistance. The additional resistance due to separation of flow and the generation of eddies is called separation drag or eddy resistance.
The frictional resistance RF is the sum total of the frictional resistance of a 2D surface of
infinite aspect ratio (surface of zero pressure gradient), RF0, and an additional component
due to the 3D effect on friction, commonly known as friction form effect, and taking k as the form factor, RF = (1 + k)RF0. When the waves generated by the ship have high wave
slope, like in full form ships moving even at slow speed, waves break and the drag is manifested as viscous resistance. Thus, if one measured the energy content of the wake behind the ship experimentally, the drag would consist of the normal viscous resistance and the drag due to wave breaking, and the remaining part of the total drag can be mea- sured by estimating the energy content in the waves, which is known as wave pattern resistance. Figure 7.1 shows the various components of hydrodynamic resistance to the forward motion of a body in water, which are discussed subsequently.
A detailed study of the different components of ship resistance is necessary to under- stand the complex phenomena involved and to design the hull form of a ship to minimize the resistance of the ship. However, for many practical purposes, it is sufficient to divide the total bare hull resistance into two components: (1) frictional resistance and (2) remain- ing components lumped together as residuary resistance, which is mainly wave resistance. The total-resistance coefficient is a function of Reynolds number and Froude number such that viscous resistance or, mainly, frictional resistance is a function of Reynolds number and the residuary resistance, mainly the wave-making resistance, is a function of Froude number. Thus,
C f R F C C C f R C f F T n n F R F n R n = = + = = where and ( , ) ( ) ( ) 1 2 where C R S V C R T T F F = =
/( / )is the total resistance coefficient /( / 1 2 1 2 * * *r 22 1 2 * * *r S V CR RR
)is thefrictional resistance coefficient and / ( = // ) is theresiduary resistancecoefficient is the wet 2* * *r S V2 S ; tted surface proportional to, L2
Rn = VL/ν is the Reynolds number, named after Osborne Reynolds known for his experi-
ments on viscous fluids among other things ν = μ/ρ is the kinematic coefficient of viscosity
Fn= /V gL is the Froude number
Total resistance
Residuary resistance
Form resistance
Pressure resistance
Wave resistance
Wave pattern resistance Wake resistance
Total resistance
Viscous resistance
Wave breaking resistance Viscous pressure resistance (including eddy resistance)
Frictional resistance Two-dimensional frictional resistance
FIGURE 7.1
As per the boundary layer theory initiated by Prandtl in 1904 when a viscous fluid flows past a solid boundary, the layer of the fluid next to the boundary sticks to it (‘no slip’ condi- tion), and the velocity of the fluid increases from zero at the boundary to nearly the value it would have had if there had been no viscosity. This change in velocity takes place in a narrow layer of the fluid next to the solid boundary. This layer is called the thin boundary layer. It is assumed that the effects of viscosity on the flow around a body are confined to the boundary layer, and that the flow outside the boundary layer is that of an inviscid fluid. This simplifies the problems of viscous fluid flow to a great extent.
At low Reynolds numbers, the flow in the boundary layer appears to take place in a series of thin layers or ‘laminas’, and the flow is described as laminar. At high Reynolds numbers, the fluid particles have a mean velocity superposed on which are small random velocity fluctuations in all directions and such a flow is called turbulent flow. As the Reynolds number increases, there is a transition from laminar flow to turbulent flow. The critical Reynolds number at which this transition occurs depends on a number of factors, includ- ing the roughness of the surface and the presence of disturbances such as eddies in the flow approaching the solid boundary. The flow around a ship is almost always turbulent because the ship Reynolds number is high and the wetted surface is comparatively rough.
A number of researchers such as R.E. Froude, William Froude, Hughes and Schoenner have worked on developing a formulation for estimating frictional resistance of ships based on Reynolds number. In 1957, the International Towing Tank Conference (ITTC) decided that in all future work, the frictional resistance coefficient for ships and ship mod- els would be calculated by the formula
CF=0 075. (log10Rn-2) .-2
This frictional resistance coefficient corresponds to the 2D frictional drag coefficient CF0
and applying the friction form factor k, the 3D frictional drag can be estimated as
CF= +(1 k C) F0.
Ship surface is not a smooth surface, and this leads to an increase in frictional drag. The effect of roughness is usually taken into account by adding a roughness allowance ΔCF to
the frictional resistance coefficient. A commonly used value is ΔCF = 0.4 × 10−3. However,
one may also use the following formula:
103 105 0 64 1 3 DC k L F= æ s è ç öø÷ - / .
A standard value of the equivalent sand roughness of a newly painted steel hull is
ks = 150 × 10−6 m (150 µm), but lower values are now routinely obtained by modern ship-
building techniques and paint technology. During the service of the ship, the hull surface becomes progressively rough due to damage to the paint coating, corrosion and erosion of the surface and ‘fouling’ by marine organisms that attach themselves to the hull, result- ing in increased resistance. This makes it necessary to dry-dock the ship at intervals to clean and repaint the hull. The rate of fouling depends on a number of factors such as the time spent in port and at sea, and the time spent in temperate waters and in tropical waters. Empirical allowances are sometimes used to allow for the increased resistance due to fouling, e.g. a drop in speed of ¼%/day in temperate waters and ½%/day in tropical waters at constant power.
