RESISTANCE 27
of R , can only be obtained by making assumptions as to the amount of frictional resistance, viscous pressure drag and eddy-making resistance, all quantities subject to considerable doubt. The wave-making resistance has been measured directly by observing the shape of the wave system astern of the model and computing its energy and the total viscous drag has been measured by a pitot tube survey behind the model (Wehausen, 1973). Both of these methods are relatively new, and there are problems in interpreting the results. In the meantime it is perhaps best to be content with com- paring the differences between the calculated and mea- sured resistances for pairs of models of the same overall proportions and coefficients but differing in those features which are likely to affect wave-making resistance.
A comparison of much of the available data has been made by Lunde, the measured C, being derived from C, on Froude’s assumption and using his skin-friction coefficients, the calculated C, being empirically cor- rected for viscosity (Lunde, 1957).
At low Froude numbers, less than 0.18, it is difficult to determine CR with any accuracy. At higher speeds, the humps at Fn of 0.25 and 0.32 and the intervening hollow are much exaggerated in the calculated curves, and any advantage expected from designing a ship to run at the “hollow1’ speed would not be fully achieved
in practice (Fig. 24). The general agreement in level of the curves over this range depends to some extent upon the form of the model, theory overestimating the resistance for full ships with large angles of entrance.
Just above a speed of Fn = 0.32 the model becomes subject to increasing sinkage and stern trim, effects which are not taken into account in the calculations.
The last hump in the C,-curve occurs a t a Froude number of about 0.5, and here the calculated value of C, is less than the measured C,, again possibly due to the neglect of sinkage and trim.
In all cases the humps and hollows on the measured curves occur at higher values of Fn than those given by theory, by amounts varying from 2 to 8 percent.
In other words, the model behaves as though it were longer than its actual length, and this is undoubtedly due mostly to the virtual lenghening of the form due to the viscous boundary layer. At very low speeds, Fn
= 0.1, the wave-making resistance varies approxi- mately as the square of the tangent of the half-angle of entrance, but its total value in terms of RT is very small. At high speeds, with Fn greater than 1.0, the wave-making resistance varies approximately as the square of the displacement, illustrating the well-known fact that at very high speeds shape is relatively un- important, the chief consideration being the displace- ment carried on a given length.
Section 5
28 PRINCIPLES OF NAVAL ARCHITECTURE
1
, I
N
-IN I1 V I-
T I I I I l l ! ! I I I I l l /
effect is found in ship models, the intercept BC being greater the fuller the model and the smaller the ratio There are three main causes of this form resistance.
The ordinate of the curve
C , , ,
applies to a flat surface having the same length and wetted area as the model and so neglects any effects due to 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 plank, the average velocity must be higher. Thus the real skin friction of a ship must be greater than that of the “equivalent plank.”
Since the pressure and velocity changes and the extra path length are greater the fuller and stumpier the form, such shapes would be expected to have greater form drag. This is borne out by experiments on bodies of revolution in air, where, for example, Young (1939) found form drag percentages varying with length- diameter ratio as follows:
L/V l’3.
L/D 10 5 3.33
Form drag, percent
. . . . .
5 17 30 Similar figures have been found for bodies of rev- olution run deeply submerged in water. For a given volume of displacement, increases of L/D ratio 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 upon the exact shape and upon the amount of appendages necessary to give directional
stability, and varies between 5 and 7. For surface ships the intercept C,, has been found to vary from 5 to 15 percent of C,,, in warships and up to 40 percent or more in full cargo ships. These increases, however, cannot be attributed wholly to curvature effects, which leads to the other causes of form effect.
In discussing wave-making resistance, it was pointed out that the existence of the boundary layer had 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, owing to the closing in of the streamlines, and the forward components of these ex- cess pressures will exert a forward thrust overcoming some of the ship’s 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 nonviscous fluid, and so is called the viscous pressure drag.
If the curvature near the stern becomes too abrupt, the bilge radius too hard, the after end sections too U-shaped or there are other discontinuities in the hull shape, the water may no longer be able to follow the hull and breaks away, and the intervening space be- tween the hull and the smooth-flowing water is filled with eddies, as illustrated in Fig. l(d). A point a t which this happens is called a separation point, and the re- sulting 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.
