rise or sink bodily and also trim. At low speeds there is a general sinkage and a slight trim by the bow as compared with the at-rest condition (Fig. 41.) As speed increases the movement of the bow is reversed and at Fn = 0.30 or thereabouts the bow begins to rise ap- preciably, the stern sinks still further and the ship takes on a decided trim by the stern (Fig. 42).
As D.W. Taylor (1943) pointed out, large trim changes or sinkage of the center of gravity are symp- toms rather than causes of high resistance. Neverthe- less they may indicate the desirability of altering the at-rest trim by shifting the center of gravity longitu- dinally. The reductions of resistance which can be ef- fected by such changes of trim as are practicable in large displacement craft are very small, but in high- speed planing craft the position of the center of gravity and the resultant still-water trim have a most impor- tant influence on performance. In both cases the pos- sible effects can be investigated on model scale.
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 a t high speeds. At low speeds the increased draft aft makes the stern virtually fuller, with a consequent increase in form and separation resistance, whereas
STRUT BARREL WITH FAIRED ENDS
-7
STRUT BARREL ENDS WITH SHARP EDGES Fig. 39 Typical strut barrel ends
Table 4-Appendage Resistance on LUCY ASHTON Model length in m
ShiD 2.74 3.66 4.88 6.10 7.32 9.14 speed in
knots increment in ship C, increment in model C,
Ratio with bossings
8 0.44 0.48 0.52 0.56 0.58 0.61 12 0.52 0.57 0.60 0.62 0.65 0.68 14% 0.10 0.12 0.14 0.16 0.17 0.20
increment in ship C, increment in model C,
Ratio with A brackets and open shafts 8 0.48 0.52 0.56 0.58 0.61 0.67 12 0.43 0.47 0.52 0.54 0.57 0.61
14% 0.33 0.37 0.41
-
0.46 0.51Table 5-Approximate Resistance of Appendages Resistance expressed as percent of bare hull resistance.
Type of ship 0.21 0.30 0.48
Value of Fn
Large, fast, 4 screws 10-16 10-16
-
Small, fast, 2 screws 20-30 17-25 10-15 Small, medium speed, 2 screws 12-30 10-23 - Large, medium speed, 2 screws 8-14 8-14
-
All single-screw ships 2-5 2-5 -
Previous Page
42 PRINCIPLES OF NAVAL ARCHITECTURE MAIN DECK
MAIN BARREL ($ PROPELLER
MODEL SCALE
-FULL SCALE
NOMINAL BOUNDARY LAYER THICKNESS Fig. 40 Nominal boundary layer thickness in way of typical appendages (Von Kerczek, et a1 1983)
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, at level trim, the wetted surface per unit of displacement is much increased, so that the frictional .resistance is increased also, but because of the finer form a t the reduced draft, the residuary re- sistance is decreased. In general, except in high-speed ships, the total resistance per unit of displacement will be greater, but because of the lower displacement the total resistance and power will be reduced, and the ship in ballast will make a higher speed a t the same power.
In ballast condition it is usually necessary to carry considerable trim by the stern in order to ensure ad- equate immersion of the propeller, and this will have similar effects to those stated in the foregoing-higher resistance a t low speeds, less at high speeds. For any ship which is likely to spend an appreciable part of her time a t sea in ballast condition, model experiments are usually made to investigate these effects.
5.6 Shallow-Water Effects. 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 con- sequent greater reduction in pressure and increased sinkage, trim 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 a t 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.
These changes have been studied by Havelock (1908) for a point pressure impulse travelling over a free water surface.
When the water is very deep, the wave pattern con- sists of the transverse and diverging waves shown in Fig. 6, the pattern being contained between the straight lines making an angle a of 19 deg 28 min on each side of the line of motion of the point.
As is discussed more fully in Chapter VIII, Vol. 111, in water of depth h the velocity of surface waves is given by the expression
(44) where L , is the length of wave from crest to crest.
