RESISTANCE OF SHIPS
RESISTANCE OF SHIPS
Prof O.P.Shah
RESISTANCE OF SHIPS OUTLINE
1. Introduction : Definition of resistance and effective power, importance of subject, brief history Newton, Euler, Leonardo da Vinci, Chapman, DAlembert, Beaufoy, Hall brothers etc. Reech and W. Froude.
2. Components of Resistance : Simplification of difficult problem. Components assumed to be independent. Different components.
3. Laws of Similarity : Use of models. Need for similarity laws. Geometrical, kinematic and kinetic similarity. Force ratios. Dimensional analysis. Practical applicationin ship resistance. Froude similarity and model testing.
4. Viscous Resistance : Froudes plank experiments. R.E. Froudes formula for f. Reynolds number. Boundary layer theory. Laminar and turbulent flow. Turbulence stimulation. Blasius and Prandtl-Karman lines. Other friction lines : Schoenherr, Prandtl-Schlichting, Hughes, and others. Form resistance Hughes, Lap-Troost, Granville. ITTC line. Grigsons formulation. Effect of roughness. Nikuradses pipe experiments. Roughness allowance. Bowdens formula. Fouling. Anti-fouling paints, SPC paints, banning of TBT.
5. Wave Resistance : Kelvin wave pattern. Ship waves. Wave interference humps and hollows. Theoretical methods. Comparison with experiments. Bulbous bows.
6. Other Resistance Components : Eddy resistance and boundary layer separation. Wave breaking resistance and vortex resistance. Appendage drag. Air and wind resistance.
7. Effect of Shallow Water : Schlichtings method. Landwebers extension.
8. Model Testing : Ship model tanks. Model size. Turbulence stimulation. Blockage. Ship model correlation.
9. Prediction of Effective Power : Methodical series and regression equation methods. Resistance of submerged bodies.
10. Hull form and Resistance11. High Speed Marine Craft
1. INTRODUCTION
Definitions
The resistance of a ship is the force that resists the motion of the ship. For the present study, the resistance R of a ship is the force that opposes the forward motion of the ship at a constant speed V in a straight line in still water. The power required to overcome this resistance is called the effective power PE :
If the resistance is in kN and the speed in m per sec, the effective power will be in kW. The speed of the ship is often given in knots : 1 knot is a speed of 1 nautical mile per hour, one (international) nautical mile is equal to1852 m, so that 1 knot is 0.5144 m per sec.
It is important to study the subject of Ship Resistance because (a) it is necessary to determine the effective power at the design speed so that an appropriate propulsion system may be fitted to the ship, and (b) it is desirable to design the hull form of the ship so as to minimize its resistance subject to the various design constraints.
1.2History
The importance of the subject was realized in ancient times and considerable effort was devoted to determining the shape of a hull form of minimum resistance. In the course of time, very efficient hull forms were developed empirically for the ships of the day. Many leading scientists and mathematicians, Newton (1642-1727) and Euler (1707-1783) among them, studied the problem. Leonardo da Vinci (1452-1509) is believed to have proposed the use of models to study the subject. Among the earliest to carry out ship model experiments was the Swedish naval architect Frederic Chapman (1721-1808). The French Academy of Sciences offered a prize for an experimental study of ship resistance, and the prize was won by a group that included DAlembert (1717-1783). Experiments were also carried out by Beaufoy in England, the Hall brothers in Scotland and Benjamin Franklin (1706-1790) in America towards the end of the 18th century. The beginning of the 19th century brought mechanical propulsion of ships to the forefront, and the problem of estimating the power required to propel a given ship at a specified speed. Early model experiments proved to be unsuccessful in providing a solution to this problem because the relation between the resistance of a model and the resistance of a ship was not properly understood. In 1832, a French naval constructor, Edmund Reech, proposed that the ratio of the resistance of a ship to the resistance of a geometrically similar model was equal to the ratio of their displacements provided that their speeds were proportional to the square roots of their lengths. However, even this did not provide correct results. Model experiments had become discredited by the latter half of the 19th century, and were in danger of being abandoned altogether when William Froude (1810-1879) proposed that by dividing the resistance into two components, each of which followed different scaling laws, the model experiment could indeed be used to predict the resistance of a ship with sufficient accuracy. The British Navy gave a grant to Froude to build a tank for model experiments and demonstrate the correctness of his proposal. Froude built his tank in Torquay and carried out various experiments, including experiments with a model of HMS Greyhound. The resistance of the ship was determined by towing HMS Greyhound by another ship, and this confirmed that the method proposed by Froude gave accurate results. Froudes ship model tank was followed by similar establishments in various parts of the world. In India, there are ship model tanks at the Central Water and Power Research Station (CWPRS), Pune, IIT Kharagpur, IIT Madras and the Naval Science and Technological Laboratory (NSTL), Visakhapatnam. The last is a very large modern establishment.
2. COMPONENTS OF TOTAL RESISTANCE
2.1Main Components
The total resistance of a ship is due to several causes and the phenomena involved are extremely complicated. 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 can be divided into the resistance of the bare hull and the resistance of the appendages such as rudders, bilge keels, stabilizer fins and sonar domes.
In this section, only the hydrodynamic resistance of the bare hull and its components are considered. Aerodynamic resistance and the resistance of appendages is taken up later.
2.2Components of Bare Hull Resistance
The resistance of the bare hull can be divided into two main components in two ways. In the first approach, the total resistance RT of the bare hull is divided into components according to their causes. When the hull moves at or near the surface of water, the motion is resisted by the viscosity of water. The motion of the hull also generates waves at the surface and this gives rise to a component of resistance. The causes of the resistance are thus the viscosity of water and the waves generated by the ship :
where RV is the viscous resistance and RW the wave resistance. It is important to note that the total resistance has been defined here as the resistance of the bare hull in water. The total resistance of the ship must of course include the aerodynamic resistance and the appendage resistance. Note also that wave resistance (or wave-making resistance) is due to the waves generated by the ship in calm water; the additional resistance due to the motion of the ship in waves already present in the sea is called added resistance in waves and is considered separately.
The second approach looks at the effect of the motion of the hull in water, viz. the creation of stresses tangential to the surface (friction) and normal to the surface (pressure) :
where RF is the frictional resistance and RP the pressure resistance.
2.3Viscous Pressure Resistance
The viscosity of water also alters the pressure distribution around the hull, and thereby causes an increase in the pressure resistance. That part of the pressure resistance that is due to viscosity is called the viscous pressure resistance RVP, and :
or :
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 slopes or buttock line slopes or discontinuities, 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.
2.4Form Resistance
The frictional resistance RF is further divided into the frictional resistance of a two-dimensional surface of infinite aspect ratio (surface of zero pressure gradient) RF0 and the form resistance RForm, which is the additional frictional resistance due to the three-dimensional shape or form of the hull. A two-dimensional surface of infinite aspect ratio is a plane surface of finite length, a thickness tending to zero and a breadth tending to infinity. This division of frictional resistance into two-dimensional frictional resistance and form resistance was necessitated by the possibility of being able to calculate RF0 theoretically.
2.5Resistance Components and Acceleration
Another way of looking at the components of resistance is by noting that force is equal to mass x acceleration, and associating the various components of resistance with the components of acceleration imparted to water by the motion of the ship :
Two-dimensional frictional resistance is due to the acceleration of water in the direction of motion.
Form resistance is due to the acceleration of water normal to the direction of motion.
Wave resistance is due to the acceleration of water in the vertical direction.
Eddy resistance (by some stretch of imagination) is due to the angular acceleration of water.
2.6Other Resistance Components
The waves generated by the ship sometimes break, and this gives rise to another component of resistance called wave breaking resistance.
In a ship with a transom stern, a part of the wetted surface is perpendicular to the direction of motion or nearly so. This gives rise to a resistance component contributed by the transom stern, and is called transom resistance.