To explain the failure of the streamlines to follow the hull it is necessary to consider the variation of pressure and velocity along the length.
The water particles immediately in contact with the hull are assumed to be carried along with it. Due to viscosity, these particles draw the next layer of par- ticles with them, and this effect spreads outwards from the hull. The spread of the boundary layer continues until the velocity of the outer particles at any point is just equal to the potential flow velocity a t this point (Fig. 26). The boundary layer gets thicker from bow to stern due to the continual entrainment of more water. Within this layer the velocity gradients are very much greater than those existing in the potential flow, and most of the fluid shear responsible for the skin friction must occur within this boundary layer. This leads naturally to the idea of a boundary layer of finite thickness, within which the influence of viscosity is important and beyond which viscosity may be ne- glected, a concept which has proved useful in analyses of various problems in aerodynamics (Prandtl, 1904).
Since the velocity in the boundary layer approaches the potential flow velocity asymptotically, its thickness is usually taken to the point where the velocity is 0.99 of that of the undisturbed fluid. The body shape defined by the outer limits of the layer may be considered to move without friction and normal pressures appear to be transmitted across the boundary layer without sen-
RESISTANCE 29
sible distortion. Fluid particles moving aft from mid- ships, relative to the body, have their velocities diminished both by the shearing stresses and by the increasing pressures. Some of them may have insuf- ficient kinetic energy to overcome the adverse pressure gradient and so come to rest before reaching the stern or even start to move forward. Following particles are then forced outward away from the body, setting up pressures which tend to move them back towards the hull, thus causing large scale vortices in the boundary layer. From this point the flow is separated from the hull and a widening band of eddying water intervenes between the hull and the smooth flow outside it. These eddies carry away the kinetic energy which has been expended in forming them and so give rise to sepa- ration drag. Sufficient is not yet known to divide the total viscous drag into its separate components, and this fact has an important bearing on the extrapolation of model results to the ship.
In addition to form and separation resistance, eddy- making resistance is also caused by struts, shafts, bossings and other appendages, as is discussed in Sec- tion 5.3.
Especially in the case of bluff hull forms the phe- nomenon of wave-breaking and wave-breaking resist- ance 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.
At very low Froude numbers, below approximately 0.10, wave-making hardly occurs and the free surface a t the stern rises to a height approximately equal to V 2 / 2 g , where V is the speed of the ship and g the acceleration due to gravity, in accordance with the Bernoulli equation. As the ship speed increases how- ever, this rise of the wave at the stern no longer occurs and instead the bow-wave breaks.
The resistance associated with wave-breaking at the stern has recently been the subject of extensive in- vestigations. Important studies on this topic were made by Taneda, et a1 (1969), who presented the results of flow observations around a fishery training ship and its corresponding scaled model. Baba (1969) was the
first to term the observed flow around the bow “wave- breaking” and presented measurements of momentum loss due to wave-breaking by means of a wake survey far behind a ship model. Baba proposed a hydraulic jump model as a means of calculating the momentum loss due to breaking waves. He showed that wave- breaking resistance may contribute an appreciable part of the total resistance of full forms. Dagan, et a1 (1969) carried out a theoretical study of the two- dimensional flow past a blunt body. The drag associ- ated with wave-breaking was obtained by calculating the loss of momentum of the flow. Ogilvie (1973) ob- tained analytical results for the case of a fine wedge- shaped bow and obtained a universal curve for the shape of the bow wave on the hull. Experimental re- sults generally confirm their predictions.
The most recent work in this area was carried out by Kayo et a1 (1981). They carried out systematic ex- periments on the effect of shear on the free surface.
They concluded that bow wave-breaking can be con- sidered to be due to flow separation at the free surface.
Bow wave-breaking can generally be avoided by re- quiring that the tangent to the curve of sectional areas a t the forward perpendicular be not too steep. Tani- guchi, et a1 (1966) derived a criterion for full form, low-speed ships that can be applied in this connection.
