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On high speed monohulls in shallow water

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 THE SECOND CHESAPEAKE POWER BOAT SYMPOSIUM
ANNAPOLIS, MARYLAND, MARCH 2010

On High Speed Monohulls in Shallow Water

Dejan Radojcic, University of Belgrade, Faculty of Mech. Engineering, Dept. of Nav. Arch., Serbia
Jeffrey Bowles, Donald L. Blount & Associates, Virginia, USA

ABSTRACT PB Brake (installed) power

PE Effective power
The hydrodynamic performance of marine craft has PD Delivered power
long been known to be influenced by water depth. When RT Total resistance
operating in shallow water at subcritical speeds (typical for Rw Wave making resistance
displacement vessels), they slow down at constant power. RF Frictional resistance
On the contrary, when operating in shallow water at RR Residuary resistance
supercritical speeds (typical for planing vessels), vessel RT Total resistance
speeds increase at constant power. Additionally, surface SWPF Ratio of Dh/D
waves generated by the hull vary radically with vessel SWRF Ratio of RWh/RW or RRh/RR
speed and water depth. T Draught
T Wave period
In recent years, mega yachts are being designed for T Thrust of a propulsor
length Froude numbers (FnL) greater than 0.4, with many t Thrust deduction fraction
operating between 0.5 and 1.0, and some have even higher V Speed (velocity) of the vessel
non-dimensional speeds. As these modern mega yachts Vw Wave speed
being delivered have overall lengths up to and often w Wake fraction
exceeding 100 meters, shallow water effects are being x Distance from track
observed by their captains in relatively deep water. Thus, it  Displacement
is the intent of this paper to refresh, for the mega and high  Vol. of water displaced at rest
speed yacht community, what defines shallow water, the D Quasi-propulsive efficiency = PE/PD
impact on performance and a general discussion on the H Hull efficiency
responsibilities for hull-generated waves and wake
J Jet efficiency
occurring due to shallow water. A power prediction
O Propeller open-water efficiency
procedure applicable in everyday engineering practice is
outlined and an underpinned by a numerical example. P Pump efficiency
R Relative rotative efficiency
NOTATION (SI Unit System) w Wavelength
 Mass density of water
B Breadth on DWL 0.7R Cavitation number at 0.7 propeller radius
CB Block coefficient  Dynamic (running) trim angle
DP Propeller diameter c Thrust loading coefficient
E Wave energy
Fnh Depth Froude number = V/(gh)1/2 Subscripts:
FnL Length Froude number = V/(gL)1/2
Fn Volume Froude number = V/(g1/3)1/2 h - finite water depth
g Acceleration due to gravity d or  - infinite water depth (deep water)
h Water depth
H Wave height INTRODUCTION
L Length on DWL
LOA Length Overall Already a decade or so ago some remarkable (although
L/1/3 Slenderness ratio probably not all commercially successful) high speed
n Propulsor shaft speed monohulls, which deserved special attention concerning
shallow water behavior, were built. For example these It should be noted that only shallow water phenomena
might include Jupiter MDV 3000 (LOA=146 m, V=40+ kts), are treated here, i.e. waterway is restricted by depth only
Corsair 11000 (LOA=102 m, V=35+ kts), Suzuran while the width of waterway is unrestricted. If the width
(LOA =187 m, V=30 kts) etc. Their predecessor Destriero would be restricted too, that would be restricted water
(LOA=67 m, V=60+ kts) was built 15 or so years ago. In phenomena where everything mentioned would be more
recent years some outstanding mega yachts have also been pronounced.
delivered such as A (LOA=118 m, V=23 kts), Ectasea
(LOA= 86 m, V=35 kts), and Silver (LOA=73 m, V=27 kts). WAVE PATTERNS

Both the length and speed of these large monohulls are The impact that shallow water operation has on vessels
growing relative to conventional values; these larger and can be explained by looking at the equation for the velocity
faster monohulls have different resistance characteristics as of surface waves:
explained by Blount and McGrath (2009) paper. Further to g w  2h 
these differences, it is likely that operators are noticing that Vw2  * tanh  
2  w 
these vessels behave differently in shallow water when
compared to navigation in deep water. What is being For extremely shallow water cases where water depth
observed is a hydrodynamic phenomenon referred to as (h) is significantly smaller compared to wave length w, the
shallow water effect. equation can be simplified, as:

Various references indicate that shallow water effects  2h  2h Vw  gh

tanh    so
 w  w
begin to show up at depth Froude numbers greater than
0.7, peak at a depth Froude number of about 0.9-1.0 and
subside at a depth Froude number of about 1.2. With this Assuming that Vw=V (i.e. the transverse waves that
information three regions of depth Froude numbers can be follow the ship have the same speed as the ship) and
identified:
introducing depth Froude number, Fnh=V/gh, it can easily
be proven that transverse waves (whose length increase as
Subcritical region (Fnh<0.6-0.7) where the effects of
ship speed increases) can follow the ship only until Fnh=1.
water depth are almost negligible Namely, Fnh=1 represents the maximum speed of waves
Critical region (0.7<Fnh<1.2) where vessel’s speed when water depth is h; their length then becomes indefinite.
reduces dramatically (at constant power) So, when Vgh (or Fnh1) transverse waves cannot follow
Supercritical region (Fnh>1.2) where speed may be the ship any more. It should be noted, however, that in
even greater than in the deep water. reality critical speed occurs around Fnh=0.95 vice the
theoretical value of Fnh=1.
According to the equation for depth Froude number,
for a speed of 30 knots which is currently not unusual, For (theoretical) speeds less than Fnh=1.0, the system
water depths below 50m (Fnh=0.7) are considered shallow consists of a double set of waves, one transverse and one
and will impact vessel performance! So, it can be easily diverging; just as in deep water. In deep and shallow
concluded that today’s larger, faster commercial vessels water, for values less than Fnh=0.4, the diverging (bow)
and mega yachts are likely to experience shallow water wave angle is 19 degrees and 28 minutes relative to the
effects more frequently and in some cases are sailing in vessel center line (according to the Kelvin ship-wave
what is (hydrodynamically) considered shallow or littoral pattern). Above Fnh0.4 the wave angle tends to increase a
waters throughout their lifetime. bit as shown in Figure 1 for Fnh=0.65.

When operating in shallow water in the critical region, As Fnh approaches 1, the angle of the waves generated
wake wash and power demand can increase significantly by the hull approaches 90 degrees relative to the center line
having a negative impact on vessel performance, the and the waves appear to travel with the hull. At this point
environment, and safety. Maneuvering, noise and vibration all of the wave energy is contained in a single crest moving
characteristics can also be affected. Consequently, the at the same speed of the vessel and only transverse waves
intention of this paper is to firstly identify the above are observed (see Figure 2).
mentioned problems and secondly, to propose approximate
guidelines for evaluation of resistance and powering When depth Froude number exceeds 1.0 the diverging
predictions for fast monohulls operating in shallow water. wave angle decreases. Transverse waves subside and the
Monohulls only are treated herein since both deep and wave system is made up entirely of diverging waves (see
shallow water data for catamarans are not readily available. Figure 3) (Figures 1 through 3 are from Nwogu, 2003).
The proposed power prediction procedure should, however, Figure 4 is a plot showing the change in the diverging wave
be applicable to all high speed vessels, if appropriate data angle with depth Froude number. Both the theoretical
is available. results and experimentally obtained values are depicted.
 WAVE WASH

Fast vessels produce wave wash that is different than

that of conventional ships and natural waves; they have
long periods and significant energy. The amplitude of the
leading wave produced by high speed craft is not so large
(when compared to storm waves, for instance) but it does
have a relatively long wave period. When these waves
reach (get into) shallow water their height increases
rapidly, often causing large and damaging surges on the
beaches. They also arrive unexpectedly, often long after
Figure 1 - Surface wave pattern when Fnh = 0.65 the high speed craft has passed out of view. This further
increases the potential danger because the large waves are
not expected by the public when they arrive.