In certain circumstances, the motion of a body in a fluid produces a force normal to the direction of motion. This is called lift. When lift is generated, there is an associated resistance or drag known as induced drag. Some types of high speed marine craft depend on the generation of lift for supporting their weight in motion, and in these craft induced drag is a component of resistance. In some marine craft, the motion of the craft generates spray and this may give rise to spray resistance, particularly if the spray strikes the hull. A ship may continuously take in large quantities of air or water from outside for some internal purpose. This air or water, assumed to be at zero velocity outside the ship, is forced to acquire the velocity of the ship when taken into the ship. The rate of change of momentum of this fluid gives rise to momentum drag.
2.7Measurement of Resistance Components
In carrying out experiments to study ship resistance, it is usual to measure only the total resistance of the ship or model. However, techniques have been devised to determine experimentally some individual resistance components for ship models. Frictional resistance can be determined by measuring the tangential stress at several points on the surface of the ship model and integrating the resulting stress distribution. The pressure resistance can be similarly determined by measuring the pressures on the hull surface. The wave resistance can be determined by calculating the rate at which the energy of the wave system generated by the ship model is increasing, since the work done by the wave resistance is theoretically equal to the energy of the waves generated by the ship. The energy of the waves is determined by the measurement of wave heights in the wave pattern behind the model. The resistance determined from the wave pattern in called wave pattern resistance, and this is slightly different from the wave resistance because of the effect of viscosity on the waves, wave breaking and other causes. The effect of viscosity is to cause the body moving in a viscous fluid to impart a momentum to the fluid in the direction of motion. The rate of change of this momentum is theoretically equal to the viscous resistance. This change of momentum is determined by measuring the velocities at several points in the wake (the disturbed fluid behind the ship model); the resistance calculated in this way is called wake resistance.
2.8The Froude Law
A detailed study of the different components of ship resistance is necessary to understand 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 : (i) the frictional resistance, and (ii) the remaining components lumped together as residuary resistance, which is mainly wave resistance. This division of the total resistance RT into frictional resistance RF and residuary resistance RR was first proposed by W. Froude, who also stated what he called the Law of Comparison :
The residuary resistances of geometrically similar ships are proportional to their displacements if their speeds are proportional to the square roots of their lengths.
i.e.
constant if constant for geometrically similar ships
where and L are the displacement and length of the ship respectively. This is now called the Froude law. Speeds of geometrically similar ships proportional to the square roots of their lengths are called corresponding speeds. A more modern approach is to call the two components viscous resistance and wave resistance. The relationship between the main components of ship resistance is indicated Fig. 1.1.
Question :
What will be the components of the hydrodynamic resistance of a ship without appendages in the following cases?
(a) The ship is moving at the surface of a viscous fluid.
(b) The ship is moving at the surface of an inviscid fluid.
(c) The ship is moving deeply submerged in a viscous fluid.
(d) The ship is moving deeply submerged in an inviscid fluid (DAlembert paradox).
Fig. 1.1 Components of Resistance
3. LAWS OF SIMILARITY
3.1Need for Laws of Similarity
Resistance experiments have occasionally been carried out with full size ships, beginning perhaps with the HMS Greyhound in 1874. Other notable full size resistance experiments involved the Imperial Japanese Navy Ship Yudachi in 1933 and the former Clyde paddle steamer Lucy Ashton in the 1950s. However, it is difficult to carry out experiments with ships because of the costs involved and because it is difficult to control the conditions of the experiment. Carrying out experiments with small scale ship models is much more convenient, but it is necessary to know how the quantities measured in a model experiment are related to the corresponding quantities in the ship, i.e. to know the laws of similarity.
3.2Conditions of Similarity
Three conditions of similarity must be satisfied in carrying out a resistance experiment with a ship model :
Geometrical similarity, which requires that the ratio of any two dimensions in the model must be equal to the ratio of the corresponding dimensions in the ship.
Kinematic similarity, which requires that the ratio of any two velocity components in the flow around the model and the corresponding velocity components in the flow around the ship must be equal, i.e. the flow patterns around the model and the ship must be geometrically similar.
Kinetic similarity, which requires that the ratio of any two forces acting on the model must be equal to the ratio of the corresponding forces on the ship.
Suppose that the forces that must be considered in studying ship resistance are inertia forces, gravity forces, viscous forces and pressure forces. If , L and T denote the fundamental dimensions of mass, length and time, these forces may be expressed in terms of the fundamental dimensions as follows :
Inertia force = mass acceleration L3 L T -2 =L4T -2
(3.1)
Gravity force = mass acceleration of gravity L3 g = g L3
(3.2)
Viscous force = coefficient of viscosity velocity gradient area
(3.3)
Pressure force = pressure area
(3.4)
In this, and are the density and coefficient of dynamic viscosity of the liquid, g the acceleration of gravity and p the pressure. The various force ratios are then :
(3.5)
(3.6)
(3.7)
Here L, V and p are a characteristic length, a characteristic velocity and a characteristic pressure associated with the ship model or the full size ship, and is the kinematic viscosity of the liquid. For the model experiment to replicate exactly the conditions of the ship, these force ratios for the model must have exactly the same values as for the ship.
3.3Dimensional Analysis
The same result may be obtained by dimensional analysis, which is a technique to obtain a partial solution to a physical problem too difficult to solve completely. The technique requires only knowledge of the physical quantities that enter the problem and provides only the form of the solution. Dimensional analysis uses the concept that every equation that represents a physical relationship must be dimensionally homogeneous. Dimensional analysis consists in listing all the variables that may be considered to be involved, writing down their dimensions in terms of the fundamental dimensions (mass, length and time in problems not involving heat and electricity), and finding a dimensionally homogeneous relationship between the variables.
The total resistance RT of a ship or model of a given geometry may be regarded as being a function of the size of the ship expressed in terms of its length L and speed V, the density and the viscosity of the liquid in which the ship is moving, the acceleration of gravity g, and the pressure p defined in some specific manner :
(3.8)
where k is a constant. This equation may be written in terms of the fundamental dimensions :
i.e.
so that
.
Solving these three equations to express a, b and c in terms of d, e and f, one gets :
so that
or
i.e.
(3.9)
This is normally written as follows :
(3.10)
or
(3.11)
where :
is the total resistance coefficient
S is the wetted surface, proportional to L2
is the Reynolds number, named after Osborne Reynolds known for his experiments on viscous fluids among other things,
is the kinematic viscosity,
is the Froude number, and
is the Euler number.
3.4Application of the Laws of Similarity
There are some important advantages of using a relationship such as Eqn. (3.11) rather than one such as Eqn. (3.8) :
Eqn. (3.11) contains dimensionless quantities the values of which are the same in any consistent system of units.
The number of independent variables in Eqn. (3.11) has been reduced to three from the six in Eqn. (3.8), so that it is necessary to vary only three variables independently in an experiment to derive the nature of the functional relationship in Eqn. (3.11).
It is difficult if not impossible to vary some of the independent variables in Eqn. (3.8) in an experiment, whereas the independent variables in Eqn. (3.11) can be varied quite easily.
What Eqn. (3.11) implies is that if the Reynolds numbers, the Froude numbers and the Euler numbers of the model and the ship are made equal, their total resistance coefficients would also be equal :
if
,
,
(3.12)
where the subscripts M and S refer to the model and the full size ship respectively. For the Reynolds numbers of the model and the ship to be equal :
(3.13)
whereas for the Froude numbers of the model and the ship to be equal :
(3.14)
Ignoring the small difference between the kinematic viscosity of the fresh water in which a model is usually tested and the kinematic viscosity of the sea water in which most ships usually move, the only way in which Eqns. (3.13) and (3.14) can be simultaneously satisfied is if the model and the ship have the same length and the same speed. If the model is to be smaller than the ship, either the Reynolds numbers of the model and the ship can be made equal or the Froude numbers can be made equal, not both.