The work carried out by Inui, et a1 (1979), Miyata, et a1 (1980) and Kayo, et a1 (1981) has resulted in a cri- terion based on the half-entrance angle of the water- line. Finally, the work carried out by Taylor, G.I. (1950) reveals that a t a certain speed the free surface be- comes unstable and breaks when the radius of cur- vature of the curved streamlines result in a value of the centrifugal acceleration V 2 / R greater than a crit- ical value. This so-called Taylor instability criterion, when applied to the case of the flow around the stem of a ship with radius R, results in the approximate expression that R 2 V 2 / 5 0 , with R in meters and V in m per sec, to avoid wave-breaking.
5.2 Air and Wind Resistance. A ship sailing on a smooth sea and in still air experiences a resistance due to the movement of the above-water hull through the
VELOCITY DISTRIBUTION IN BOUNDARY LAYER,
Fig. 26 Schematic diagram of boundary-layer flow
U‘ = Velocity at any point on hull in potential flow without viscosity
The velocity in hoiiridary layer approaches U’ asymptotically, and the thickness of layer, 6, is usually measured 1.0 the point where the velocity is 0.99 lJ’
30 PRINCIPLES OF NAVAL ARCHITECTURE
P i CENTER OF WIND FORCE
B A
Vs = S H I P SPEED VR =RELATIVE WIND VT :TRUE WIND
VS Fig. 27 Diagram of wind force
air. This resistance depends upon the ship’s speed and upon the area and shape of the upper works.
When a wind is blowing, the resistance depends also upon the wind speed and its relative direction. In ad- dition, the wind raises waves which may cause a fur- ther increase in resistance. The effects of waves are dealt with in Chapter VIII, Vol. 111.
The “true” wind is termed to be the wind which is due to natural causes and exists at a point above the sea whether or not the ship is there. Zero true wind is still air. The “relative” or “apparent” wind is the vectorial summation of the velocities and directions of the ship and the true wind (see Fig. 27).
Because of the many functions which superstruc- tures have to fulfill, they cannot be adequately stream- lined, and in any case this would be effective only in winds from nearly dead ahead. The reduction in total ship resistance which can be realized by such means is therefore relatively small.
Most of the resistance of superstructures is due to eddy-making, and therefore varies with the square of the speed, and the effects of Reynolds number changes can be neglected. For a ship moving in still air the air resistance can therefore be written as
RA, = coefficient
x
ipA,Vz (24) whereAT = transverse projected area of above-water hull V = ship speed
and the coefficient will have a value depending on the shape of the hull and erections.
An extensive study of the resistance of ships’ super- structures has been made by Hughes (1930). Models were made of the above-water hull and erections, and were towed upside down in water at different speeds and at different angles to simulate various relative wind strengths and directions. Three models were used, representing a typical tanker, cargo ship and Atlantic liner, small structures such as railings and rigging being omitted.
The simulated relative wind velocity V, was deter- mined, Fig. 27, and the total force F acting on the model was measured, together with its direction and point of application. For a given arrangement at a constant angle 8 of the relative wind off the bow, the value of F/(V,)’ was found to be constant for all speeds up to those at which wave-making began to be important.
A typical plot of Fl( V,)’ and a is shown in Fig. 28, where a is the angle between the centerline of ship and the resultant wind force. The value of Fl( V,)’ is a maximum when the relative wind is on the beam, with the maximum area presented to the wind. I t does not correspond with maximum wind resistance to ahead motion, since it is acting approximately at right angles to the direction of motion. As shown later, max- imum wind resistance occurs when the relative wind is about 30 deg off the bow.
I L ’ ;INC
u.