Consequently, wash restrictions were implemented on

several sensitive high speed craft routes. During the last
20-years of evolution, wash restrictions were first based on
speed limits, then wave wash heights, and ultimately by the
limitation of energy produced by wash at a certain distance
from the vessel’s track. According to the latest findings
both wash height and energy are important; see for instance
Cox (2000) and Doyle et al. (2001).

Concerning wash in the sheltered waters, it is only a

Figure 2 - Surface wave pattern when Fnh = 0.90 vessel’s divergent waves which are relevant. A visual
indicator of wave wash size is usually its height only;
however the wave period seems to be the critical factor
regarding damage.

Deep Water

As mentioned above, wave wash restrictions are now

based on the energy in the wave train. By using this
approach the wave height and period are taken into
consideration. For example, the State of Washington
restricts wave wash energy, E, to values of less than 2450
J/m at a distance of 300 m off the vessel’s track, or 2825
Figure 3 - Surface wave pattern when Fnh = 1.50 J/m at a distance of 200 m off the vessel’s track.
(Figures 1 through 3 are from Nwogu, 2003)
The distances from the vessel are included in the
requirements because wave height diminishes as the lateral
distance from the track increase. The decay rate in far-field
(distances beyond two waterline lengths) may be obtained
from the relation 1/x0.33, where x is distance perpendicular
to ship track. It should be noted, however, that the wave
period is nearly constant as distance x changes.

The calculations of wave wake energy per linear

length of wave front is given by the following equation, in
which the period, T, is associated with the maximum wave
height:
E = (g2H2T2)/16 = 1960H2T2 J/m .

Figure 4 - Impact of depth Froude number on diverging A numerical calculation example is provided in
bow wave angle Appendix 1 for reference.
Shallow Water height is a function of slenderness ratio L/1.3. By
applying low wash design principles, the wave height
The characterization of shallow water waves is more might be reduced but the wave period is not affected (Cox,
complicated because wave period also varies with distance 2000). Moreover, since hull length directly influences
from the sailing line. Longer and faster waves travel on the wave period, increasing length is less effective than
outside of the wash and have a larger Kelvin angle than the reducing displacement. Therefore, deep water wave height
shorter and slower waves. When the waves are in very essentially varies directly with displacement, while the
shallow water and the supercritical region, the first wave in period remains essentially constant. Characteristics such as
the group is usually the highest. However, as depth trim, sinkage, and transom immersion are also influential
increases, the second or third wave typically becomes the on wave wash height, but are secondary to length and
highest. displacement. Hull section shape has little effect, as shown
in Figures 7 and 8.
The appropriate measure of wave wash in shallow
water seems to be both the wave height and wave energy.
As expected, the largest waves occur around Fnh=1 as
shown in Figure 5. Most of the energy is contained in a
single long-period wave with little energy decay at a
distance. The decay rate in shallow water is smaller than in
deep water and is a function of h/L ratio; the hull form
itself has very little impact. Further, the decay ratio at
critical speeds is different than that in supercritical region,
as shown in Figure 6. This is a contributing factor to
unexpectedly large waves in shallow water at a larger
distance from a vessel’s track. If ratio h/L>0.5, the waves
are more or less the same as in deep water.

Figure 5 - Variation of wave height and energy with depth

Froude number (measurements recorded at xL),
(Doyle, 2001)

Low Wash Hulls

Naval architects are nowadays trying to identify low

wave wash hull form characteristics, which is not as simple Figure 6 - Decay rate of wave energy and height with
as it sounds. Generally, low wash ships have an increased distance from vessel track (Doyle, 2001)
length and decreased displacement, i.e. far-field wave wash
 dramatically change compared to that in deep water.
Namely, resistance (RTh) shows a pronounced peak
(maximum) at the critical Fnh of about 0.95, due mainly to a
dramatic increase of wave making resistance (RWh) evident
by the growth of transverse waves. Operation in the
supercritical region (Fnh1), which is characterized by the
formation of diverging waves only, results in a reduction of
resistance compared to deep water. As the water becomes
shallower for constant speed, or speed increases for
constant depth, the effects explained above become more
pronounced. The frictional resistance also changes slightly
due to changes of trim and sinkage, and therefore wetted
Figure 7 - Wash wave trace for chine and round bilge hull surface areas, but these effects are secondary to the changes
for the same vessel design (Phillips and Hook, 2006) in wave making resistance.

It is worthy of noting that in shallow water actually

only one resistance component – the wave making
resistance (RW) – changes dramatically from its deep water
characteristics. Consequently, this phenomenon may be
well expressed through the ratio of shallow water wave
resistance to deep water wave resistance, i.e.
SWRF=RWh/RWd. Following this logic, three speed regions
may be detected as shown in Figure 9:

Subcritical region (according to the ITTC Fnh<0.7)

where the effects of water depth are almost
Figure 8 – Wash wave trace from same vessel traveling negligible (SWRF is around 1)
forward and in reverse (Phillips and Hook, 2006) Critical region (0.7<Fnh<1.2) where RWh increases
dramatically (SWRF is greater than 1)
Consequently, for low wash the following is important: Supercritical region (Fnh>1.2) where RWh may be
- Speeds corresponding to FnL=0.35-0.65 should be smaller than RW (SWRF is a bit smaller than 1).
avoided
- Displacement should be as low as possible while
length should be as large as possible.

So, if all important parameters are kept the same, so

called low wash hulls produce wave patterns that are not
much different than that of ordinary hull forms.
Furthermore, according to Cox (2000) there is no sufficient
evidence for claims that catamaran, multi-hull vessel, or
any other form is significantly better than monohulls
(provided comparison is made between comparable
designs). According to PIANC 2003, high speed craft
wave wash cannot be reduced just by optimizing the hull
form and various design ratios since wave period generally
increases with speed and doesn’t decay quickly, which is
particularly important for navigation in shallow water. Figure 9 - Resistance as function of V and h