As h / L , increases, tanh 27rh/L, approaches a value of unity, and for deep water this leads to the usual expression
(V,) = (gLW/2.rr) tanh 2.rrh/LW
(VJ2 = gL,/27r (45)
As the depth h decreases, and the ratio h / L , becomes small, tanh 2.rrh/LW approaches the value 2.rrh/LW1 and for shallow water the wave velocity is approxi- mately given by the equation
( VJ2 = gh (46)
The wave pattern for the pressure point goes through a critical change when V =
,,@
(see Fig. 43).RESISTANCE 43
2 4 6 0 10 12 14
SHIP SPEED IN KNOTS
I I d l l l I . I C . * k . . L
0.03 0.06 0.09 0.12 0.15 0.18
SCALE OF Fn
Fig. 4 1 Changes in sinkage and trim with speed for T.2 Tanker model. Ship dimensions: 155.4 X 20.7 X 9.2 m according to Norley (1948)
For speeds less than V = ,,@) the system consists of a double set of waves, transverse and diverging as in deep water, advancing with the pressure point a t velocity V. For values of V less than about 0.4
a,
the pattern is enclosed between the straight lines hav- ing an angle a = 19 deg 28 min to the centerline, as for deep water. As V increases above this value, the angle a increases and approaches 90 deg as V ap- proaches
,/&
Fig. 43.The pressure point is now generating a disturbance which is travelling a t the same speed as itself, and all the wave-making effect is concentrated in a single crest through the point and a t right angles to its direction of motion. This pattern agrees with observations on models and ships when running at the critical velocity in shallow water. The whole of the energy is trans- mitted with the wave, and the wave is called a wave of translation.
When V exceeds
a,
a begins to decrease again, the wave system being contained between the lines given by sin2a = g h / (v2,
Fig. 43. It now consists only of diverging waves, there being no transverse wavesor 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.
The effect upon resistance due to these changes in wave pattern in shallow water has been investigated by Havelock (1908) for a pressure disturbance of linear dimension 1 travelling over water of depth h. The re- sistance curves are reproduced in Fig. 44. Each curve is marked with the value of the ratio of depth of water h to the characteristic length of the disturbance 1, that marked CCI being for deep water. When the ratio h / l is 0.75, there is a marked peak at a speed corresponding to a value of V / a = 0.86. Since = 0.866, this corresponds to a value of unity for V/m, so that the peak corresponds to the speed of the wave of trans- lation for that particular depth of water, or the critical speed. A t this speed the resistance is very much greater than in deep water, but ultimately at a suffi- ciently 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
0
t Q 3 0 u)
0 a
k! 0.01 t
U Y
u)
t W t.
u)
I
a
0.02 L 1 I I
0 0.06 0.12 0.18 0.24 0.30 0.34 0.36 0.42 0.48 Fn
Fig. 4 2 Curves of stern sinkage or squat in unrestricted water depth according to Miller (1 963)
44 PRINCIPLES OF NAVAL ARCHITECTURE
$
= 0.40zh
= 0.99= 1.4
6
theoretical considerations and on model experiments carried out in the Hamburg and Vienna tanks.
Typical frictional and total resistance curves for deep water are shown in Fig. 45 to a base of speed.
At any particular speed V , in deep water they are R, and R, respectively. At this speed the wave pattern generated by the ship will have a wave length L , given by
vm2 = gLw/27r (47)
In water of depth h the same wave length L , would be generated at some lower or intermediate speed V,, where
V," = (gLW/27r) tanh 27rh/Lw (44) and the ratio of the two speeds is
(48) V,/V, = (tanh 27rh/LW)v2
= (tanh gh/Vwz)%
A curve of V,/V, is shown to a base of V,/m in Fig. 46. The reduction in speed on this account is
v,
- V, =sc
in Fig. 45, and Schlichting assumed that the wave- making resistance in shallow water at speed V, would be the same as that at speed V , in deep water. The total resistance a t speed V, would then be found a t point B by adding the wave-making resistance R,, to the appropriate frictional resistance at this speed, R,.
The line AB is in fact parallel to EF.
There is a further loss in speed SV, 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 V, = V, - SV,.
Schlichting investigated this reduction in speed by
Fig. 43 Effect of shallow water on wave pattern
the relation between power and speed.
Speeds below and above V =
m
are referred to as subcritical and supercritical, respectively. Nearly all displacement ships operate in the subcritical zone, the exceptions being destroyers, cross-channel ships and similar types. It is seen from Equation (32) that as the depth of water decreases the speed of a wave of given length decreases also. Thus to maintain the same wave pattern a ship moving in shallow water will travel at a lower speed than in deep water, and the humps and hollows in the resistance curve occur a t lower speeds the shallower the water.An analysis of shallow-water effects was made by Schlichting (1934). I t covered the increase in resistance in shallow water a t subcritical speeds, not the decrease at supercritical speeds, and was for shallow water of unlimited lateral extent. The analysis was based on
0 4 06 0 8 10 1.2 1.4 1.6
-
V,G
Fig. 44 Effect of shallow water on wave resistance
R, = wave resistance I = characteristic length h = depth of water
RESISTANCE
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 factor controlling 8Vp was the ratio
R T = TOTAL RESISTANCE R F = FRICTIONAL RESlSTANCE
&X/h
where Ax = maximum cross-sectional area of the hull and h = depth of water. Fig. 46 shows the curve of V J V , against
Kx/
h derived by Schlichting from his model tests and also the relation between V, and V , for different depths of water h. I t should be noted that the ratio V,/V, is sensibly unity for values of V,/less than 0.4, so that in this region the effect of shallow water on the wave-making part of the resist- ance is unimportant. If in Fig. 45 the distance 8Vp is now set out horizontally from B to give BC, C will be a point on the curve of total resistance in shallow water. The corresponding speed is V,. This construc- tion can be made from a number of points to obtain the whole curve.