In an inviscid fluid, the pressure distribution around a curved body is such that there is no resistance. The effect of viscosity in the fluid causes a gradual decrease in the pressure around the body in the direction of flow compared to the pressure distribution in the inviscid flow, and this results in the component of resistance called the viscous pressure resistance. If the body is streamlined, the viscous pressure resistance is small and need not be consid- ered separately but included in form resistance. With a body having a large curvature in the afterbody, i.e. a body which is ‘bluff’ and not streamlined, the particle may come to rest and its flow is reversed by the adverse pressure gradient. Fluid particles moving in the reverse direction meet the particles moving from forward to aft, pushing them away from the surface of the body, and an eddy is created between the surface of the body and the flow moving from forward to aft, and a vortex is shed. The extent of boundary layer separation and the mag- nitude of eddy resistance depend on a number of factors apart from the shape of the curved surface. Separation is more likely to occur in laminar flow and low Reynolds numbers than in turbulent flow and high Reynolds numbers. A high hydrostatic pressure reduces separation.
In the case of bluff hull forms, the free surface flow ahead of the bow becomes irregular and complex even at very low Froude number, usually leading to breaking waves at the bow. The resistance associated with wave breaking has been the subject of extensive inves- tigation. Bow wave breaking is considered to be due to flow separation at the free surface, and it can generally be avoided by requiring that the tangent to the curve of sectional areas at the forward perpendicular is not too steep.
The movement of the hull through water creates a pressure distribution, i.e. areas of increased pressure at bow and stern and of decreased pressure over the middle part of the length. Since the free surface is a surface of constant pressure, this pressure difference generates waves. There is greater pressure acting over the bow, as indicated by the usually prominent bow wave build-up, and the pressure increase at the stern, in and just below the free surface, is always less due to the build-up of the boundary layer at the stern. The resulting added resistance corresponds to the drain of energy into the wave system, which spreads out at the stern of the ship and is continuously recreated. This is the so-called wave-making resistance. The result of the interference of the wave systems originating at bow, shoulders (if any) and stern is to produce a series of divergent waves spreading out- wards from the ship at a relatively sharp angle to the centre line and a series of transverse waves along the hull on each side and behind in the wake.
The presence of the wave systems modifies the skin friction and other resistances, and there is a very complicated interaction among all the different components. Submerged bod- ies just below the surface of water also create wave systems and, therefore, experience wave- making resistance. However, as the depth of submergence increases, wave making reduces. An early idea of the ship wave pattern was given by Lord Kelvin (1887–1904) by consider- ing a pressure point travelling over the water surface. The Kelvin wave pattern consists of 1. A transverse wave system and
2. A divergent wave system
The meeting point of the transverse and divergent waves is a high point. The transverse waves move in the same direction of the ship and with the same speed. If the wave length is λ, then
l= 2pV2
or l p p L V gL Fn = é ë ê ê ù û ú ú = 2 2 2 2
where L is the length of the ship. Since the divergent waves move at an angle θ to the direc- tion of the ship, the speed of these waves is (V cos θ) and hence the wave length λ′ is
l¢ =2p( cos )V q2 =l 2q
g cos
The wave-making resistance increases with ship speed. But since this is the integration of the longitudinal pressure components developed by the wave system, the increase in wave resistance is undulatory in nature. When there is a crest in the wave profile in the forepart and a trough in the aft part, the wave-making resistance is high. But when there are crests near both fore and aft ends, the longitudinal pressure components in the fore and the aft tend to cancel and this resistance increase is reduced. Therefore, based on ship length and Froude number, there are humps and hollows in the wave resistance curve. If n is the number of wave crests in the ship length L, the hollow and hump speeds can be shown to occur at Fn given in Table 7.1.
Normally, the first bow wave crest occurs around the quarter of a wavelength aft of the bow. For high-speed vessels (say, planing craft), e.g. Fn = 1.5, the wavelength is more
than 14 times the ship length and the first wave crest occurs at about 3.5 times the ship length behind the bow. Therefore, at high speeds, the water surface along the ship length is almost horizontal.
