International Journal of Physics, 2013, Vol. 1, No. 2, 22-27
Available online at http://pubs.sciepub.com/ijp/1/2/1
© Science and Education Publishing
DOI:10.12691/ijp-1-2-1
Performance Assessment of the Waterjet Propulsion
System through a Combined Analytical and Numerical
Approach
Parviz Ghadimi*, Roya Shademani, Mahdi Yousefi Fard
Department of Marine Technology, Amirkabir University of Technology, Tehran, Iran
*Corresponding author: pghadimi@aut.ac.ir
Received December 13, 2012; Revised April 21, 2013; Accepted April 24, 2013
Abstract The application of waterjets is rapidly growing and they are increasingly being chosen for propulsion in
high-speed crafts. Waterjet as a propulsion system of a vessel is also favorable when it comes to maneuvering,
appendage drag, draft and fuel consumption at high speeds. Furthermore, waterjet system has recently gained more
credibility for its acceptable efficiency because of the advent of more efficient and large pumps. This type of
propulsion system consists of many components working together harmoniously, thus establishing a complex system.
A significant problem facing designers when predicting performance of the waterjet is the interaction between the
hull and the waterjet. This paper describes the powering performance of a vessel equipped with a waterjet system.
The interaction between the hull and waterjet is studied in order to predict the powering characteristics. The work
starts with an introduction of the waterjet and a review of its current status in design and analysis. Subsequently,
hydrodynamic properties of its components are computed and interactions among them are analyzed. Finally,
numerical computation is performed for acquiring pressure distribution by a two dimensional computer code in the
suction area of the waterjet inlet to predict the possible occurrence of cavitation for the inlet duct.
Keywords: waterjet, inlet flow, numerical analysis, cavitation, propulsion system
1. Introduction
In 16‟Th century, Toogood and Hays for the first time
proposed waterjet propulsion system, as reported by J.S.
Carlton [1]. At that period of time, waterjet propulsions
were used in high-speed pleasure craft and work boats.
However, in recent years, this system has been considered
for large high-speed crafts. Accordingly, many huge
waterjet units have been used in wide range of ships such
as passenger and naval crafts.
The waterjet propulsion is a complex system. On the
contrary, the screw propellers are simpler, lighter and
more efficient than waterjet system. However, the arrival
of more efficient pumps, the necessity for timely
delivering the critical commercial cargoes, and the
required maneuverability for particular vessels have made
the usage of waterjets more attractive.
It is normal to divide this type of propulsion system into
a hull and a waterjet. It has been demonstrated that
waterjet-hull interaction can affect the overall efficiency
more than 20% [2]. Usually, waterjet system is broken
down into subsystems and an explicit modular approach is
applied to analyze them. In order to assess the interaction
between the hull and the jet, a parametric method is used.
In 1980, an early contribution related to the jet-hull
interaction is attributed to Etter et al. [3]. A complete
review of the existing relations for waterjet-hull
performance is presented by Allison [4]. Van Terwisga [2]
proposed a parametric propulsion prediction method for
the waterjet driven craft. Several numerical methods were
devoted to study the flow behavior in every part of
waterjet system by many researchers whose works were
submitted to the ITT Conference [5].
A complete design method that utilized the numerical
scheme for the analysis of system parts has been used by
CCDOTT1 [6]. N.W.H. Bulten [7] used CFD methods to
design and analyze the whole jet system. Also, a flushtype waterjet propulsion unit was designed with different
inner diameter impellers by Moon et al. [8]. In a related
work, analysis of a waterjet axial pump was performed
numerically [9]. They investigated the performance of the
axial-flow-type waterjet based on the variation of the
impeller tip clearance.
The main distinction between the procedure
implemented here and the previous methods is that the
present method applies the empirical, analytical and
numerical methods simultaneously to reach a conceptual
design of the waterjet system. By numerical analysis, inlet
design parameters that describe model geometry have
been specified. Moreover, a simple 2-dimensional inlet
model has been used here instead of a complex 3-D
geometry. By this method, we have an acceptable
propulsion system in a short time. Accordingly, a
complete jet system design code has been developed that
combines the empirical, analytical and numerical methods.