Consider a ship of length 100 m with a speed of 10 m per sec, and a model of length 4 m. If the Reynolds number of the model is to be equal to the Reynolds number of the ship, the speed of the model should be :
m/s
(3.15)
At such a high speed, even a 4 m model would have a very high resistance, and an experimental facility capable of such high speeds and forces would not be practicable. On the other hand, if the Froude numbers of the model and the ship are to be made equal, the model speed should be :
m/s
(3.16)
which is easy to achieve. Since both the Reynolds numbers and the Froude numbers of the model and the ship cannot be made equal, and it is almost impossible to make the Reynolds numbers equal, only the Froude number of the model is made equal to the Froude number of the ship in carrying out resistance experiments with ship models.
Dynamic similarity also requires the Euler numbers of the model and the ship to be equal. If the pressure p is taken as the hydrostatic pressure, which is normally permissible, geometrical similarity and Froude similarity automatically ensure the equality of Euler numbers :
(3.17)
where hM and hS are the depths of immersion of corresponding points in the model and the ship respectively, by geometrical similarity, and .
One may therefore write :
(3.18)
which, following Froude, can be written as :
(3.19)
where :
is the viscous resistance coefficient, assumed to be a function of Reynolds number only for a given geometry, and
is the wave resistance coefficient, assumed to be a function of Froude number only for a given geometry.
Therefore, if geometrically similar ships (geosims) move at speeds such that their Froude numbers are equal, their wave resistance coefficients will also be equal. Noting that in geometrically similar ships, the wetted surface S is proportional to the square of the length L and the displacement volume is proportional to the cube of the length,
constant implies constant,
and
constant implies constant since, with k1, k2, k3, k4 and k5 as constants,
constant.
Thus, another expression of the Froude law is that for geometrically similar ships, the wave resistance coefficient is constant if the Froude number is constant.
3.5Use of the Froude Law
The Froude law may be used for the determination of the resistance of a ship from the measured resistance of its geometrically similar model provided that a method can be found to determine the viscous resistances of the model and the ship :
- The model total resistance RTM is measured at a speed VM.
- The model viscous resistance RVM at the speed VM is calculated by some independent means.
- The model wave resistance at the speed VM is obtained : .
- The ship wave resistance at the corresponding speed is obtained using the Froude law :
.
- The ship viscous resistance RVS at the speed VS is calculated.
- The total resistance of the ship at the speed VS is obtained : .
This procedure may also be expressed in terms of the resistance coefficients :
(3.20)
The procedure of calculating the resistance of the ship from the resistance of the model requires a method for calculating viscous resistance. Methods of calculating the viscous resistance are considered in the next chapter.
4. VISCOUS RESISTANCE
4.1Froude Plank Experiments
To determine the resistance of a ship from the resistance of its model, it is necessary to divide the total resistance into frictional resistance and residuary resistance (or viscous resistance and wave resistance), and to have a method of determining the frictional resistance or viscous resistance. This is what was proposed by William Froude in 1868 when he stated his law of comparison.
Froude then went on to develop a method for determining the frictional resistance of a ship or model. For this purpose, he carried out resistance tests in a tank with a series of wooden planks of lengths varying from 2 feet to 50 feet at speeds varying from 1.5 ft per sec to 13.3 ft per sec. The planks had surfaces covered with varnish, paraffin, calico, and fine, medium and coarse sand. Froude found that the resistance of a plank could be expressed by the formula :
(4.1)
where f was a friction coefficient dependent on the roughness of the surface, S the wetted surface and V the speed, n being 1.83 for smooth surfaces and 2.00 for rough surfaces. Froude proposed that the frictional resistance of a model or a ship could be taken to be equal to the resistance of an equivalent plank, i.e. a plank of the same length and wetted surface as the model or ship.
The results of W. Froudes smooth plank experiments were later re-analyzed by his son, R.E. Froude, resulting in the formula :
(4.2)
with RF in lbs, S in ft2 and V in knots and the friction coefficient for smooth surfaces in sea water given by :
(4.3)
L being the length of the model or the ship in feet. For fresh water :
(4.4)
There are metric or SI equivalents of these formulas, but the Froude formula for estimating frictional resistance is mostly of historical interest, and conversion of the expressions in Eqns. (4.3) and (4.4) is unnecessary.
4.2Boundary Layer Theory
Based on the concept of dimensional analysis, modern methods for estimating the frictional resistance use formulas of the type :
(4.5)
The Boundary Layer Theory initiated by Prandtl in 1904 led to attempts to develop theoretical methods for determining the frictional resistance of plane surfaces. When a viscous fluid flows past a solid boundary, the layer of the fluid next to the boundary sticks to it (no slip condition), 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 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. Fig. 4.1 shows some features of a boundary layer on a plane surface. 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 upon a number of factors including 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. In a ship model moving at the same Froude number as the ship, the flow may be laminar because the Reynolds number is much lower and the model surface is smooth.
Fig. 4.1 Boundary Layer on a Plane Surface
The Boundary Layer Theory can be used to determine the frictional resistance of two-dimensional plane surfaces. Consider a plate of length L and breadth B moving at a velocity U in a viscous fluid of density. Let the thickness of the boundary layer at a distance x from the leading edge be and the velocity at a distance y from the surface of the plate be u. The mass of the fluid flowing per unit time between y and is :
(4.6)
and the change in velocity of this mass of fluid over the distance x is . The change in momentum per unit time of the fluid that has occurred in the boundary layer is the frictional resistance of the plate (considering one side of the plate) over the distance x :
(4.7)
In laminar flow, the velocity in the boundary layer can be taken to have a parabolic velocity profile in the boundary layer, with , and , so that
(4.8)
Substituting this in Eqn. (4.7), one obtains :
(4.9)
In laminar flow past a plane surface, the thickness of the boundary layer is found to increase with distance downstream from the leading edge according to the following relation :
(4.10)
The frictional resistance of a plane surface of length in laminar flow is then given by :
so that :
(4.11)
More accurate calculations lead to the Blasius friction line for laminar flow :
(4.12)
In turbulent flow, the velocity u in the boundary layer can be taken to vary from 0 at to at according to the one-seventh power law :
(4.13)
The boundary layer thickness has been found to vary with distance x from the leading edge of the plane surface according to the formula :
(4.14)
The frictional resistance of the plate is then obtained as :
(4.15)
The two-dimensional frictional resistance coefficient is thus obtained as :
(4.16)
This friction formula was first obtained by Prandtl and von Karman, the coefficient being changed from 0.072 to 0.074 for a better fit with experimental data, and is known as the Prandtl-Karman friction line. The Blasius laminar friction line and the Prandtl-Karman turbulent friction line are shown in Fig. 4.2, along with lines along which transition from laminar flow to turbulent flow can possibly occur.
Fig. 4.2 - Laminar, Transition and Turbulent Friction Lines
Several such turbulent friction lines have been derived using the Boundary Layer Theory based on somewhat more complex considerations of the velocity profiles in the boundary layer instead of the simple power law used in deriving the Prandtl-Karman line. Two friction lines that have been widely used in the study of ship resistance are the Prandtl-Schlichting line popular in Europe :
(4.17)
and the Schoenherr line (American Towing Tank Conference or ATTC line) used in America :
(4.18)
Many such friction lines have been proposed. Fig. 4.3 shows some of these friction lines.
When calculating the frictional resistance of a model, these friction lines may be used as they are. However, when calculating the frictional resistance of the ship, it is necessary to add a roughness allowance to the value of CF0, a commonly used value being 0.0004.
Fig. 4.3 Turbulent Friction Lines
4.3Form Resistance
The frictional resistance of a curved body is different from the frictional resistance of a plane surface (flat plate) because the velocity distributions are different. The difference between the frictional resistance of a curved surface and that of a plane surface is called form resistance since it depends upon the shape or form of the surface. A method to determine the form resistance of ships was proposed by Hughes.