I t should be noted that a t point C the total resistance
R,= RES,STANCE
45
W 0
z
k-
u) W K LL 0 W -1 V v )
a
e
a
in shallow water at speed
v‘
is less than that in deep water at speed V , -point A. If it is desired to find the speed in shallow water for the same total resist- ance, this will be given approximately by the point H.8 V = 8C
+
SV, (49)l---vm---
Fig. 45 Determination of shallow water resistance by Schlichting’s method
These percentages are given in contour form in Fig.
47.
Schlichting’s work is not theoretically rigorous, but it may be looked upon as a good engineering solution of a confused and complicated problem. In particular, the assumption of equal wave resistance in deep and The total speed loss is
which can be expressed in percentage terms as 8V/V, x 100 = ( V , - V , ) / V , x 100
___
Fig. 46 Curves of velocity ratios for calculating resistance in shallow water (Schlichting)
46 PRINCIPLES OF NAVAL ARCHITECTURE
qh
Fig. 47 Schlichting's chart for calculating reduction in speed in shallow water
sv
Total loss of speed 6 V is given in contours as percentage of speed, -
v,
x 100shallow water when the lengths of the ship-generated waves are the same is open to question. The waves will be steeper and the resistance therefore higher in shallow water, which means that the speed deduced for point C will be somewhat too high. This will partly offset the fact noted above that for the same total resistance the speed should be somewhat higher than that given by the point C, and with all the unknown factors in the problem, C probably gives a close esti- mate of the shallow-water speed.
As an example of the use of the contours, consider the ore-carrier given on the SNAME Resistance Data Sheets No. 9 (Section 61.7, p. 397, of Saunders, 1957).
The ship has dimensions 112.8 m (370 ft) x 19.5 m (64 ft) x 5.3 m (17.5 ft). Assuming a deep-water speed of 13 knots, the speed in water 7.3 m (24 ft) deep, un- restricted laterally, is required
A , = 103.2 m2;
K,
= 10.16 and a X / h = 1.392V , / a = (13 x 0.5144)/(9.81
x
7.3)' = 0.790 andV,'/gh = (0.79)' = 0.624
From the contours in Fig. 47, the speed loss 6V/ V ,
= 20.3 percent, and the ship speed = 10.35 knots.
When the shallow water is restricted laterally, as in an estuary, river or canal, the increase in resistance or the loss of speed will be enhanced. Landweber (1939)
published the results of experiments on the resistance of a merchant ship model in a number of different sized rectangular channels, all at speeds below the critical speed. An analysis of the data suggested to Landweber an extension of Schlichting's method for predicting shallow-water resistance to the case of lateral restric- tion also, i.e., in channels.
Since the speed of translation of a wave in a channel depends only on the depth of water, h, as it does in unrestricted water, it seemed reasonable to assume that Schlichting's method of correcting for the wave- making part of the resistance would still be valid. The speed correction for the displacement flow would, how- ever, have to be modified to take into account the additional restriction introduced by the lateral limita- tion of the channel walls. In shallow water of unlimited width the speed reduction is a function of
Kx/
h, and Landweber sought a similar parameter which would also introduce the width of the channel, b. He found this in terms of the hydraulic radius of the channel,R H *
This ratio is in common use in practice of hydraulics, and is defined as
(50) area of cross section of channel
wetted perimeter
R H =
For a rectangular channel of width b and depth h R, = bh/(b
+
2h)When b becomes very large, R H = h, this corresponds
RESISTANCE 47
to the case of shallow water of unlimited width.
the hydraulic radius is
When a ship or model is in a rectangular channel, (51) R H = (bh - A,y)/(b
+
2h+
p )where
A , = maximum cross-sectional area of hull p = wetted girth of hull at this section.
From the model results, Landweber was able to de- duce a single curve giving the ratio V,/ V, in terms of
fix/
RH for use in restricted, shallow channels. This curve is shown in Fig. 48. It is also reproduced on Fig.46, and it will be seen that it does not quite agree with that derived by Schlichting, a fact which can be ac- counted for by Schlichting's neglect of the effect of the width of the tank in which his experimental data were obtained. Saunders prefers to use Landweber's curve for both unlimited shallow water or restricted water, as the case may be, entering with the value of
ax/
h ora,/
RH as necessary.For completeness and convenience in use, therefore, the curve of V,/ Vm to a base of Vm
/a
is repeated on Fig. 48.To illustrate the case of resistance in restricted chan- nels, consider the ship of the example, given previ- ously, moving in a channel having the section shown in Fig. 49; see Saunders 1957, Vol. 2, section 61.7.