1
Center for the Commercial Deployment of Transportation
Technologies
International Journal of Physics
By considering the ship geometry limitations and
hydrodynamic properties, this software proposes suitable
jet conceptual design parameters. Consequently, by using
simple 2-D domain for inlet duct and computed pressure
distributions for different inlet angles, an optimum inlet
has been proposed which leads to minimum losses and
maximum efficiency.
23
jet
IVR
Waterjet propulsion system has three main components:
Inlet, Pump and Nozzle. Figure 1 shows typical waterjet
arrangement while Figure 2 demonstrates the idealized
profile of the jet system. Sea water enters the system with
the velocity Vin and leaves it with a different velocity V jet .
The mass flow rate of the water through the waterjet is
given by
A jetV jet
m
density of water, respectively. The thrust produced by the
system is equal to the rate of change of momentum:
T A jetV jet V jet Vin
(2)
Hence, the effective propulsion power delivered by the
system is given by
S V jet Vin
PE TVS mV
(3)
Vin
VS
(6)
The inlet has an optimal inlet velocity ratio at which jet
has minimum losses at that point. This optimum point has
been found by studying a range of velocity ratio and
computing related system efficiency. Head losses in the
duct inlet and outlet are functions of squared velocity [1]:
(1)
A jet and are the area of nozzle outlet and
(5)
This indicates that the jet efficiency is equal to the ratio
of useful power to the total power absorbed by the system.
Because of the effect of hull and geometry of the inlet
duct, the inlet velocity is different from ship velocity.
Hence, the inlet velocity ratio is defined as follows:
2. Basic Theory of System
where
PE
PE Losses
hInlet Vin2 2 g
(7)
2
hOutlet V jet
2g
(8)
where and are empirical coefficients. These
parameters can be assumed as fixed values depending on
the inlet and outlet geometries [1].
Power input to the system (received by the pump) is
given by
1 2
Ppump m V jet
Vin2 g h j hloss (9)
2
where h j is the vertical distance between the outlet nozzle
and the water line. By separating system losses and
applying loss coefficients, we obtain
Ppump
By using equation (10) and assuming the velocity ratio
as VS V jet , equation (5) can be written as
Figure 1. Typical waterjet general arrangement [1]
j
Figure 2. Idealized waterjet arrangement [1]
In Figure 2, the energy equation between points 1 and 2
can be written as
P1 V12
P V2
H P 2 2 H hloss
g 2g
g 2g
where
H P : Pump head
H : Vertical distance between inlet and outlet
hloss : Total head loss.
The jet efficiency is defined as
m 2
2
[V jet 1 1 1 Vs 2 2 gh j ] (10)
2
(4)
2 (1 (1 ) )
1 (1 )(1 )
2
2
(11)
2 ghi
Vs2
2
Jet efficiency can be calculated from equation (11).
Other design parameters have been found based on the
velocity ratios. Figure 3 shows that jet outlet diameter
increases at high velocity ratio, while the maximum value
of jet efficiency is observed to occur at high velocity ratio.
Thus, to achieve high efficiency, a large jet diameter
should be selected. However, the aft geometry of ship
makes limitations for jet diameter. Therefore, a specified
jet diameter is selected and other design parameters are
assessed based on this main restriction. On the other hand,
the optimum velocity ratio is found by setting the
derivative of equation (11) with respect to equal to
zero to obtain:
opt
where
is given by
1 1
1
(12)
24
International Journal of Physics
1 1
2
2 gh j
1 1
VS2
(13)
2
By using the pump and hull efficiencies, the total
efficiency is defined by the following equation:
OA Pump .Hull . jet
Hull
(14)
1 t
1
(15)
where Hull , and t are the hull efficiency, wake and
thrust deduction factors, respectively. Wake factor can be
estimated by the boundary layer theory which will be
explained in the next section.