Hughes carried out a series of resistance experiments with planks and shallow draught pontoons of varying length-breadth ratios. From his experiments, he concluded that the frictional resistance coefficient of a two-dimensional plane surface of infinite aspect ratio is given by :
(4.19)
He also concluded that the ratio of the frictional resistance coefficient of a two-dimensional plane surface of finite aspect ratio to the frictional resistance coefficient of a two-dimensional plane surface of infinite aspect ratio at the same Reynolds number is a constant that depends only on the aspect ratio and is independent of the Reynolds number.
Hughes suggested that, by an analogy with the frictional resistance coefficients of plane surfaces of finite and infinite aspect ratios, at equal Reynolds numbers the ratio of the frictional resistance coefficient of a three-dimensional body such as a ship to the frictional resistance coefficient of a plane surface of a two-dimensional surface of infinite aspect ratio is a constant that depends on the form of the body but is independent of the Reynolds number. This ratio is called the form factor :
both at the same .
(4.20)
Other methods of determining form resistance have been proposed by Lap and by Granville among others.
The form factor may be determined by empirical formulas or by a method proposed by Prohaska. Prohaskas method is based on the theory that the wave resistance coefficient is proportional to the fourth power of the Froude number for a given form, i.e. :
(4.21)
so that :
(4.22)
or, more generally,
(4.23)
The form factor can be obtained by fitting the model experiment data to these equations.
4.4The ITTC Line
In 1957, the International Towing Tank Conference (ITTC) decided that in all future work, the frictional resistance coefficient for ships and ship models would be calculated by the formula :
.
(4.24)
This ITTC 1957 Friction Line is not a two-dimensional friction line but a model-ship correlation line. It is the Hughes friction line with a built-in form factor. The ITTC 1957 friction line is now the standard method for calculating the frictional resistance. However, based on the results of ship model correlation in the years after 1957, it is now common to use a form factor with the ITTC friction line. There have also been proposals to adopt some other formulations for the frictional resistance of ships and ship models, including a method proposed by Grigson that involves a correction to the ITTC line.
4.5Viscous Pressure Resistance
The pressure distribution in the flow around a curved body is different from that around a plane surface that has a zero pressure gradient. However 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 inviscid flow, and this results in the component of resistance that is called viscous pressure resistance. If the body is streamlined, viscous pressure resistance is small and need not be considered separately but included in form resistance. The flow of an inviscid fluid and that of a viscous fluid past a streamlined body is shown in Fig. 4.4.
Fig. 4.4 Flow of a Fluid Past a Curved Surface
If the body has a large curvature or slopes in the afterbody, or there are discontinuities in its surface, the flow cannot follow the surface and separates from it and eddies are created between the surface of the body and the separated flow. This gives rise to eddy resistance or separation drag. The phenomenon of flow separation may be explained with the help of the Boundary Layer Theory.
A fluid particle arriving near the forward end of the body has some kinetic energy by virtue of its velocity. The favourable pressure gradient in the forward part of the body causes an acceleration in the fluid particle and an increase in its kinetic energy. In the after part of the body, the fluid particle moves in an adverse pressure gradient and some of its kinetic energy is used in working against the pressure gradient. In an inviscid fluid, the kinetic energy gained by the particle in the forward part of the body is used up in the afterbody. In a viscous fluid, the kinetic energy of the fluid particle close to the surface (i.e. in the boundary layer) is partly used to overcome the frictional resistance. With a streamlined body, the initial kinetic energy of the particle and that gained in the favourable pressure gradient in the forebody are sufficient to overcome both the frictional resistance and the adverse pressure gradient and carry the particle beyond the after end of the body. With a body that has a large curvature in the afterbody, i.e. a body that is bluff and not streamlined, the kinetic energy is completely used up before the fluid particle reaches the after end, the particle comes 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. At the point at which boundary layer separation starts the velocity profile has a zero gradient normal to the surface, ahead of the separation point this gradient is positive, and aft of this point the gradient is negative. Fig. 4.5 illustrates the phenomenon of boundary layer separation. The separation zone is a region of low pressure and therefore a cause for high resistance.
Fig. 4.5 Boundary Layer Separation
The extent of boundary layer separation and the magnitude of eddy resistance depend upon 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. Boundary layer separation can be reduced or eliminated by boundary layer suction, but this has not been used in ships.
4.6 Effect of Roughness
Froudes plank experiments included experiments with planks coated with fine, medium and coarse sand grains. From his experiments, Froude concluded that frictional resistance is proportional to where is equal to 2 for rough surfaces, whereas it is less than 2 for smooth surfaces. Nikuradses roughened pipe experiments not only confirmed Froudes result that for rough surfaces the frictional resistance is proportional to , but also provided further insight into the resistance of rough surfaces.
At low Reynolds numbers when the flow is laminar, the effect of roughness is negligible and the surface behaves like a smooth surface. As the Reynolds number increases, there is a transition from smooth flow to the flow past a rough surface in which the frictional resistance coefficient becomes independent of Reynolds number, i.e. the resistance becomes proportional to . The Reynolds number at which the transition starts and the value of the frictional resistance coefficient depend upon the relative roughness of the surface , where is the length of the surface and its equivalent sand roughness. A formula for the frictional resistance coefficient of rough surfaces based on Nikuradses work is :
(4.25)
This behaviour of rough surfaces is explained through the Boundary Layer Theory. In laminar flow, the roughness of the surface has negligible effect on its resistance and the surface behaves like a smooth surface. In turbulent flow, there is a laminar sub-layer within the turbulent boundary layer. So long as the roughness elements are within this laminar sub-layer, they do not affect the flow. The thickness of the laminar sub-layer decreases with increasing Reynolds number, and as the roughness elements begin to protrude beyond the laminar sub-layer they begin to affect the flow. When the Reynolds number becomes large, the thickness of the laminar sub-layer becomes very small compared to the height of the roughness elements and the surface behaves like a fully rough surface and the frictional resistance is proportional to .
This is important for ship model testing. The Reynolds number of a ship model is small and it is therefore easy to make its surface hydrodynamically smooth. The Reynolds number of the ship is large and it is very difficult to make its surface behave like a smooth surface. It is therefore necessary to allow for the roughness of the ship surface when calculating its resistance.
The surface of a ship does not normally behave like a fully rough surface, i.e. its frictional resistance coefficient is not independent of Reynolds number. The effect of roughness is usually taken into account by adding a roughness allowance to the frictional resistance coefficient. A value that is commonly used is . However, one may also use the following formula :
(4.26)
A standard value of the equivalent sand roughness of a newly painted steel hull is m (150 microns), 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 rougher due to damage to the paint coating, corrosion and erosion of the surface and fouling by marine organisms that attach themselves to the hull, resulting 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 upon a number of factors such as the times spent in port and at sea, and 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 per cent per day in temperate waters and per cent per day in tropical waters at constant power.
Fouling is minimized by the use of anti-fouling paints that contain an ingredient that dissolves slowly in the water and is poisonous to marine organisms. Self polishing co-polymer (SPC) coatings not only prevent fouling but become smoother during the service of the ship. Unfortunately, these coatings contain a powerful biocide, tri-butyl tin (TBT), and have now been banned because of the excessive harm that they do to marine ecology. Anti-fouling paints using copper based biocides continue to be used and newer anti-fouling paints have been developed that prevent marine organisms from attaching themselves to the hull without poisoning them.
5 WAVE RESISTANCE
5.1 Ship Waves
A ship moving on the surface of the sea experiences frictional resistance and eddy making, separation, and viscous pressure drag in the same way as does the submerged body. However, the presence of the free surface adds a further component. The movement of the hull through water creates a pressure distribution similar to that around the submerged body; i.e., areas of increased pressure at bow and stern and of decreased pressure over the middle part of the length.
But there are important differences in the pressure distribution over the hull of a surface ship because of the surface wave disturbance created by the ships forward motion. 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 than around a submerged body. The resulting added resistance corresponds to the drain of energy into the wave system, which spreads out astern of the ship and has to be continuously recreated. Hence, it has been 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 outwards from the ship at a relatively sharp angle to the centreline 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 bodies 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.