0 w -I 4 V v)
D:W. Taylor (1943) suggested that the air resistance 1.0 of ordinary ships in a head wind could be assumed
equal to that of a flat plate set normal to the direction of motion and having a width B equal to the beam of the ship and a height equal to B / 2 . From experiments in air, he derived a resistance coefficient of 1.28, so that
2F
0.e2
E
0.6r
E 04R A A = 1-28 x
X
PAT ( VR)z V u.= 1.28 x
’/z
X 1.223 x XB2 x ( V R ) z (25) where VR is the “apparent” wind velocity, or wind velocity relative to the ship, in m per sec, B is in m, p0.2
g 0’
0 t
= 0.783 x % B z x (V,)z
In
I I I I I I
0 30 60 90 120 150 180
DIRECTION OF RELATIVE WIND
is in kg/m3 and R A A is in newtons. In still air, V, =
V. Fig. 28 Resultant wind force and center of pressure
( DEGREES OFF BOW 1
RESISTANCE 31
A A
LONGITUDINAL PROJECTED AREA A, TRANSVERSE PROJECTED
AREA A T = 0.3 A, + A 2
Fig. 29 Areas for air resistance
The center of wind force is close to the bow for winds nearly head on, moving aft with increasing val- ues of 8 to a point near the stern when the wind is nearly astern.
For a beam wind, most of the area, both main hull and superstructure, is normal to the wind, and has the same resistance value, so that the effective area is approximately equal to the longitudinal projected area A L , Fig. 29. For a head wind, the main hull below the weather deck has a much lower specific resistance than the frontal area of the superstructures. Hughes found this ratio to be 0.31,0.27 and 0.26 for the tanker, cargo ship and liner, respectively. For practical purposes the
"equivalent area" can thus be found by adding 0.3 of the projected main hull area to the projected super- structure area, giving the transverse projected area
A T (Fig. 29).
Hughes developed an equation F = Kp( V,)' ( A , sin2 8
where again F is in newtons, V, is in m / see, p is the mass density of air = 1.223 kg/m3,
and A , and AT are in square meters. He found that K had a value of approximately 0.6 for all values of 8, varying between 0.5 and 0.65.
For a head wind, 8 = a = 0, and the wind resistance will be
+
A T cos' 8)lcos ( a - 8) (26)R A A = F = KpA,( VR)2, with VR in meter per second, so that
1.0
0.8
3
8
v) 0.6 Y.
3 - I 0.4
a 0.2
0 30 60 120 150 180
80 = DIRECTION OF RELATIVE WIND IN DEGREES OFF BOW Fig. 30 Resistance coefficients for relative wind ahead or astern. All models
with normal superstructures
which is nondimensional in consistent units. Putting K = 0.6
R A A =z 0.734AT( VR)" (27)
which is practically the same as Taylor's expression (25) although the area A , is somewhat different.
For small angles of wind off the bow or stern, the wind force in the line of the ship's motion will be approximately F cos a. Values of F cos alpAT( VR)', or K cos a, for varying values of 8 are shown in Fig.
30 for a tanker, a cargo ship and a liner with normal superstructures. These curves show that, although cos a decreases with an increasing angle of the apparent wind off the bow or stern, the experimentally deter- mined values of F increase so rapidly, because of the rapid increase of area presented to the wind as 8 de- parts from 0 or 180 deg, that the product F cos a increases with 8 and the maximum resistance to ahead motion on all three types of ship occurs when the wind is about 30 deg off the bow. This has been confirmed by full-scale data obtained on the German ship Hum- burg by Kempf, et a1 (1928). The ahead resistance is given by
R A A = F COS a = KpAT( V R ) ~ COS a (28a) With R A A in newtons, VR in m/sec, and p = 1.223
R A A = 1.223 A,( VR)'(K cos a ) (28b)
where ( K cos a ) is the ordinate from Fig. 30 at the desired value of 8.
For a head wind, a = 0, and K from Fig. 30 is about 0.6, so that
R A A = 1.223 x 0.6A,(lr,)'
= 0.734 AT( VR)2 in agreement with Equation (25).
For flat plates of area A , normal to the wind, it was found the value of RIA,( V,)' to be 0.710 in agreement with Hughes' values derived from tests in water.
In a second paper, Hughes (1932) investigated the effects of changes in shape, type and arrangement of superstructures, measuring their single and combined resistances by attaching them to the underside of a raft rather than to a specific hull. The resistance of superstructures can be reduced either by a reduction