1.0
to the inclusion of hull boundary layer flow. The amount
of the hull boundary layer flow which is included in the
inlet flow is important as it can impact the size,
performance, and propulsive efficiency of the waterjet
pump and must be taken into consideration during the
design phase of a waterjet pump.
Prediction of the inlet momentum velocity requires a
thorough understanding of the boundary layer velocity
profile which is instrumental for estimation of the
boundary layer thickness, .
Svenson et al. [10] presented data for hull boundary
layer thickness for high Reynolds number applications, as:
0.27 Lx Re
0.9
6
x
（16）
where
Rex
4.0
Efficiency
Jet Diameter
1
U Lx
3.5
（17）
0.8
3.0
2.5
0.6
0.5
2.0
0.4
1.5
Jet Diameter [m]
Efficiency
0.7
Accordingly, velocity at any point in the boundary layer
region is given by:
U y
U
0.3
1.0
0.2
n
(18)
0.5
0.1
0.0
where
0.0
0.3
0.4
0.5
0.6
Velocity Ratio
0.7
0.8
0.9
n log10 Rex
Figure 3. Efficiency and jet diameter as a function of velocity ratio
Figure 4. shows how the pump head is inversely
proportional to the mass flow rate of the jet system.
Therefore, for a specified velocity ratio, a pump should be
selected as such that its head and mass flow rate
correspond to those values obtained from the plots in
Figure 4 at that designated velocity ratio. On the other
hand, the engine and propulsion system spaces have
dimensional limitations, since mixed flow pumps is a
better option than other types like axial pumps.
300
500
Head
Mass Flow
450
250
350
300
150
250
200
100
150
Mass Flow [Ton/s]
400
200
Head [m]
1
100
50
50
0
0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Velocity Ratio
(19)
A standard method of expressing the impact of the inlet
momentum velocity is the inlet wake fraction or factor
given as:
1
Um
U
(20)
which can be conversely written as:
U m 1 U
(21)
where U , is the uniform speed at far stream (in this case
is considered to be the ship-speed VS ) and U m is the
velocity inside the boundary layer (but here, it is the flow
speed at the inlet due to the boundary layer). Figure 5
shows the computed values of wake parameter and the
boundary layer thickness as functions of ship velocity. It
is quite obvious that these two significant parameters
decline as ship velocity increases. However, the value of
wake parameter is about 0.9 and this parameter could be
considered as a constant quantity.
Figure 4. System head and flow rate as a function of velocity ratio
4. Validation of Design Parameters
3. Assessment of Wake Parameter
The waterjet inlet provides flow from beneath the ship
to the waterjet pump. The waterjet inlet is located near the
stern area of the ship, and as a result inflow to the flush
waterjet inlets will include a significant amount of hull
boundary layer flow. This means that for a flush waterjet
inlet, the flow entering the waterjet inlet has a momentum
velocity, U m , which is less than the ship speed, VS , due
As mentioned before, main results of the current code
in the form of mass flow rate, pump head and power have
been compared with the available results of design
software and model test [6]. A range of data made
available by the current code was compared with that
provided by the CCDOTT software [6]. Furthermore,
comparison was made between the results of the current
code with the CCDOTT experiment for the design point
shown in figures 6, 7, and 8. „WJ‟ sign in comparison
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figures shows the results of the present method. Figure 6
shows the comparison of the mass flow rate for the
experimental design point, the current method and fast
ferry computed results which was offered by CCDOTT [6].
Also, figures 7 and 8 present the same comparisons for
system head and power, respectively. These figures
demonstrate that the existing errors between the results of
the current code and the experimental data are generally
lower than 5%. Based on the presented validations, it can
be concluded that the off-design results are quite reliable.