Fig. 5.1(a) - Kelvin Wave Pattern due to travelling Wave
Fig. 5.1(b) - Kelvin Wave Pattern due to travelling Wave
Fig. 5.2 - Bow and Stern Wave Systems (Schematic) of a Surface Vessel
5.2 Kelvin Wave Pattern
An early idea of the ship wave pattern was given by Lord Kelvin (1987) by considering a pressure point travelling over the water surface. The Kelvin wave pattern (Figs. 5.1a and 5.1b) consists of
(a) a transverse wave system, and
(b) a divergent wave system
The meeting point of the transverse and divergent waves is a high point. In Fig. 5.2, a bow wave system and a stern wave system generated by the forward motion of the ship is shown. The transverse waves move in the same direction of the ship and with the same speed. If the wave length is then,
or,
where L is the length of ship.
Since the divergent waves move at an angle to the direction of the ship, the speed of these waves is and hence the wave length is
=
=
5.3 Wave Interference
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 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 below:
Table 1 - Humps and Hollows in Wave Resistance
nHollow speed
Hump speed
1
2
3
4 4/1 0.798
4/5 0.357
4/9 0.266
4/13 0.221 4/3 0.461
4/7 0.362
4/11 0.235
4/15 0.206
Normally, the first bow wave crest occurs around quarter of a wave length aft of the bow. For high speed vessels (say, planning craft), e.g. Fn = 1.5, the wave length 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 , the marine craft spans two or more waves, changes in draught and trim are small and the drag is predominantly frictional. As 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 resistance 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. , 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 resistance 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 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 displacement, which requires to be supported by buoyancy, becomes important. A volume Froude Number is defined as
At speeds higher than those corresponding to Fn = 0.95, the bottom and aft should be designed for planning, i.e. the lift generated at the boat bottom should support the weight and the boat C.G. must rise up so that there is an effective reduction of wetted surface and hence, frictional resistance. Flow is made to separate at the side as well as at the stern. This is the fully planning region when, the residuary resistance increases very slowly with speed. This is shown in Fig. 4. The development of flow from displacement mode to fully planning mode is discussed in detail in (Savitsky, 1976).
5.4 Bulbous Bows
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 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 multihull ship like a catamaran, the wave system internal to the hulls are superposed. If this superposition is such that the internal wave system is reduced in height, the total wave resistance becomes less than twice that of either monohull. 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 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 obtain 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.
Fig. 4 Speed Power Trends for different Ship Types
5.5Theoretical Formulation and Approximation
5.5.1Equation of motions
The equations of motion for an incompressible Newtonian fluid may be written as
(5.5.1)
where,
q=ju + jv + kw
F=iX + jY + kZ
P=pressure
=
=density
=kinematic viscosity
The derivation of the foregoing equations, known as the Navier-Stroke equations for an incompressible Newtonian fluid with constant viscosity, may be found in many basic references (Schlichting 1968).
When gravity is the only body force exerted, a body force potential may be defined such that
and
where h is the height above a horizontal datum.
Then Equation (1.1) reduces to
(5.5.2)
The Navier-Stroke equations evolved over a period of 18 years starting in 1827 with Navier and culminating with Stokrs in 1845.
If L is a characteristic length scale over which the velocity varies in magnitude by U, equations (5.5.1) are expressed in dimensionless form by resort to L and U then it is seen that the ratio , which is Reynolds number, represent the ratio of the inertial to viscous forces. In a wide class of flows the Reynolds number is very large and the viscous terms in the above equations are much smaller than the remaining inertial terms over most of the flow field.
5.5.2Equation of continuity
The conservation of mass for an incompressible fluid requires that the volumetric dilatation be zero, i.e.
(5.5.3)
Equation (5.5.3) is invariant, i.e., independent of the choice of coordinates.
The solution of 4 unknowns (u, v, w and p) becomes fully determined when the initial and boundary conditions are specified. The Kinematic boundary condition for a nonporous wall is that the normal and tangential components of the velocity relative to the boundary must be zero. From these, the body and free surface boundary conditions and condition at liquid infinity can be written.
5.5.3 Rotational and irrotational flows
The rates of rotation of a fluid particle about the z, y, z axes are given by (Schlichting 1968)
(5.5.4)
(5.5.5)
(5.5.6)
They are components of the rotation vector . The flows for which curl are said to be rotational because each fluid particle undergoes a rotation, in addition to translations and pure straining motion. In the regions of flow where curl q = 0, a real fluid exhibits an irrotational or inviscid-fluid like behaviour since the shear stress vanishes. Rotation is related to two fundamental concepts, namely, circulation and vorticity. Circulation, , is defined as the line integral of the velocity vector taken around a closed curve, enclosing a surface S.
(5.5.7)
According to Stokes theorem,
(5.5.8)
and therefore, Eq. (1.7) may be written as
(5.5.9)
in which twice the components of rotation vector appear. They are said to be the components of the vorticity vector such aht and .
In real fluids vorticity may be generated redistributed, diffused, and destroyed since frictional forces are not conservative. In other words, vorticity is ultimately dissipated by viscosity to which it owes its generation. For example, the vorticity found in a vortex about four diameter downstream from a circular cylinder is about 70 per cent of the vorticity produced at the separation point (Bloor and Gerrard 1966). The remainder is partly diffused and partly cancelled by the ingestion of fluid bearing oppositely signed vorticity.
5.5.4Velocity potential
Irrotational motion exists only when all components of rotation vector are zero, i.e.,
(5.5.10)
It is then possible to devise a continuous, differentiable scalar function such that its gradients satisfy Eqn. (5.5.10).
In cartesian and cylindrical polar coordinates, the velocity components are thus gives by
(5.5.11)
i.e.
(5.5.12)
respectively. Evidently, Eqs. (5.5.11) and (5.5.12) satisfy Eqs. (5.5.10) automatically, i.e. potential flow is irrotational. It is also true that a potential exists only for an irrotational flow.
The introduction of into the continuity equation (1.4) results in a second order linear differential equation, known as the Laplace equation.
(5.5.13)
5.5.5Eulers equation and their integration
The assumption of zero shear enables one to reduce Eqs. (5.5.1) to
(5.5.14)
These are the celebrated Euler equations and have been obtained by Euler about 100 years before the evolution of the Navier Stokes equations.
The use of the conditions of irrotationality and the force potential enable one to reduce the three Euler equations into one equation,
where, q2 = u2 + v2 + w2 and F (t) is an arbitrary function of time only. Frequently, F (t), is absorbed into since this does not affect the physical quantities of interest.
For steady flows , then
This is the familiar Bernoulli equation and enables one to determine the pressure distribution once the potential function and hence the velocity distribution are obtained from the solution of Laplaces equation.
The boundary conditions can be written as
where, n is the unit normal vector at the boundary point drawn outward of the body and into the fluid. is the derivative in the direction of n.
5.5.6Singularity in the flow
Suppose there is a free stream flowing in direction x with its potential given by , now we put an object in this flow. Consequently, the flow around the object will be distributed and tend to be the same free stream flow as one goes away from the object. This object in the flow can be termed as singularity in the stream. Once set of problems in naval hydrodynamics is to find this disturbance, i.e., the modified flow when the geometry of the object is known, e.g. forward motion of a ship in an otherwise calm sea. The reason why this object is called a singularity is that the governing equation of the flow which is Laplace equation will not be required to be satisfied in the space occupied by this object. From the mathematical consideration it is possible to introduce such singularity in the flow in the form of singular solutions of Laplace equation. These singular solutions violate continuity equation and Laplace equation can not be satisfied at singular points of the flow described by these solutions. But if these singular points can be located inside the object or maximum on the surface of the object where Laplace equation need not be satisfied then the introduction of such singular solution and superposition of flow created by them, with or withoug a free stream can lead to some meaningful physically realistic situations. The well known singularities or singular solutions which are used for this purpose are source, sink, doublet or dipole and vortex.