The cross-sectional area of the canal
= [(76.2 x 10.67)
+
(10.67)'/2= 968.3 m2
+
(18.43 x 10.67/2)]and the wetted perimeter is
= [76.2
+
10.67 cosec 45"+
10.67 cosec 30"]= 112.6 m
The maximum cross-sectional area of the ship is 104.0 m2 and the girth 30.17 m. Hence
R, = (968.3 - 104.0)/(112.6
+
30.17)= 864.3/142.77 = 6.05 m was the ratio
&,/RH = m / 6 . 0 5 = 1.69
The equivalent depth of the canal for calculating the critical wave speed is given by
Cross-sectional area 968.3
Width at water surface - 105.3 - 9.20 m To find the speed of the ship in the canal when the resistance is equal to that in deep water at, say, 8 knots
V , = 8 knots = 4.115 m/sec and
CURVE 'FO V TO BASE OF
vcu G
(SCHLICTING, FIG. 46) 096
I ti
CURVE OF -J-" TO BASE OF
V,, = SPEED IN RESTRICTED CHANNEL VW= CORRESPONDING SPEED IN DEEP WATER VI = SCHLICHTINGS INTERMEDIATE SPEED
h = DEPTH OF WATER
RH= HYDRAULIC RADIUS OF CHANNEL A X = MAXIMUM SECTION AREA OF H U L L
! 0.4 0 6 10 12 14
Fig. 48 Curves of velocity ratios for calculating resistance in restricted channels (Landweber)
48 PRINCIPLES OF NAVAL ARCHITECTURE
I-
105.3mL 19.57m
5.33 m
-
10.67mL b
I
t 1
10.67rn 7 6 . 2 m - - 18.43 m
-I- Fig. 49 Cross-section of a canol
V , / @ = 4.115149.81 x 10.67 = 0.402 From Fig. 46 the ratio of V,/ V , is unity, and V, = V,, so that there is no effect of wave-making and all the speed reduction is due to potential flow changes.
f l x / R , = 1.67, V,/ VI = 0.783 From Fig. 48 for
and therefore, since V, = V ,
V,, = 0.783 x V , = 0.783 x 4.115 = 3.222 m / s e c
= 6.26 knots
or a reduction of 22 percent.
Incidentally, if the curve obtained by Landweber is used for the unrestricted shallow-water case, as sug-
0.07
I I I \ I I I 1
0 0 6
0 05
0 0 4
003
0 02
I I I I I I
1.2 1.3 1.4 1.5 1.6 1.7
O L 1.0 1.1
DEPTH OF WATER DRAFT
Fig. 50 Sinkage of bow a t FP in shallow water
gested by Saunders, instead of that published by Schlichting, the speed loss for this ship in shallow water is 18.6 percent instead of the 20.3 percent found previously, the speed being 10.58 knots instead of 10.35.
When large, medium-speed ships or very high-speed ships such a s destroyers have to run measured mile trials, the question often arises of finding a course over which the water is deep enough to ensure that the effects on resistance and speed will be negligible or within stated limits. Conversely, if such a course is not available in a particular case, it is desirable to be able to correct the trial results to obtain the probable performance in deep water.
Both these problems can be solved by the methods described here, and a number of such cases are given by Saunders (1957), Chapter 61, together with charts to assist in their more rapid solution.
The effect of shallow water on some typical mer- chant ship forms has been investigated a t DTRC by running resistance and propulsion tests on models (Norley, 1948). These represented a Liberty ship, a Victory ship, a T-2 ocean-going tanker and a T-1 inland tanker.
The increase in resistance and shaft power, together with sinkage and trim, were measured over a range of speeds 1 a number of depths of water from 6.71 m in the case of the T-1 tanker up to deep water.
The detailed results for each model are given in the report. Figs. 50, 51 and 52 show generalized plots of the sinkage a t bow and stern and the increase in PD and revolutions per minute to a base of the ratio of ship draft to water depth. The sinkage increased with speed and with decrease in water depth, that a t the bow being greater at all speeds up to the maximum reached, for which Vlm = 0.149. There were indi- cations that the sinkage would be greater with larger ratios of beam to draft, but further model tests would be necessary to confirm this trend. There appeared to be a real danger of the ship striking the bottom if the