The off-design results become particularly important when
the high-speed craft tends to reach a maximum speed over
the main hump of the speed-resistance curve, because the
power which is produced by the engine in off-design
condition is sometimes lower than the required value
25
separation could occur at two points at different inlet
velocity ratios [11]. Figures 10 and 11 show the positions
of the cavitation and separation phenomena at low and
high inlet velocity ratios, respectively. The probability of
cavitation occurrence was shown by the current work to
be a function of the inlet angle, ramp radius among other
design parameters. In the meantime, Inlet angle was found
to be the most influential parameter which affects the
cavitation occurrence and as such, the best inlet angle was
determined. Based on this observation, it was decided that
controlling of the cavitation occurrence be done through
variation of the inlet angle while keeping other parameters
constant.
70
CCDOTT
WJ
Design Point
65
0.096
1.10
1.00
δ [m]
0.95
Head Rise
60
1.05
0.094
0.092
55
50
δ [m]
Wake
0.90
0.090
0.85
ω
45
0.80
40
0.088
0.6
0.75
0.62
0.64
0.66
0.68
0.7
0.72
Velocity Ratio
0.70
0.086
Figure 7. Comparison of system head results
0.65
0.084
0.60
0
10
20
30
40
50
60
59000
Ship Velocity (Knot)
WJ
Design Point
58500
58000
Figure 5. Boundary layer thickness and wake factor as a function of ship
velocity
Power [HP]
95
CCDOTT
90
57500
57000
56500
56000
WJ
55500
Design Point
85
55000
Flow Rate
80
54500
75
54000
0.600
70
0.610
0.620
0.630
0.640
0.650
0.660
0.670
0.680
0.690
0.700
0.710
Velocity Ratio
65
Figure 8. Comparison of system power results
60
55
50
0.6
0.62
0.64
0.66
0.68
0.7
0.72
Velocity Ratio
Figure 6. Comparison of flow rate results
5. Analysis of the Inlet Design Parameters
The current code is capable to compute the inlet area
and pump diameter from momentum equation and
geometry limits. Based on these computed results,
empirical theory was used to predict the system losses and
basic geometrical parameters. It was found that elliptical
inlets have more advantages than the rectangular inlets.
Experiments show that elliptical inlet geometry with
aspect ratio of 1.3 is the best inlet form for flush type jet
[2]. Figure 9 shows the parameterized geometry of the
inlet profile. Obviously, a fair inlet profile has many
advantages, but sometimes transom geometry and aft form
of ship impose limitations. To survey the inlet profile
shape and to analyze the flow over this domain, some
geometrical parameters have been defined. Cavitation and
Figure 9. Inlet design parameters
Figure 10. Cavitation (point A) and separation (point B) at low Inlet
velocity ratio (IVR)
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International Journal of Physics
Figure 14 presents the cavitation number at the centerline
of the system for two different inlet angles. These figures
could furthermore indicate how and where the cavitation
could occur. Figure 14 shows how the system will operate
at a design speed range without cavitation at the inlet ramp
when the inlet angle becomes 26 degrees.
Figure 11. Separation (point A) and cavitation (point B) at high Inlet
velocity ratio (IVR)
6. Numerical Analysis of the Inlet Duct
p pv
1 U 2
2
(22)
When “ 0 ”, cavitation starts. This implies that high
cavitation numbers give less risk for cavitation. Figure 13
shows contours of cavitation numbers in XY plane while
2
Reynolds Averaged Navier Stokes
Figure 12. Comparison of minimum pressure for different inlet angles
Figure 13. Cavitation number around inlet duct (Inlet angle 26°)
1.4
1.2
1
0.8
σ
When designing a high-speed waterjet inlet, it is
difficult to achieve an optimum design that is both
efficient and has low drag. This is because cavitation
occurs in the water that is unfavorable to both drag and
efficiency.