5.5.7 Source
Consider a point from which fluid is emanating at constant rate uniformly in all directions. Such a point is called a source. Let a source be situated at the origin. Consider a sphere of radius, r, with its centre at the origin, the volume rate of flow across this sphere
where, ur is radial velocity
Integrating once gives
is the velocity potential due to am source located at the origin. The streamlines are radially outward. For a source located at any points, x1, y1, z1.
The velocity potential will become
where,
Here m is called the strength of the source.
It may be mentioned here that source potential as given above satisfies Laplace equation as well as continuity equation everywhere except at the point where the source is located. Therefore, this is a singular solution. We can obviously use this solution provided the singular point is not in the flow or fluid domain. (Fig. 5.1).
5.5.8Sink
A negative source is called a sink, i.e., a point which is absorbing fluid from all the sides at a uniform rate. The potential for a sink can be written by changing m to m. Hence, for a sink
The streamlines are radially inward. (Fig. 5.1).
Fig. 5.1 Source and Sink
Fig. 5.1 Vortex
5.5.9Pulsating source and sink
If the rate of emanating fluid or absorbing is varying with a frequency, it is called a pulsating source/sink. Here m will require to be changed to or like.
5.5.10Doublet or dipole
Consider a sink at the point x = -a, y, z and a source at x = +a, y, z. The strength in both the cases is say m. Now if we start bringing the source and sink closer such a way that as the distance between them reduces, strength m increases such that the product remains constant. In the limiting case, the combination is called a doublet of strength . The potential for a doublet can be obtained as follows:
=
=
=
For a being small, using standard procedure
=
In the limiting case of
So,
This can also be written as
-
as
The streamlines of a doublet are coaxial circles from source to sink.
5.5.11Semi-infinite half-body generation
If the source potential is differentiated with respect to r obtain the radial velocity and integrated over the surface of a sphere centered upon the origin, it follows that the rate of flux Q of fluid emitted from the source is precisely equal to m. This parameter is known as the source strength. If m is negative, the flux direction is irreversed and the singularity is called a sink. Mathematically, the distinction between a source and a sink is simply the sign of the strength m, generally we shall use the term source with distinction.
The streaming flow past a semi-infinite half-body can be developed by superposing the source potential and a free stream so that
The resulting flow is axysymmetric about the x-axis, and the streamlines in the x-y place are as shown in Fig. 5.3. Differentiation of the above with respect to x indicates a stagnation point at x = ()1/2, where q = 0. Here flow is deflected around the source; thereafter the outer flow upstream of the stagnation point continues downstream, but with a permanent deflection from the x-axis due to the fluid emanating from the source. Although the inner flow does not correspond to the physical domain of the fluid, it is of interest because it reveals how the source serves to generate the body. Thus, fluid originally emitted from the source tends to oppose the incoming stream and produces the stagnation point, but ultimately all of the inner flow is diverted downstream to infinity. Since the rate of flux emitted by the source is m, and since far downstream this fluid must move with velocity U for the pressure to balance across the dividing streamline, the cross-sectional area of dividing streamline far downstream is equal to m/U. The resulting half-body is semi-infinite in extent.
Fig. 5.3 Streaming Flow Past a Semi-infinite Half body Generated by a Point Source at the Origin. The Body is Axisymmetric about the x-axis and corresponds to the Position of the Dividing Streamline
5.5.12Finite closed body generation
To represent the more practical situation where the body is closed and of finite length, we need to introduce not only a source but also a sink of opposite strength, located so that the fluid emitted by the source will be absorbed into the sink. The sink clearly must be located down-stream of the source, and if these two singularities are situated symmetrically about the origin, x = a, the potential will be
Differentiation with respect to x, with y = z = 0, reveals that the stagnation points are located at x , where, is determined from the equation
The streamlines associated with the above define a Rankine ovoid (Fig. 5.4). The maximum radius b can be found from continuity, since the flux passing across the plane x = 0, inside a circle of radius b will be equal to the flux emitted from the source. Thus, with x = 0,
The resulting flow, shown in Fig. 5.4, is similar to that actually observed for streamlined axysymmetric bodies. From Bernoullis equation one can compute the pressure distribution on the body; it will take a maximum value at the two stagnation points and a minimum at the central plane x = 0, where the velocity is a maximum.
Fig. 5.4 Streaming Flow Past a Rankine Ovoid, or Source-Sink Combination
We might proceed to construct more general axysymmetric bodies by distributing sources and sinks continuously along the body axis. This is a practical method for determining the flow characteristics of bodies of revolution, especially if they are relatively slender. In stead, let us focus our attention on the opposite extreme, where the separation between the two singularities as they are brought together, their strengths m must increase at the same time, for otherwise they will cancel out in the limit when they coincide. This, it is necessary to make the product , a constant, with the result.
=
=
=
The last term is called a dipole or doublet, and the constant is its moment (Section 5.5.10). If we examine the resulting flow, from the combination of this dipole with the uniform stream, in a spherical coordinate system, where, , above equation takes the form
Since the derivative with respect to the radius r vanishes on , the above equation give the flow of a uniform stream of velocity U, past a sphere of this radius.
5.5.13(a) Thin ship theory
One type of body geometry for which the source potential known is the thin non-lifting planar surface that can be associated with a symmetrical thin hydrofoil moving at zero angle of attack, and also with a ship hull of small beam. For these situations, the body surface is to be the first approximation, a flat sheet of small thickness and the source potential itself satisfies the condition of zero normal velocity on the sheet provided sources are located on the sheet and there are no other boundaries (as the flow from the source will be in the plane of the hydrofoil or ships central line plans, and the velocity normal to the boundary will be zero).
Thus thin bodies of this type can be represented hydrodynamically by a centre line plane distribution of simple sources, of strength proportional to the normal velocity on the body surface, provided only that the flow is symmetrical with respect to thin centre plane. This approximation forms the basis of Michell theory of wave resistance of thin ships. Two important topics: the use of a source distribution to represent the thickness effects of thin wing or hydrofoils and the Michell theory of wave resistance of thin ships. If body represented by , source strength at any point in C. L. is proportional to .
(b) Slender ship theory
Here the bodys breadth and depth both in assumed to be small compared to length and sources are distributed on a line represented by the body. If sectional area , source strength at any point in this line is proportional to .
(c) Source distribution on body surface (Hess & Smith)
Source of strength proportional to waterline slope can be distributed over the body surface so that body boundary condition can be satisfied exactly.
(d) CFD methods
All the above methods suffer from calculation failure at the singularity itself and particularly at ends of the body. Mathematical solution have focussed an approximation for this with exact boundary condition satisfaction. The advent of modern high-speed computers have given rise to many CFD techniques.
The viscous flow calculation is more involved. One way to do this calculation is to compute the potential flow and velocities which can form the starting point of viscous flow calculation. Numerous research workers have attempted a complete viscous flow solution using CFD techniques. But a general solution is still a long way.
Therefore, experimental techniques are still the main source of realising drag prediction of underwater bodies.
6. OTHER RESISTANCE COMPONENTS
6.1Eddy Resistance and Boundary Layer SeparationBesides the frictional and wave making resistance, it was pointed out before that several other components contribute to the resistance of a ship such as eddy resistance, viscous pressure drag, separation resistance, and wave breaking resistance.
The turbulent frictional belt around a ship consists of eddies or vortices, so that all forms
of frictional resistance are really due to eddy making. However, the term is usually applied to the resistance due to eddy formation or disturbed streamline flow caused by abrupt changes of form, appendages or other projections, and excludes tangential skin friction. When the total model resistance is measured over a range of speeds and plotted as the coefficient against (in the figure), the curve will be of the general shape shown in Figure 6.1.