SHALLOW WATER INFLUENCE ON RESISTANCE The increase of wave-making resistance, or SWRF, in

the critical region is of primary importance for fast vessels
In shallow water, vessel resistance is very much and depends mainly on the ratio of h/L (where L is vessel’s
different than in deep water, and often may play a waterline length). This dependence is well illustrated by a
significant role in vessel design. Namely, due to the 3D diagram given in Figure 10 (Hofman and Radojcic,
phenomena explained above – growth of transverse waves 1997; Hofman and Kozarski, 2000), where FnL is length
up to Fnh0.95 (theoretically Fnh=1) which is then followed Froude number based on ship waterline length. The
by a complete loss of transverse waves (Fnh1)– a vessel’s information is depicted differently in the so-called shallow
resistance, sinkage and trim (which are all interrelated) water resistance charts shown in Figures 11 and 12. Figure
12 provides the same information as Figure 11, but has Fnh Fh
1.40
instead of FnL on the vertical axis (the relationship between super-critical region
two Froude numbers is FnL=Fnh*(h/L)). Figure 11 is a 1.20
3.0
2.0
1.5
contour plot of the surface illustrated in Figure 10, where 1.0

the critical region is indicated by gray scaling. Operation 1.00

in the black and dark gray zones should be avoided. 0.80

critical region
5.0

4.0

All three diagrams are obtained by relatively 0.60

complicated theoretical calculations (see Hofman and sub-critical region

0.40
Radojcic, 1997; Hofman and Kozarski, 2000).
Nevertheless, the diagrams shown are universal, simple and 0.20

therefore useful since the influential parameters that are

tied together are only L, h and V. Other ship parameters, 0.00
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

such as hull form, section shape, beam etc. are insignificant h/L
and may be neglected. Figure 12 - Shallow water resistance chart

With the purpose to support the above-mentioned, the Blount and Hankley (1976) in their analysis of the
Michlet software (Version 8.07,  Leo Lazauskas) was shallow water testing of a Series 62 model by Toro noted
employed to calculate surface wave patterns for various that resistance is unaffected by water depth when water
length and depth Froude numbers. The resulting wave depth is greater than 80% of the vessel overall length (see
patterns are shown in Appendix 2. Figure 13). One additional factor to remember concerning
the operation of planing craft in shallow water is that the
resistance hump (“hump speed”) typically occurs at a lower
speed in shallow water than it does in deeper water (Toro).

Because the impact that shallow water operation has

on resistance can be so significant, it is imperative that a
yacht master should know when a vessel is operating in
shallow water so the potential impacts are understood.
With regard to whether the vessel is operating in the
subcritical, critical, or supercritical region, only the depth
Froude number needs to be known. It is possible to write
the equation for depth Froude number in two forms,
consolidating constants and conversion factors to provide

Figure 10 - Surface plot of SWRF

(Figures 10 to 12 are from Hofman and Radojcic, 1997)

FL
0.70

super-critical region

0.60 1.5
2.0
3.0
4.0

0.50

critical region

0.40

1.0
0.30
5.0

sub-critical region
0.20

0.10

0.00
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

h/L
Figure 11 - Shallow water resistance chart Figure 13 - Influence of water depth on the effective power
for a Series 62 hull form (Blount and Hankley, 1976)
simple equations that can be easily calculated on a pocket Similar to the variation of resistance due to shallow
calculator for voyage planning or to check specific water effects, the variation of quasi propulsive efficiency is
operating conditions. also affected by the ratio of vessel length to water depth as
Fnh  0.164 *V / h 2 or V  6.1 * Fnh * h 2 V kts, h m depicted in Figure 16 (from Filipovska, 2004). As with
1 1

resistance, the ratio of L/h is the most influential parameter

These equations were used to generate the chart shown in for propulsive efficiency in shallow water.
Figure 14 for quick reference.
80

70

60

50
VESSEL SPEED (kts)

SUPERCRITICAL REGION
40
CRITICAL REGION
30
SUBCRITICAL REGION
20 Fnh = 1.20

Fnh = 0.70
10

0
0 10 20 30 40 50 60 70 80 90 100 110
WATER DEPTH (m)

Figure 16 – Typical surface plot of quasi propulsive

Figure 14 - Quick-reference chart indicating region of efficiency (Filipovska, 2004)
operation
To expand on the cause for this phenomenon, we must
The other and by far the simplest item often used to look at the definition of quasi-propulsive efficiency as
consider when determining if shallow water effects are D=OHR, where H=(1-t)/(1-w). Both thrust deduction
present is the ratio of the water depth to the vessel overall fraction (t) and wake fraction (w) have pronounced peeks
length. Shallow water effects may be noticeable when a in the critical region which is shown in Figure 17.
vessel is operating in water depths less than h/LOA< 0.80. Generally, in the critical region, the wake fraction curve
has a hollow while thrust deduction fraction curve has a
SHALLOW WATER INFLUENCE ON PROPULSIVE peak, resulting in a pronounced hollow in hull efficiency
COEFFICIENTS H. It should be noted, however, that experimental data
regarding the change in propulsive coefficients due to
Variations in the quasi-propulsive efficiency, D, in operation in shallow water are not readily available.
shallow water are exactly opposite to resistance, i.e. around 0.3
the critical depth Froude number ηD decreases compared to 0.2

the value in deep water. That is, a plot of ηD as a function 0.1

of Fnh has a pronounced hollow around the critical speed as

w

shown in Figure 15. The reduction in d is primarily related -0.1

-0.2
to increased propulsor loading due to the increased -0.3
resistance (resulting in a decrease in O), but is influenced 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 Fnh

by other factors as well (Hofman and Radojcic, 1997; 0.6

Radojcic, 1998). Note in the figure that the critical speed 0.5

for d occurs slightly sooner than it does for resistance, i.e. 0.4
t

it is around Fnh=0.85 for D. 0.3

0.2

0.1
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 Fnh

0.6

0.5

0.4
n

0.3

0.2

0.1
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 Fnh

Figure 15 - Quasi propulsive efficiency of a river Figure 17 - General trends for w, t, and D in extremely
fire-fighting boat (shown in Figure 18) shallow water
SHALLOW WATER INFLUENCE ON POWERING The net result is that due to both an increase in
PREDICTIONS resistance and a reduction in quasi-propulsive efficiency, it
might happen that vessel’s speed in the critical region is
Figure 18 provides curves (model test data) for substantially lower than is expected and/or that power
delivered power, shaft speed, vessel trim, and vessel demand is substantially increased. This is explained
sinkage at various water depths for a 27.5 meter river graphically in Figure 19. However, when operating in the
vessel propelled by three screw propellers. This super-critical region the reverse occurs; the required power
comprehensive figure illustrates the following shallow to achieve a specific speed may be smaller than in deep
water impacts: water due to smaller resistance and somewhat larger
propulsive efficiency.
 The trim () and sinkage of the hull varies with
water depth. Cavitation considerations, which are important for all
 The hump speed (vessel speed at which maximum propulsors, are especially important when considering
trim occurs) is less in shallower water. shallow water operation. In the worst-case scenario, the
 The propulsion power significantly increases in thrust loading, on a propulsor sized for deep water
the critical region as water depth decreases. operation, can increase substantially enough due to shallow
 Propeller shaft speed (n) at a given vessel speed water resistance to cause the onset of excessive cavitation
increases as water depth decreases (indicating a and thrust breakdown. The achievable shallow water
reduction in efficiency) vessel speeds may be significantly less than in deep water
cases because thrust breakdown causes the quasi-
propulsion efficiency to drop dramatically.