Up to speeds corresponding to Fn≤ 0.27, the length of the marine craft spans two or more
waves, changes in draught and trim are small and the drag is predominantly frictional. As the speed increases, wave-making resistance increases and above Fn = 0.36, it increases at
a very fast rate. At Fn = 0.40, when the ship length equals the wavelength, the wave resis-
tance is maximum and virtually forms a barrier to the speed of displacement vessels. This is primarily because the increased velocities around the hull form result in negative pressure causing the stern to settle deeply in water and trim by stern. If the boat is to be driven in the high-speed displacement mode, i.e. 0.40 ≤ Fn≤ 0.95, it is necessary to change the stern shape
to reduce separation drag and also reduce the build-up of negative pressure. This is achieved by designing a wider, flatter and broader stern than before. Then the wave or residuary resis- tance barrier is crossed and wave resistance is no longer an important factor. The frictional resistance, however, remains a dominant factor. At these speeds, the flat bottom of the aft body may generate some lift force, which may support some weight. Around this speed, some
TABLE 7.1
Humps and Hollows in Wave Resistance n
Hollow Speed Hump Speed λ/L Fn λ/L Fn
1 4/1 0.798 4/3 0.461
2 4/5 0.357 4/7 0.362
3 4/9 0.266 4/11 0.235
lift is generated and this range is also known as ‘semi-planing’ region. At high speeds, length loses its importance as a principal parameter for resistance, and weight, which requires to be supported by buoyancy, becomes important. A volume Froude number Fn∇ is defined as
F V
g
nÑ =
Ñ ( 1 3/)
At speeds higher than those corresponding to Fn = 0.95, the bottom and aft should be
designed for planing, i.e. the lift generated at the boat bottom should support the weight and the boat’s centre of gravity must rise up so that there is an effective reduction in wet- ted surface and, hence, frictional resistance. Flow is made to separate at the side as well as at the stern. This is the fully planing region when the residuary resistance increases very slowly with speed. Wave resistance is almost negligible with some drag due to spray. The development of flow from displacement mode to fully planing mode is discussed in detail by Savitsky (1964, 1985). If the vessel is supported by hydrofoils, at high speeds, the lift gen- erated by submerged or surface-piercing hydrofoils can support the entire weight of the ship and the vessel comes out of water, thus reducing viscous drag and limiting it to drag of foils and struts connecting the vessel and foils. Figure 7.2 shows the changes in resis- tance pattern with increasing speed in displacement mode, planing mode and hydrofoil- borne mode. Larsson (2010) has shown diagrammatically the percentages of components of resistance of surface ships, which is reproduced in Figure 7.3.
When two wave systems meet together, a resultant wave system is created. Mathematically, the resultant wave height can be obtained by linear superposition. Simply stated, if two wave crests meet, a higher wave is generated, and if a wave crest meets the trough of another
Semi-planing Planing Hydrofoil Displacement mode 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 Froude number 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 To tal resistance (kN ) FIGURE 7.2
Fo rm eff., friction Wa ve breaking 2. 2×1 0 –3 2. 3×1 0 –3 8. 1×1 0 –3 5. 4×1 0 –3 C =T Wa ve patter n Fo rmeff., pressur e (viscous pressur e re sistance) Residuary resistanc e 15 5 2.5 5 15 10 2.5 10 60 57.5 5 2.5 5 25 35 7.5 15 22.5 20 (Spr ay ) (A pp end.) 5 7.5 Roughness Visc ous resistance 65
Flat plate friction
Tanker Container ship Fi shing ve ssel Pl aning bo at 2.5 Wave FIG U R E 7 .3 Per ce nt ag es o f r es is ta nce c om pon en ts o f s u rf ace s h ips .
wave, a relatively low-height wave system is generated. An example of application of this principle is the bulbous bow of a ship where the forward wave crest due to bow and the wave trough due to the immersed bulb interact to reduce wave-making resistance. In a mul- tihull ship such as a catamaran, trimaran or pentamaran, the wave system in between the hulls is superposed. If this superposition is such that the internal wave system is reduced in height, the total wave resistance becomes less than the sum of the wave resistance of individual monohulls. This nature of superposition is dependent on the hull separation. Therefore, in a catamaran vessel, hull separation is very important. A properly chosen hull separation can reduce the total resistance considerably. If the displacement volume of each hull of a twin-hull vessel is pushed below the waterline such that the waterplane becomes thin, one obtains a small waterplane area twin-hull (SWATH) vessel. Because of the thin waterplane, the waves generated are small, and judicious hull separation distance can reduce waves further. Hence, resistance becomes predominantly viscous. Care must, however, be taken to reduce flow separation. But due to an increase in wetted surface, the frictional resistance is high, and so these vessels are generally not driven at high speed.
Surface ships are normally designed for moving at a level trim condition. Even if there is a resultant vertical component of pressure distribution around the hull form, the resulting trim in running condition is negligible and its effect on resistance can be ignored. However, in ballast and light-load conditions, there may be considerable aft trim, which may cause flow separation and increased drag. So if a surface ship has to spend a large portion of its life in partly loaded or ballast condition, the hull form design should be done such that there is no increased drag due to trim. On the other hand, slight trim of a planing vessel is advantageous for generating lift, which must be provided at the design stage itself.
A ship sailing on a smooth sea and in still air experiences air resistance, but this is usually negligible, and it may become appreciable only in high wind. Although the wind speed and direction are never constant and considerable fluctuations can be expected in