Fig. 6.1 Elements of Model Resistance
Also shown is a curve of the coefficient of frictional resistance for a smooth flat plate in fully turbulent flow. The intercept between the curves of for the flat plate and for the total model resistance is the so called residuary resistance coefficient. In a typical case the curve at the very low values of is almost parallel to the curve but some distance above it. Since the primary component of the coefficient varies roughly as the fourth power of the speed, the wave making resistance at very low values of must be vanishingly small, and so the intercept (BC in Figure 6.1) cannot be attributed to this cause. If a curve is drawn parallel to the curve of , the intercept FG represents the wave making resistance coefficient . On this assumption, the intercept FE (=BC) must be due to some other cause, and this is the form resistance.
There are three main causes of this form resistance. The ordinate of the curve applies to a flat surface having the same length and wetted area as the model and so neglects any effects due to the curvature of the hull. This curvature affects the pressure distribution along the length, causing the velocity to increase along most of the middle part and to decrease at the ends. The former effect outweighs the latter. Also, since the path along a streamline from bow to stern is longer on a shaped body than on a flat plate, the average velocity must be higher. Thus, the real skin friction of a ship must be greater than that of the equivalent flat plate. Since the pressure and velocity changes and the extra path lengths are greater the fuller and stumpier the form, such shapes would be expected to have greater form drag. This has been verified by experiments on bodies of revolution run deeply submerged. For a given volume of displacement, increases in the length to diameter ratio L/D beyond a certain point, while it may still reduce the form drag, will increase the frictional resistance because of the greater surface area and so in terms of total resistance there will be some optimum value of the L/D ratio. The value depends on the exact shape and on the amount of appendages necessary to provide directional stability and control, and varies between 5 and 7. For surface ships the intercept CRM has been found to vary from 5 to 15 percent of in naval vessels and up to 40% or more in full cargo ships. These increases, however, cannot be attributed solely to curvature effects, which leads to the other causes of form effect.
The existence of the boundary layer has the virtual effect of lengthening the form and reducing the slopes of the after waterlines. This is a region where the normal pressure on the hull is higher than the static pressure and the forward components of these excess pressures will exert a forward thrust overcoming some of the ships resistance. The presence of the boundary layer reduces these forward components, resulting in an increase in resistance as compared with that which would be experienced in a non viscous fluid, and is called the viscous pressure drag.
If the curvature near the stern becomes too abrupt, the water may no longer be able to follow the hull and breaks away, and the space between the hull and the smooth flowing water is filled with vortices, as illustrated in Figure 6.2. A point at which this happens is called a separation point, and the resulting resistance is the third element of form drag, called separation resistance. Separation of this kind can also affect the pressure distribution on the hull, and so modify the viscous pressure drag. In addition to form and separation resistance, eddy making resistance is also caused by struts, shafts, bossings and other appendages.
Fig. 6.2 Separation and Vortices
6.2Wave Breaking Resistance and Vortex Resistance
Especially in the case of bluff hull forms the phenomena of wave breaking and wave breaking resistance have to be considered as well. For this type of hull the flow ahead of the bow becomes irregular and complex, usually leading to a breaking wave, mentioned in the previous section as well. At very low Froude numbers, below approximately 0.10, wave making hardly occurs and the free surface at the stern rises to a height approximately equal to , where is the speed of the ship and g the acceleration due to gravity, in accordance with Bernoullis equation. As the ship speed increases however, this rise of the wave at the stern no longer occurs and instead the bow wave breaks. The resistance associated with wave breaking has been the subject of extensive investigations. 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 be not too steep. At a certain ship speed the free surface becomes unstable and breaks when the radius of curvature of the curved streamlines results in a value of the centrifugal acceleration greater than a critical value. This is the so called Taylor instability criterion (1950), and when applied to the case of the flow around the bow of a ship with radius , results in the approximate expression that , with in meters and in m/sec, to avoid wave breaking
6.3Air and Wind Resistance
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 a storm, constant speed and direction are usually assumed. Even in a steady wind the speed of the wind varies with height above the sea. For consistency therefore the speed is quoted at a datum height of 10m. Near the sea surface the wind is considerably slower than at and above the datum height. According to Davenport the variation of speed with height can be sufficiently represented by
where is the datum height, is the mean wind speed at the datum height, and is about
7.5 for the atmosphere (this is like the 7th power law in turbulent boundary layers). The axial wind force (wind resistance) is given in terms of a coefficient which is expressed as
where is the transverse projected area of the ship. The axial wind force coefficient
is function of the relative wind angle and typically it varies between as varies
from 0 to 180 degrees. The above force is generally insignificant except when the ship is stopped in a wind or during low speed maneuvering. The wind side force is computed on
the basis of the lateral (side) projected area , and is given by the expression
The variation of with the relative wind angle is generally more or less sinusoidal, and
the maximum value of about 0.8 occurs near 90 degrees (beam wind). The yaw moment
generated by the wind is
where is the length overall and the moment coefficient is
,
where is the center of pressure, which typically varies between of the ships length.6.4Added Resistance due to WavesWinds are seldom encountered at sea without windgenerated waves, sometimes from distant storms. Such waves approaching the ship from ahead can cause appreciable added resistance, in part from the diffraction effect of the moving hull on the encountered waves and partly from the indirect effect of the heaving and pitching motions caused by the waves. In beam and quartering seas, there may be heavy rolling and some yawing, both of which will add to the resistance. Required rudder action, in particular during tight manoeuvres, may also make a significant addition.6.5Appendage resistanceThe principal appendages in ships are the bilge keels, rudders, bossings or open shafts and struts. All these items give rise to additional resistance, which is best determined by model experiments. For rudders this can also be calculated from a knowledge of their shape, using drag coefficients for airfoils of similar characteristics and Reynolds numbers appropriate to their speed and length. The correlation of model measurements to the ship is a difficult question which is not yet satisfactorily solved. The model appendages themselves are very small, so that the Reynolds numbers based upon their speed and dimensions are also small, and scale effect is likely to be important. This is especially so with struts and open shafts. Some tanks have adopted the practice of measuring the increase in CTM on the model due to appendages, and adding only half of this to the total bare hull ship resistance coefficient. Other tanks make no such reduction, adding the full value of the increase in CTM to the ship bare hull resistance, so that the designer must be aware of the specific towing tank techniques. As a means of making approximate estimates of appendage resistance for design purposes, Table 2 quotes overall figures derived from model tests, no reduction being made for scale effects. The appendage resistance is expressed as percent of bare hull resistance.
Table 2 Appendage Resistance of various Ship Types
6.6Trim effects
Due to the change in pressure distribution around a ship at different speeds, it will rise or sink and trim. At low speeds there is a general sinkage and a slight trim by the bow as compared with the at rest condition, Figure 6.3. As speed increases the movement of the bow is reversed and at about the bow begins to rise appreciably, the stern sinks still further and the ship takes on a decided trim by the stern, Figure 6.3. In the average merchant ship form, additional trim by the stern in the at rest condition usually results in an increase in resistance at low speeds and a decrease at high speeds. At low speeds the increased draft aft makes the stern virtually fuller, with a consequent increase in form and separation resistance, whereas at high speeds this is more than offset by the reduction in wave making due to the finer entrance in the trimmed condition. In ballast condition it is usually necessary to carry considerable trim by the stern in order to ensure adequate immersion of the propeller, and this will have similar effects to those stated in the foregoing higher resistance at low speeds, less at high speeds. For any ship which is likely to spend an appreciable part of her time at sea in ballast condition,
model experiments are usually made to investigate these effects.
Fig. 6.3 - Changes in Sinkage and Trim with Speed
7. SHALLOW WATER EFFECTS
7.1Schlichtings Method
The resistance of a ship is quite sensitive to the effects of shallow water. In the first place there is an appreciable change in potential flow around the hull. If the ship is considered as being at rest in a flowing stream of restricted depth, but unrestricted width, the water passing below it must speed up more than in deep water, with a consequent greater reduction in pressure and increased sinkage, trim usually by the stern, and resistance. If in addition the water is restricted laterally, as in a river or canal, these effects are further exaggerated. The sinkage and trim in very shallow water may set an upper limit to the speed at which ships can operate without touching bottom. A second effect is the changes in the wave pattern which occur in passing from deep to shallow water. When the water is very deep, the wave pattern consists of the Kelvin transverse and diverging waves shown in Figure 5.1a, the pattern being contained between the straight lines making an angle of 19 deg 28 min on each side of the line of motion.