Figure 19 - R and T(1-t) versus V in shallow and deep

water showing that the achievable speed in shallow water
might be much lower than in deep water, i.e. V4h vs. V4∞
(Hofman and Radojcic, 1997)

A simple check procedure for the thrust loading limits

of a propeller is presented by Blount and Fox (1978).
Equations for several different propeller designs are given
that establish a border representing the maximum thrust
loading limit as a function of cavitation number. Once the
propeller operating conditions are calculated they are
plotted on the same axes as the thrust loading limit. If the
propeller operating conditions approach and become
Figure 18 - Power-speed diagram (with RPM, trim and asymptotic to the thrust loading limit, it is indicative that
sinkage) of a fire-fighting river boat operating in different cavitation is present (See Figure 20).
water depths (Heser, 1994)
 1
Zone II for a short period of time at lower speeds, has to be
MAXIMUM THRUST LOADING
LIMIT TAKEN FROM BLOUNT &
FOX (1978)
changed for vessels intended to operate in shallow water.
Larger water jets with higher cavitation margins must be
CALCULATED PROPELLER
chosen which means, that an installation tailored for
OPERATING CHARACTERISTICS
shallow water is going to be heavier and more expensive.
0.1
c

CURVE INDICATES
INCEPTION OF CAVITATION
In situations where significant cavitation (or thrust
breakdown) does not occur when operating in shallow
THRUST BREAKDOWN
IS PRESENT
water (critical region), the vessel’s speed could still be
significantly limited as the main engines often do not have
enough torque available at low engine speeds to allow them
0.01
0.01 0.1 1 to reach rated speed and power, i.e. the engines are then
0.7R
overloaded. Further discussion of engine loading (or
overloading) is beyond the scope of this paper even though
Figure 20 - Plot of propeller performance indicating
it is very important for all propulsors in general and
thrust loading limits
especially for propellers. Blount and Bartee (1997) provide
a good explanation of engine loading and overloading.
In the case of water jets, a cavitation check is
performed by simply plotting the shallow water resistance
FULL SCALE TRIALS
curve on the water jet performance map to determine if the
curve crosses into the cavitation region. Figure 21 shows a
The above conclusions, some of them based on theory,
generic performance map with a deep water and shallow
sound good enough provided that the changes explained
water resistance curve overlaid on it. The contours of
actually occur when operating in shallow water. Each of
efficiency included in this figure support the claim that the
the two full scale trial cases presented below represent
efficiency decreased in shallow water operation. Three
different vessel sizes, section shapes, and vessel speeds to
cavitation zones are also shown in the figure:
show that the impacts are observed by craft of different
types.
 Zone I - unrestricted operation
 Zone II – limited operation allowed The first vessel is a hard chine, flat bottom planing
 Zone III – operation not recommended – thrust craft with an overall length of 10.5 meters that utilizes
breakdown likely. submerged propellers for propulsion. Figure 22 (from
Blount and Hankley (1976) shows that when water depth is
less than 80% of LOA, the vessel power demand increases
relative to deep water when operating at displacement
speeds and that the vessel power demand decreases relative
to deep water when operating at planing speeds (note
similarity with Figure 13).

Figure 21 - Typical water jet performance map (RR-

KaMeWa prospectus, Allison 1993)

From Figure 21 it is obvious that for properly sized

water jet for deep water, in shallow water: a) zone III may
be easily reached, and b) operation at poor efficiency
around the hump is unavoidable. Therefore, the usual Figure 22 - Ratio of shaft power in shallow water to deep
practice of choosing the smallest water jet size which meets water for a fast craft operating in different water depths
the thrust design point at high speed, and which operates in (Blount and Hankley, 1976)
 Figure 23 illustrates the low speed performance in the familiar deep water evaluation methods, and b)
subcritical region of a water jet propelled, round bilge, 46- replacement of specific computer routines (when
meter motor yacht when operating in shallow water at necessary) either in deep or in shallow water.
maximum power. The data in this case is limited, but
clearly shows increase in resistance associated with It seems, however, that a reliable method for the
operation in shallow water. evaluation of propulsive coefficients (and particularly
quasi-propulsive efficiency) in shallow water does not (yet)
exist. Namely, it is not only that shallow water propulsive
coefficients in critical and supercritical region are rare and
are still missing for fast shallow water hull forms (usually
river vessels with unique form, i.e. extremely low draught
with tunnels and relatively large L/B and B/T ratios), but
they differ considerably from fast sea going vessels. It is
obvious, therefore, that the propulsive coefficients of
dissimilar hull forms also differ either in deep or in shallow
water. The above-mentioned is actually the main reason
that evaluation of SWPF (ratio of quasi-propulsive
coefficients in shallow and deep water) explained in
Figure 23 - Plot of the maximum vessel speed versus water Radojcic (1988) paper is abandoned here. Instead, the
depth for a displacement yacht open water efficiency is calculated for the vessel in the
specific operating condition and the changes on the
propulsive factors are ignored. In this manner the
detrimental impact of increased propulsor loading on
POWERING PREDICTIONS IN SHALLOW WATER propulsor efficiency (the most significant factor) is
calculated relatively accurately.
It is well known that a strong interaction exists
between resistance and propulsion, particularly for high Resistance Evaluation of High Speed Vessels in Deep
speed craft, so that an integrated approach or even Water
integration of the whole ship design synthesis would be
desirable, as discussed by, for example, Allison and Deep water hull resistance may be obtained in various
Goubault (1975) or Radojcic (1991). Nevertheless, ways, including use of specific model tests, or appropriate
traditional division into resistance and propulsion systematic series model data and regression equations.
evaluation is accepted here as the data available are not Although deep water resistance is not the subject of this
sufficient to support a one-problem approach. So, deep study, for the sake of completeness a few prediction
water and then shallow water resistance (based on methods are mentioned here.
Radojcic, 1998) should be evaluated first and then the
quasi-propulsive efficiency is determined based on the Resistance in the planing and semi-planing (or semi-
actual thrust loading of the propulsor in the shallow water displacement) regime for the hard chine hulls may be
condition. This modular approach, however, allows easier calculated according to Savitsky (1964), Hadler (1966),
updating when new information is obtained, which in this Hadler and Hubble (1971), modified Savitsky method
case is surely needed. presented by Blount and Fox (1976), Hubble (1981) or the
regression equations by Radojcic (1991). Original NPL
A reasonably good estimation of the shallow water series (Bailey, 1976), or regression equation derived by
resistance, and in particular of the dominant wave-making Radojcic et al. (1997 and 1999) may be used for hulls with
resistance component (in the critical and supercritical round bilge, while for double-chine hulls the Radojcic et al.
region), may be obtained through the application of theory, (2001) paper would be appropriate. The Andersen and
as for instance the Srettenski integral (thin ship Guldhammer (1986) and Holtrop (1984) methods may be
approximation which is similar to well known Michell used for fast ships with conventional hull forms. For
integral for deep water) – see Hofman and Radojcic (1997) slender displacement hull forms, Cassella and Paciolla
and Hofman and Kozarski (2000). However, the approach (1983) or Fung and Leibman (1995) may be used. The
accepted here is simpler and may be used in everyday original data of Series 62, 65, NPL, 64, etc., or the
engineering practice as is based on an undemanding regression equations derived from the data, may also be
evaluation of SWRF (ratio of residuary resistance in used. In this respect, see also the recent Blount and
shallow and deep water). Thus final results primarily rely McGrath (2009) paper. The calculation of factors
on deep water data (which are more reliable) and possible secondary to the bare hull resistance, such as the appendage
inaccuracies in (unreliable) SWRF should not influence resistance, wind resistance, added resistance due to waves
shallow water power evaluation to a great extent. Besides etc., is also addressed in some of the papers identified
being intuitive, this approach enables: a) employment of above.
Propulsive Efficiency Evaluation of High Speed Vessels The powering performance (prediction of speed) of a
in Deep Water water jet propelled vessel can be determined by finding the
speed at which the resistance curve and the power contour
Vessels Driven by Propellers of interest intersect.
For quasi-propulsive efficiency evaluation (D=ORH)
it is important to know the open water efficiency of the
Resistance Evaluation of High Speed Vessels in Shallow
propeller O, relative rotative efficiency R and hull Water
efficiency H (consisting of thrust deduction fraction and
wake fraction). R and H can be evaluated as per Blount The approach presented here is not new and is based
and Fox (1976) paper for hard chine craft, per Bailey (1976 on the fact that L/h is the dominant factor and that other
or 1982) for round bilge craft, and per Holtrop (1984) for parameters (L/T, CB, hull cross-section, etc.) are of
fast displacement ships. Of course there are other sources secondary importance. It is also assumed that shallow
which may be consulted, as well as accurate (but water influences only residuary resistance, while frictional
expensive) model tests. Open water efficiency can be component is essentially unchanged. Shallow water
approximated from characteristics of the propellers with residuary resistance may be calculated from SWRF given
the segmental type sections (KCA, AEW, MA, etc.) even if by RRh/RRd = f(Fnh, L/h), which is obtained either from
custom fixed pitch or controllable pitch propellers are used. model tests, or calculated from linearized wave theory for
The Wageningen B-Series propeller data can be used for the resistance of a ship in deep and shallow water. In both
slower vessels. Regression equations for AEW and KCA cases results are expressed in terms of a ratio of shallow to
series are respectively given in Blount and Hubble (1981) deep water residuary resistance (i.e. SWRF), rather than the
and Radojcic (1988), for example. absolute resistance.