Cavitation generally occurs when the local pressure on
a body moving in a fluid drops to or below the vapor
pressure of the fluid. When reaching the vapor pressure,
small “bubbles” or cavities are produced. These cavities
will collapse when they reach a higher-pressure region and
cause a small “water hammer” to form. This phenomenon
is called cavitation. Cavitation can produce the negative
effects of noise, vibration, and erosion or damage to the
inlet, and therefore, must be avoided in order to safeguard
the efficiency and the drag.
RANS2 code has been used to find pressure distribution
around the inlet duct. Two-dimensional geometry of
waterjet inlet duct has been modeled. Additionally, a
rectangular region is constructed around the hull to
represent the boundaries of the ocean. Since the ocean is
not actually bounded, those boundaries will have to either
be placed adequately far away from the ship hull and
defined as a wall, or placed closer to the hull to reduce the
size of the problem and defined as an opening.
Once the modeling was accomplished, an inlet duct was
analyzed for four different inlet angles ranging from 26 to
34 degrees, and minimum pressure at the given inlet
angles were compared against the numerical findings of
CCDOTT group [6]. This comparison in Figure 12 shows
a good agreement between the validation base and
numerical calculations. Selection of the inlet angle was
done based on two different design considerations;
suitable length of propulsion system and avoidance of
cavitation. Finding the optimum inlet angle requires a trial
and error progress. Accordingly, a large angle was
selected and possibility of cavitation was examined,
needless to say that assuming a proper safety factor is very
important in this progress.
Cavitation number is defined as:
0.6
0.4
Inlet Angle=28
0.2
0
Inlet Angle=34
-0.2
0
2
4
6
8
10
12
14
X[m]
Figure 14. Cavitation number on top inlet wall at two inlet angle (see
Figure 9)
3-D analysis of inlet duct which was done by CCDOTT
shows that the minimum pressure occurs at the centerline.
On the other hand, the results of the current study have
good conformity with those of the mentioned 3D analysis.
Thus, Figure 14 could be considered as a reliable source
for designing the inlet duct.
7. Conclusion
Simultaneous accomplishment of the conceptual and
basic design as well as the analysis of waterjet propulsion
system in a single process is a cumbersome task. Many
analytic and empirical methods have been proposed to
evaluate the influential parameters of this system.
Numerical methods are used for the analysis of the system
International Journal of Physics
performance, albeit not in a design process. In the current
work, a particular method has been presented which
covers all stages of the design and the analysis of waterjet
propulsion system which has also been validated by
reliable results. Two different codes have been developed,
one for the analytical assessment of the main waterjet
propulsion system parameters and the other of numerical
investigation of the inlet duct impacts.
In the current scheme, a practical method is used to
predict the powering characteristics of systems. Prediction
of the performance of the waterjet system at the design
point starts with determination of the required thrust, jet
diameter, shaft horsepower, and RPM which is done by a
developed computer program. During the detailed
hydrodynamic optimization study, the RPM, radial blade
loading distribution, and other parameters of the waterjet
were varied to arrive at the optimum design for this power
level.
Computed values of flow rate, head rise, and power
versus the velocity ratio were compared against earlier
reported experimental and numerical results. These
comparisons demonstrated good agreements indicating
that the separate handling of the waterjet and hull
designing process appears to be quite possible. The
adopted approach leads to a set of parametric relations that
describes the interaction between the hull and the waterjet
system. Because of this modular approach, the results can
simply be refined during the design process of the vessel.
Empirical relations have been used for preliminary
estimation of the required power for the vessel. From this
calculated thrust and the assessed internal losses in the
waterjet system, the required pump power can be found.
Possible cavitation at the inlet duct leads to erosion or
vibrations which must be avoided. In this paper, cavitation
characteristics at the inlet were found by looking at the
pressure distribution of the water around the inlet opening.
Accordingly, the possibility of cavitation was investigated
and controlled by close scrutiny of the cavitation number.
Along the same line, optimum inlet geometry was found
based on the observation of pressure distributions at
different inlet angles. This whole process was achieved by
a 2-D numerical code written for the flow investigation at
the inlet of the waterjet system.