The velocity of surface waves in water depth is given by the expression
In deep water increases and approaches a value of unity, and the wave velocity for deep water is given by
where is the wave length,
As the depth decreases, and the ratio becomes small, approaches the value , and for shallow water the wave velocity is approximately given by the expression
.
The wave pattern for a moving pressure point goes through a critical change when , see Fig. 7.1. For speeds less than , the system consists of a double set of waves, transverse and diverging as in deep water, advancing with the pressure point at velocity V. For values of V less than about , the pattern is enclosed between the straight lines having an angle = 19 deg 28 min to the centerline, as for deep water. As V increases above this value, the angle increases and approaches 90 deg as V approaches . The pressure point is now generating a disturbance which is traveling at the same speed as itself, and all the wave making effect is now concentrated in a single crest through the pressure point and at right angles to its direction of motion. The pattern agrees with observation on models and ships when running at the critical velocity in shallow water. The whole of energy is transmitted with wave, and the wave being called a wave of translation. When V exceeds the angle begins to decrease again, the wave system being contained between the lines given by = .
It now consists only of diverging waves, there being no transverse waves or cusps. The two straight lines themselves are the front crests of the diverging system, and the inner crests are concave to the line of advance instead of convex as in deep water.
Fig. 7.1 Effect of Shallow water on Wave Pattern
The effect on resistance due to these changes was investigated by Havelock (1908) for a pressure disturbance of linear dimensions , travelling over water of depth , is shown in Fig. 7.2. It can be seen from the figure that the peaks occur at about , the critical speed for that particular water depth. At this speed the resistance is much greater than in deep water, but ultimately at a sufficiently high speed it becomes less than in deep water. This depth effect has an important bearing on full scale ship trials, and can cause misleading results on the relation between power and speed.
Fig. 7.2 Effect of Shallow water on Wave Resistance
Speeds below and above are referred to as subcritical and supercritical, respectively. Nearly all displacement ships operate in the subcritical zone, the exceptions being fast naval ships. As depth of water decreases, it can be seen from the equation
,
the speed of the wave of given length also decreases. Thus to maintain the same wave pattern, a ship moving in shallow water will travel at a lower speed than in deep water, and humps and hollows in the resistance curve occur at lower speeds than shallow water. The ship speed loss
,
where is the speed at deep water, and at water depth , expressed in percentage terms as , is shown in Fig. 7.3 where Ax is the maximum cross sectional area of the hull.
When the ship is operating in shallow water and in restricted channels, the corresponding speed loss can be evaluated from Fig. 7.4 where denotes the hydraulic radius of the channel defined as
For a rectangular channel of width and depth
When becomes very large, , and this corresponds to the previous case of shallow water of unrestricted width. When a ship or model is in a rectangular channel, the hydraulic radius is
,
where is the maximum cross sectional area of the hull, and the wetted girth of the hull at this section.
From model tests, Landweber ( ) was able to deduce a single curve in terms of for use in restricted, shallow channels
Fig. 7.3 Chart for Calculating Reduction in Speed due to Shallow Water
Fig. 7.4 Curves for Calculating Resistance in Restricted ChannelsAn analysis of shallow-water effects was made by Schlichting (1934). It covered the increase in resistance in shallow water at subcritical speeds, not the decrease at supercritical speeds, and was for shallow water of unlimited lateral extent. The analysis was based on theoretical considerations and on model experiments, carried out in Hamburg and Vienna Tanks, in deep and shallow water, using geosim models to detect any laminar flow on the one hand and tank wall interference on the other. He found that the principal factor controlling was the ratioTypical frictional and total resistance curves for deep water are shown in Figure 7.5 to a base of speed. At any particular speed, the wave pattern generated by the ship in deep water will have a wave length, , given by
In water of depth to the same wave length would be generated at some lower intermediate speed , where
The ratio of the two speed is . A curve of is shown to the base in Figure 7.4.
The reduction in speed on this account is in Fig. 7.5, and Schlichting assumed that the wavemaking resistance in shallow water at speed would be the same as that at speed in deep water. The total resistance at speed would then be found at point B by adding the wave-making resistance to the appropriate frictional resistance at this speed, . The line AB is in fact parallel to EF. There is a further loss in speed because of the increase in potential or displacement flow around the hull due to the restriction of area by the proximity of the bottom, giving as the final speed . Schlichting investigated this reduction in speed by model tests in deep and shallow water, using geosim models to detect any laminar flow on the one hand and tank wall interference on the other. He found that the principal controlling factor for is was the ratio where is the maximum cross-sectional area of the hull
and is the depth of water.
Fig. 7.5 Determination of shallow water resistance by Schlichting's method
7.MODEL EXPERIMENTS
7.1Prediction of Resistance from Model Experiments
It may be difficult to estimate the resistance of small vessels accurately by theoretical means. Therefore, it is customary to run model experiments and extrapolate the experimental results to full scale for resistance prediction. As has been discussed above,
Therefore, model and ship/submerged body should have the same Rn and Fn for kinematic similarity. But that is not possible in a water tank where Rn similarity will require very high model speed/water speed. (In wind tunnels this is possible). Therefore, Froude similarity is followed in model tests. This procedure is very well described in Lewis, 1988. The accepted basis of prediction rests on the assumptions made by William Froude:
(1)the total resistance can be divided into two major components, frictional and residuary;
(2)since the residuary resistance RR (predominantly wave resistance) is a function of Froude number, geometrically similar bodies will have the same specific residuary resistance coefficient CR at the same Fn, where.
On the basis of the above, William Froude stated the law of comparison that states The residuary resistance of geometrically similar ships is in the ratio of the cube of their linear dimensions if their speeds are in the ratio of the square roots of their linear dimensions. These speeds are called the corresponding speeds. The total resistance coefficient CT is written as
But
=
Hence
=
=
=
If both Rn and Fn for model and ship were same, then CT for model and ship would have been same. Then
=
and
=
where, suffix m represents model values and suffix s represents ship values. These two identities, particularly, Reynolds number equality cannot be attained for a smaller model since the speed of the towing carriage is limited. Froude realised that the frictional and residuary resistance components do not obey the same law and out of this necessity made ship model testing a practical tool by proposing what is now called Froude similarity.
Model testing procedure and prediction of full-scale resistance is done in the following manner:
(1)A model is prepared, which is geometrically similar to the ship such that all linear dimensions are in the same ratio, say,
A high level of dimensional accuracy is required to be maintained on the model. The model scale must be chosen such that the model is not too small for practical measurement nor too long so that speed required is outside the capability of towing carriage. To avoid tank wall and bottom effects or blockage, generally model length < depth of water, h, in the tank, the model midship cross-section is x towing tank cross-sectional area. Model speed should be such that
(2)The model is towed in the tank by the towing carriage at the corresponding speed. Then,
=
or,
=
or,
=
or,
=
During the experiment one must ensure that there is fully turbulent flow over the complete model. This is done by attaching turbulence stimulators like trip write, stud, etc. on the forebody. The total resistant RTm is measured and wetted surface Sm is estimated. Then,
(3)The three-dimensional frictional resistance coefficient CFm is calculated from any standard friction formulation described previously for Reynolds number, which is
Then residuary resistance coefficient CRm is obtained as
=
(4) Following the Froude law of similarity
=
The three-dimensional frictional resistance coefficient CFs is calculated using the same friction law as in (3) above. Then,
=
(5)The effective power PE for ship can now be calculated
=
=
=
=
The model must be towed in such a manner that there is no force or moment imposed on the model b