Vessels Driven by Water Jets Published model tests for shallow water are rare.
For fast vessels with speeds above 30 knots water jets Sturtzel and Graff (1963) describe shallow water model
may be the preferred choice for propulsor. Quasi- tests conducted for 15 different round bilge hulls
propulsive efficiency for water jets is slightly different than encompassing a wide range of L/B, B/T, CB, and L/1/3.
propellers and is D=PRHJ, where P is pump However, a single diagram for SWRF is given, shown in
efficiency and J is jet efficiency. R and H have the same the right side of Figure 25. This diagram is a starting point
name and meaning as for propellers, but not the same for development of the simple resistance prediction method
values. H and J depend on the specific loading of the used here (from Radojcic, 1998), Equations 1 to 3 were
water jet. Reduced efficiency is directly related to used to develop the graph on the left side of Figure 25.
increased loading – a phenomenon which will occur during
shallow water operation (See Figure 24). Note the
reduction in efficiency at the same vessel speed when only
two of the three water jets are used.

Water jet performance data are usually not readily

available, so estimation of the quasi-propulsive efficiency
without consulting the water jet manufacturer is not
recommended. However, efficiency increases with ship
speed and is typically 0.60 and 0.70 for vessel speeds of 30
and 40 knots respectively, as given for example by
Svensson (1998) for high speed ferries (see Figure 24).

Figure 25 – Approximated (by EQ 1-3) (left) and original

Figure 24 - KaMeWa (now Rolls-Royce) water jet sea trial (right) SWRF of Sturtzel and Graff series
results (Svensson, 1998) (Radojcic, 1998)
 RRh/RRd = a+b*Fnhc EQ 1 value. The part of the resistance curve for the subcritical
a = EXP [(-0.00370+0.00265*(L/h))/ regime for shallowest possible water depth h1 may be
(1-0.33444*(L/h)+0.03037*(L/h)2)] calculated according to Equation 1.
b = 1/[-3.5057+0.0312*(L/h)2+14.7440/(L/h)]
c = 2.0306+10.1218*(atan[((L/h)-4.6903)/
0.7741]+/2)/
(RRh/RRd)max = 0.97476+0.01495*(L/h)3 EQ 2
(RRh/RRd)critical = (RRh/RRD)max-(L/h)/20 EQ 2a
(Fnh)max = 0.92226-0.30827/(L/h)1.5 EQ 3
(Fnh)critical = 0.95 EQ 3a

A SWRF could also be obtained from linearized wave

theory with a simplified hull, taken from, for example,
Millward and Sproston (1988), Millward and Bevan
(1985), or it may be calculated separately. SWRF as a
result of these calculations for a T/L=0.03 is illustrated in
Figure 26. SWRF for other hull forms would be similar
(Hofman and Radojcic, 1997).
Figure 27 Resistance in various water depths (with
maximal and critical peaks)

The equations are relatively simple and are suitable for

programming enabling approximate evaluation of shallow
water resistance; hence complicated theoretical approaches
(too complex for daily practice) are avoided. It should be
noted, however, that the above equations evaluate
resistance increments in the subcritical region only since
the primary problem for a vessel is to overcome the large
resistance hump around the critical speed; resistance
decrements in the supercritical regime are neglected since
they are less important.

Shallow water resistance increments (+) and

decrements (), valid for a particular sea route, are
schematically shown in Figure 27. Resistance decrements
Figure 26 - SWRF obtained from linearized wave theory
in supercritical speed regime will allow a bit higher speed
with a simplified hull (source: M. Hofman)
than in deep water, or the same speed will be maintained
with lower rpm and power consumption. On the other
hand, if propulsors are selected according to deep water
The depth of sea route may vary, between say, depth
resistance, then resistance increments in the critical region
h1 and hn, or maybe between h1 and deep water (h=), in will pose serious problems, such as reduced service speed,
which case resistance curves for several water depths are cavitation, engine overloading, vibrations, etc.
required. Using SWRF for ratios L/h1 to L/hn enables
calculation of Rh1 to Rhn. It is sufficient, however, to Propulsive Efficiency Evaluation for High Speed
connect the peaks of resistance humps, i.e. to take the Vessels in Shallow Water
envelope of Rh1 to Rhn, as shown in Figure 27. This
envelope may be calculated from (RRh/RRd)critical and Propulsive coefficients will change in shallow water as
(Fnh)critical, given by equations 2a and 3a. It should be noted discussed previously. Obviously, shallow water corrections
that resistance peaks called critical peaks (denoted "o") do to propulsive coefficients are necessary but are not readily
not coincide with SWRF peaks called maximal peaks available. There have been some attempts to determine a
(denoted ""). Both are shown in Figure 27. In the SWRF general approach for evaluation of SWPF as for instance in
graph (Figure 25), the values corresponding to critical Radojcic (1998) and Filipovska (2004). These approaches
peaks are a bit lower than the maximal peaks. Differences were based on Lyakhovitsky’s idea (see Lakhovitsky
between critical and maximal peaks (although practically 2007), but without success due to insufficient data, so
negligible) are explained in Hofman (1998) paper. further research in this area is recommended.
Obviously, (L/h)/20 in Equation 2a is some form of a Consequently, due to insufficient data supporting values for
correction factor while (Fnh)critical=0.95 is an approximate SWPF, the recommended approach for predicting powering
performance is to recalculate the open water efficiency for This means that fast littoral and inland vessels should
the propulsor based on the resistance curve once it has been be designed (matched, adapted) according to the water
adjusted for shallow water. This approach accounts for the depth. Consequently, the right choice of vessel speed and
change in open water efficiency due to increased thrust waterline length should be decided in the very early design
loading on the propulsor, but does not attempt to address phases, as there isn’t any possibility to improve poor
changes in H or R. For propeller driven vessels, this is performance later on.
done using the procedure defined in Blount and Fox
(1976), using the shallow water resistance without altering For littoral and inland vessels that operate in shallow
H and R. For water jet propelled vessels, this is done by waters, propulsor size selection must be tailored to
overlaying the shallow water resistance curve on the same operation in these conditions. For a properly sized
performance map provided by the manufacturer for deep propulsor for deep water, when operating in shallow water,
water operation (which includes an assumed H and R) – poor efficiency around the hump is unavoidable. In any
see Appendix 3. case, propulsors that are adaptable to (or are less affected
by) changes in resistance should be considered, as for
CONSIDERATIONS FOR VESSEL DESIGN FOR instance controllable pitch propellers, water jets, or
SHALLOW WATER OPERATION probably even special shrouded propellers.