Numerical findings of the current 2-D code compared
with the earlier results of the 3-D modeling show an
excellent match between the pressure distributions. As a
result, one may conclude that a lengthy process of 3-D
computation can indeed be avoided and that similar results
can easily be achieved by a 2-D analysis which is a less
laborious process and saves much computational time.
Main geometrical parameters for the design of waterjet
propulsion system have been determined by the present
method. By utilizing these important parameters, a
designer is able to predict the waterjet performance on a
ship before arriving at the basic design stage. Needless to
say that there are other considerations that must be taken
into account which include necessary number of jets,
position of the propulsion system, and the effect of the
system on compartmental arrangement of the ship.
Nomenclature
m :
ρ:
Ajet :
Vjet :
T:
Vin :
PE :
VS :
HP :
hloss :
ηjet :
IVR :
hInlet :
hOutlet :
ξ:
ψ:
PPump :
hj :
ω:
µ:
ηPump :
ηHull :
t:
δ:
LX :
Rex :
υ:
y:
U∞ :
Um :
σ:
p:
pv :
27
Mass flow rate
Density
Area of nozzle outlet
Jet velocity
System thrust
Inlet velocity
Effective power
Ship speed
Pump head
Total head losses
Jet efficiency
Inlet Velocity Ratio
Inlet head losses
Outlet head losses
Inlet losses coefficient
Outlet losses coefficient
Pump power
jet height from waterline
Wake factor
Velocity ratio
Pump efficiency
Hull efficiency
Trust reduction factor
Boundary layer thickness
Distance from bow
Reynolds number
Kinematical viscosity
Vertical distance from hull
Uniform flow velocity (ship speed)
Flow speed at the inlet due to boundary layer
Cavitation number
Fluid pressure
Fluid vapor pressure
References
Carlton.J.S. (John S.), “Marine Propeller and Propulsion”, Oxford,
Butterworth-Heinemann, 1994T.J.C. Van Terwisga, “A
Parametric Propulsion Prediction Method For Waterjet Driven
Craft”, Fast‟97 Conference, paper No. 151, 1997.
[2] T.J.C. Van Terwisga, “A Parametric Propulsion Prediction
Method For Waterjet Driven Craft”, Fast‟97 Conference, paper No.
151, 1997.
[3] Etter, R.J., Krishnamoorthy, V. and Sherer, J.O., “Model Testing
of waterjet Propelled Draft”, Proceedings of the 19 th ATTC, 1980.
[4] Allison, J.L., “Marine Waterjet Propulsion”, SNAME Annual
meeting, New York, 1993.
[5] ITTC, “The Specialist Committee on Waterjets”, 22th ITTC,
International Towing Tank Conference, 1998.
[6] Stanley Wheatley, “Development of a High-speed Sealift Waterjet
Propulsion System”, Final Report, Center for the Commercial
Deployment of Transportation Technologies California State
University, Long Beach Foundation, September 30, 2003.
[7] Bulten, N.W.H., “Numerical Analysis of Waterjet Propulsion
System”, PhD Thesis, Technical University of Eindhoven, 2006.
[8] Moon-Chan Kim, Ho-Hwan Chun, “Experimental Investigation
into the performance of the Axial-Flow-Type Waterjet according
to the Variation of Impeller Tip Clearance”, Ocean Engineering 34,
pp.275-283, 2007.
[9] Hong Gao, Wanlai Lim, Zhaohui Du, “Numerical Flow and
Performance Analysis of a Waterjet Axial Flow Pump”, Ocean
Engineering 35, pp.1604-1614, 2008.
[10] Svensson, R., etal., “Trial Results Including Wake Measurements
from the World‟s Largest Waterjet Installation”, RINA
International Conference on Waterjet Propulsion, Amsterdam, 2223 October 1998.
[11] Donald M. MacPherson, “A Universal Parametric Model for
Waterjet performance”, FAST '99, Aug 1999.
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