As mentioned above, shallow water resistance may be If vessels are to operate in shallow water, complex
much larger around critical speed than is in deep water, propulsion plants (engines, gearbox ratios, propellers,
however for supercritical speeds it is only a bit lower. water jets) may be required according to the constraints
Therefore, a real problem for vessels intended to sail at dictated by the expected water depth. Flush type water jets
supercritical speeds is to overcome large resistance hump have an advantage of being in the hull and not bellow the
around the critical speed. An interesting paper on this hull as are, for instance, various kinds of propellers. This
subject was presented by Heuser (1994). inherently reduces the vessel’s draught. Low-draught
propellers are typically surface piercing propellers but they
According to Hofman and Radojcic (1997), the only are not “elastic” to cope with large resistance and speed
way to avoid the negative influence of water depth on variations (except if equipped gearboxes with multiple gear
resistance is to avoid the critical region itself (See Figures ratios). Particular attention should be paid to intermediate
10 to 12). Obviously, the largest power increments and speeds, i.e. the critical region, and in this respect the
vessel generated waves occur when water is very shallow margins suggested by Blount and Bartee (1997) should be
(low h/L ratio) and when Fnh0.95 or FnL0.4 and thus consulted.
having a design point in (or around) these conditions
should be avoided. As the water becomes deeper and h/L Actually, all known “deep water” approaches for
increases, a somewhat higher FnL becomes critical. So improving performance, and in the first place “the longer,
when h/L0.5 (practically deep water) speeds the better” theory, are less effective in shallow water and
corresponding to FnL0.5 form the wave lengths that are are even detrimental in extremely shallow water (as ratio of
equal to ship length; this is a well known deep water h/L becomes the most influential parameter). The only
phenomenon that should be avoided. In other words, when measure that really “works” in all regimes is to reduce
h/L increases from 0.1 to 0.5, the corresponding high- displacement (weight) as much as possible, but usually that
resistance Froude numbers1 are FnL=0.3-0.5 and FnL=0.3- is easier to say than to achieve.
0.7 (with peaks at FnL0.4 and FnL0.5), respectively. So, a
CONCLUSIONS
high-speed vessel which will successfully sail in all water
depths – shallow and deep – must be able to operate in the
The following conclusions can be drawn from the
supercritical regime, i.e. above FnL0.7. This vessel also
work contained herein:
must be able to accelerate through the critical regime
 Shallow water effects can be noticed whenever
rapidly. Typical wave pattern of a supercritical vessel (i.e.
intended to operate in the supercritical regime) is depicted h/LOA<0.80 or Fnh>0.6-0.7
in Figure A3.  Three speed regions may be detected:
- subcritical region (Fnh<0.7) where the effects of water
To reduce wave wash, the depth–critical speed range depth are negligible (SWRF=1)
to be avoided (Cox, 2000) should be more than 75% and - critical region (0.7<Fnh<1.2) where wave-making
less than 125% of the speed corresponding to a depth resistance, hence resistance increases dramatically
Froude number of 1.0. High-wash speeds generally (SWRF>1)
correspond to FnL=0.35-0.65. - supercritical region (Fnh>1.2) where resistance may
be smaller than in deep water (SWRF<1)
 SWRF can be predicted with engineering accuracy,
1 while SWPF requires further research, i.e water depth
In the context of this paper speed that matches the high-resistance
Froude number might be called sustained speed.
 effects on propulsive factors are not investigated nor Blount, D., Hankley. D., “Full Scale Trials and Analysis of
defined accurately enough High Performance Planing Craft Data”, Trans. SNAME
 By far the largest power increment and vessel 1976.
generated waves occur when h/L ratio is low and
Fnh0.95 or FnL0.4 Blount, D., Fox, D., "Design Considerations for Propellers
 A high-speed vessel which will successfully operate in in a Cavitating Environment", Marine Technology, Vol. 15,
all water depths must be able to operate in the No. 2, 1978.
supercritical regime, i.e. at speeds above FnL0.7.
 High speed craft wave wash cannot be reduced just by Blount, D., Hubble, N., “Sizing Segmental Section
optimizing the hull form since wave period generally Commercially Available Propellers for Small Craft”,
increases with speed and doesn’t decay quickly. The SNAME Propellers ’81 Symposium, Virginia Beach, 1981.
wave wash decay is smaller in shallow than in deep
water and is a function of h/L. The appropriate Blount, D., Bartee, R., "Design of Propulsion Systems for
measure of wave wash in shallow water is wave height High-Speed Craft", Marine Technology, Vol. 34, No. 4,
and wave energy. 1997.
 Design speed and waterline length, must be carefully
selected for vessels designed for littoral and shallow Blount, D. L., McGrath, J. A., “Resistance Characteristics
water operations. Traditional “deep water thinking” of Semi-Displacement Mega Yacht Hull Forms”, RINA Int.
that “longer is better” might be counter to improving Conf. On Design, Construction & Operation of Super and
performance in shallow water. Mega Yachts, Genova, 2009.
 Reduction of displacement (weight) is the only
Cassella, P., Paciolla, A., "Evaluation of DTMB's Series 64
measure that can effectively lower both, resistance and
Hull Power by Means of Regression Analysis", Tecnica
wave wash.
Italiana, No. 3/4, 1983 (in Italian).
 Propulsor type and size should also be tailored to the
shallow water requirements/operation. Cox, G., “Sex, Lies, and Wave Wake”, RINA Symp.
Hydrodynamics of High Speed Craft: Wake Wash &
ACKNOWLEDGEMENTS Motions Control, London, 2000.

The authors would like to thank Donald L. Blount for Doyle, R., Whittaker, T., Elsasser, B., “A Study of Fast
his kind assistance with identifying reference and providing Ferry Wash in Shallow Water”, FAST 2001, Southampton,
a keen eye and mind in the editing process. 2001.

REFERENCES Filipovska, M., “Analysis of Shallow Water Influence on

Propulsive Coefficients”, Diploma thesis, University of
Allison, J., "Marine Waterjet Propulsion", SNAME Belgrade, Faculty of Mech. Engng, Dept of Naval
Transactions, Vol. 101, 1993. Architecture, Belgrade, 2004 (in Serbian).

Allison, J., Goubault, P., "Waterjet Propulsion for Fast Fung, S., Leibman, L., "Revised Speed-Dependent
Craft - Optimized Integration of Hull and Propulsor", FAST Powering Predictions for High-Speed Transom Stern Hull
1995, Lubeck –Travemunde, 1995. Forms", FAST 1995, Lubeck-Travemunde, 1995.

Andersen, P., Guldhammer, H., "A Computer-Oriented Hadler, J., Hubble, N., "Prediction of the Power
Power Prediction Procedure", CADMO Conference, Performance of the Series 62 Planing Hull Forms", Trans.
Trieste, 1986. SNAME, 1971.

Bailey, D., "The NPL High Speed Round Bilge Hadler, J., “The Prediction of Power Performance on
Displacement Hull Series", RINA Maritime Technology Planing Crafts”, Trans. SNAME, 1966.
Monograph No 4, 1976.
Heuser, H., "Inland and Coastal Vessels for Higher
Bailey, D.,"A Statistical Analysis of Propulsion Data Speeds", 21st WEGEMT, Duisburg, 1994.
Obtained from Models of High Speed Round Bilge Hulls",
RINA Symp. on Small Fast Warships and Security Vessels, Hofman, M., Radojčić, D., “Resistance and Propulsion of
London, 1982. Fast Ships in Shallow Water”, Monograph, University of
Belgrade, Faculty of Mechanical Engineering Dept of
Blount, D., Fox, D., "Small Craft Power Prediction", Naval Architecture, Belgrade, 1997 (in Serbian).
Marine Technology, Vol. 13, No. 1, 1976.
Hofman, M., "On Optimal Dimensions of Fast Vessels for Radojcic, D., “Power Prediction Procedure for Fast Sea-
Shallow Water", PRADS '98, The Hague, 1998. Going Monohulls Operating in Shallow Water”, The Ship
for Supercritical Speed, 19th Duisburg Colloquium, 1998.
Hofman, M., Kozarski, V., “Shallow Water Resistance
Charts for Preliminary Vessel Design”, I.S.P. Vol. 47, No. Radojcic, D., Princevac, M. Rodic, T., “Resistance and
449, 2000. Trim Predictions for the SKLAD Semidisplacement Hull
Series”, Oceanic Engineering Int., Vol. 3, No. 1, 1999.
Holtrop, J., "A Statistical Re-Analysis of Resistance and
Propulsion Data", I.S.P. Vol. 31, No. 363, 1984. Radojcic, D., Grigoropoulos, G. J., Rodic, T., Kuvelic, T.,
Damala, D.P. “The Resistance and Trim of Semi-
Hubble, N., "Planing Hull Feasibility Model", Report of Displacement, Double-Chine, Transom-Stern Hull Series”,
DTNSRDC/SPD-0840-01, 1981. FAST 2001, Southampton, 2001.

Lyakhovitsky, A., “Shallow Water and Supercritical Savitsky, D. “Hydrodynamic Design of Planing Hulls”,
Ships”, Backbone Publishing Company, Hoboken, NJ, Marine Technology, Vol. 1, 1964.
2007.
Sturtzel, W., Graff, W.,"Investigation of Optimal Form
Lewthwaite, J., “Wash Measurements on Inland Design for Round-Bottom Boats", Forschungsbericht des
Waterways using the WAVETECTOR Buoy”, RINA Landes Nordrhein-Westfalen, Nr. 1137, 1963. (in German).
Conference on Coastal Ships & Inland Waterways 2,
London, 2006. Svensson, R., "Waterjets Versus Propeller Propulsion in
Passenger Ferries", Vocational Training Centre,
Millward, A., Bevan, G., "The Behavior of High Speed Hongkong, 1998.
Ship Forms when Operating in Water Restricted by a Solid
Boundary", RINA W2 paper issued for written discussion, Toro, A., “Shallow-Water Performance of a Planing Boat”,
1985. Trans. SNAME, 1969.

Millward, A., Sproston, J., "The Prediction of Resistance of

Fast Displacement Hulls in Shallow Water", RINA
Maritime Technology Monograph No. 9, 1988.

Nwogu, O., “Boussinesq Modeling of Ship-Generated

Waves in Shallow Water”, PIANC USA Annual Meeting,
Portland, 2003.

PIANC 2003 – “Guidelines for Managing Wake Wash

from High Speed Vessels”, Report of WG 41 of
International Navigation Association, Brussels, Belgium.

Phillips, S., Hook, D., “Wash from Ships as they approach

the Coast”, RINA Conference on Coastal Ships & Inland
Waterway II, London 2006.

Radojcic, D., "Mathematical Model of Segmental Section

Propeller Series for Open-Water and Cavitating Conditions
Applicable in CAD", SNAME Propellers '87 Symposium,
Virginia Beach, 1988.

Radojcic, D., "An Engineering Approach to Predicting the

Hydrodynamic Performance of Planing Craft Using
Computer Techniques", Trans. RINA, Vol. 133, 1991.

Radojcic, D., Rodic, T., Kostic, N., "Resistance and Trim

Predictions for the NPL High Speed Round Bilge
Displacement Hull Series", RINA Conference on Power,
Performance and Operability of Small Craft, Southampton,
1997.
APPENDIX 1 - Wave wake energy evaluation (numerical example)
A generic wave trace measured at 50 m from the vessel’s track is depicted in Figure A1. It shows a maximum wave
height of 0.26 m and an associated period of 6.7 sec.

Figure A1 - Wave wash trace at 50 m off vessel track

At this distance, the wave energy can be calculated as follows:

E = 1960H2T2 = 1960 x 0.262 x 6.72 = 5947 J/m  5950 J/m .

For comparison we can look at the wave energy present at a distance of 200 m off of the vessel’s track. Using the deep
water relationship for decay rate developed by the University of Southampton (Lewthwaite 2006), the wave height at a
distance of 200 m can be calculated as follows:

H200 = HX / (200/x)0.35 = 0.26 / (200/50) 0.35 = 0.16 m .

Consequently, the wave energy at 200 m may be evaluated as

E = 1960H2T2 = 1960 x 0.162 x 6.72 = 2254 J/m  2250 J/m .

Therefore it can be concluded that the energy is below the 2825 J/m restriction of the State of Washington. However,
if the same wave would be traveling in the shallow water the decay rate would be different, and definitely smaller than in
deep water (see Figure 6), so wave energy at a distance of 200 m would most probably be higher than the allowable limit.

APPENDIX 2 - Influence of length Froude number and depth Froude number on wave pattern and height

To investigate the combined influence of length Froude number and depth Froude number on wave height, the Michlet
Software - Version 8.07 ( Leo Lazauskas) was employed to calculate surface wave patterns for various length Froude
numbers and depth Froude numbers.

An 86 x 11.5 m NPL hull form, generated with Delft Ship, was used in the calculations. The full scale dimensions
were selected to correspond to the Yacht Ectasea. The results are presented in Figure A2. The images in the center vertical
column are for a constant length Froude number (FnL 0.43), while depth Froude number increases from 0.65 to 1.5
(corresponding to Figures 1 through 3 of main text). On the other hand, the images in the horizontal row have a constant
depth Froude number (Fnh=0.90), while length Froude number increase from 0.26 to 0.61.

The progression from the top to bottom of the vertical figures illustrates the wave pattern changes associated with
transitioning from the subcritical regime to the supercritical regime. Relative to this, the horizontal figures, all evaluated
for the same depth Froude number, depict somewhat different wave patterns and heights with the different length Froude
numbers. Diverging bow wave angle (Kelvin angle) however, is the same for all horizontal figures. The middle figure has
the maximum wave height as Fnh=0.9 and FnL0.4.

Wave pattern of the same hull form as discussed above but for much higher Fnh and FnL (hence the supercritical
regime regardless of water depth) is depicted in Figure A3.
 Fnh=0.65 FnL=0.44 h/L=0.465
(V=12.88 m/s h=40 m)

Fnh=0.90 FnL=0.26 h/L=0.081 Fnh=0.90 FnL=0.43 h/L=0.233 Fnh=0.90 FnL=0.61 h/L=0.465

(V=7.46 m/s h=7 m) (V=12.61 m/s h=20 m) (V=17.83 m/s h=40 m)

Fnh=1.5 FnL=0.43 h/L=0.081

(V=12.43 m/s h=7 m)

Figure A2 - Wave patterns and heights at different length and depth Froude numbers

Fnh=2.5 FnL=0.71 h/L=0.081 (V=20.72 m/s h=7 m)

Figure A3 - Typical wave pattern of a ship at supercritical speeds

APPENDIX 3 - Numerical Example

The purpose of this appendix is to provide a numerical example demonstrating the application of the resistance and
powering prediction process recommended in this paper. A powering prediction is provided for water jets. The calculations
are performed for a vessel similar to the yacht Ectasea, but with less installed power. Input parameters for predicting the
resistance of the round bilge hull form is as follows:

 = 1880 t h = 15 m (shallowest expected on route)

LWL = 77 m
LCB = 36.3 m PB = 2 x 14 MW

Deep water resistance RTd is calculated from the original NPL systematic series model data using the Froude
extrapolation method. The ITTC 1957 model-ship friction line was used with CA = 0.0000 (see Table Figure A1). The
appendage resistance was calculated according to RAPP = (RF + RRd) * 0.10.

RRh/RRd (SWRF) is determined from the Equation 1 in the main text. The equation is used to calculate the shallow
water resistance in subcritical range only for speeds up to Fnh = 0.90 for the assumed shallowest expected water depth on
route.
FnL V Fnh RRd RF RAPP RTd RRh/RRd RRh RTh
- kts - kN kN kN kN - kN kN
0.30 16.0 0.67 131 59 19 210 1.28 168 246
0.35 18.7 0.78 202 80 28 310 1.70 344 452
0.40 21.4 0.89 342 102 44 488 2.89 987 1133
0.45 24.0 1.01 583 127 71 780
0.50 26.7 1.12 817 154 97 1069
0.55 29.4 1.23 938 185 112 1235
0.60 32.1 1.34 990 217 121 1328
0.65 34.7 1.45 1039 252 129 1421

Table A1 - Deep water and subcritical shallow water resistance (shallowest water depth h=15 m, L/h=5.1)

To develop a conservative resistance ‘envelope’, the resistance curve based on shallowest water depth and critical
peaks for other, deeper water depths (h>15 m, corresponding to L/h<5) should be determined. Critical peaks are calculated
from Equation 3 in the main text (see Table A2). The data from Tables A1 and A2 are then combined to yield the
resistance envelope curve as shown in Figure A4.

Maximal Peak s

L/h (Fnh)max h V (RRh/RRd)max RRd RF RAPP RTh

- - m kts - kN kN kN kN
5 0.8950 15.4 21.4 2.844 346 102 44 1129
4 0.8841 19.3 23.6 1.932 545 123 66 1242
3 0.8633 25.7 26.6 1.378 815 154 96 1373
2 0.8136 38.5 30.7 1.094 969 201 117 1379

Critical Peak s

L/h (Fnh)critical h V (RRh/RRd)critical RRd RF RAPP RTh

- - m kts - kN kN kN kN
5 0.9500 15.4 22.70 2.594 457 114 57 1357
4 0.9500 19.3 25.38 1.732 714 141 85 1462
3 0.9500 25.7 29.31 1.228 937 184 112 1448
2 0.9500 38.5 35.89 0.994 1062 269 133 1459

Table A2 - Shallow water resistance maximal and critical peaks for incremental L/h values
 1600
CRITICAL PEAKS

1400

MAXIMAL PEAKS
1200

HULL RESISTANCE (kN)

1000
SHALLOW WATER
RESISTANCE

800
RESISTANCE @ L/h = 4

600
SUB CRITICAL DATA RESISTANCE @ L/h = 3

400 V (kts) RT (kN)

16.0 246
DEEP WATER RESISTANCE 18.7 452
21.4 1133
200 22.7 1357
25.4 1462
29.3 1448
35.9 1459
0
5 10 15 20 25 30 35 40 45
SPEED (kts)

Figure A4 – Resistance envelope curve

Deep water and shallow water resistance curves are plotted on the water jet performance map as shown in Figure A5.
A twin water jet application was selected, the maximum predicted speed in each case is identified by the intersection of the
maximum power contour (14 MW) and the relevant resistance curve. The results indicate the speed loss when operating in
shallow water with a depth of 15 m is predicted to be about 7 knots relative to deep water performance! The OPC’s
associated with deep and shallow water operation are 0.54 and 0.42 respectively; representing a 22% reduction in
efficiency! Also note the cavitation margin – the resistance curve lies in the Zone 2 cavitation region and its proximity to
the Zone 3 curve suggests the water jets are on the verge of thrust breakdown at about 23 knots.
1600
Zone 3 Zone 2
Zone 1
2 * 22000 BKW
1400 P = 2 X 14mw
NET THRUST & HULL RESISTANCE (kN)

V = 21 KTS
RTh = 1100 KN 2 * 20000 BKW
3% mechanical losses included

d = 0.42
1200 2 * 18000 BKW

2 * 16000 BKW

1000 P = 2 X 14mw
2 * 14000 BKW
V = 27 KTS
RTd = 1100 KN
d = 0.54 2 * 12000 BKW
800
2 * 10000 BKW
SHADED AREA REPRESENTS
600 INCREASE IN RESISTANCE
2 * 8000 BKW
SHALLOW WATER RESISTANCE

2 * 6000 BKW
400

2 * 4000 BKW
DEEP WATER RESISTANCE
200
2 * 2000 BKW

0
5 10 15 20 25 30 35 40 45
SPEED (kts)

Figure A5 – Plot of deep and shallow water resistance on a typical Rolls-Royce water jet performance map

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