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Applied Ocean Research 35 (2012) 14–24

Contents lists available at SciVerse ScienceDirect

Applied Ocean Research

journal homepage: www.elsevier.com/locate/apor

Semi-empirical methods for determining the efflux velocity from

a ship’s propeller

W. Lam a,b,∗, G.A. Hamill b,1, D.J. Robinsonb,1, S. Raghunathan b,1

a Department of Civil Engineering, Faculty of Engineering, University ofMalaya, 50603 Kuala Lumpur, Malaysiab Queen’s University Belfast, Northern Ireland BT95AG, United Kingdom

a r t i c l e i n f o

Article history:

Received 22 October 2010Received in revised form 3 January 2012

Accepted 4 January 2012

Available online 2 February 2012

Keywords:

Ship’s propeller jet

Efflux velocity

Seabed scour

a b s t r a c t

The present study proposed the semi-empirical methods for determining the efflux velocity from a ship’s

propeller. Ryan [1] defined the efflux velocity as the maximum velocity taken from a time-averaged

velocity distribution along the initial propeller plane. The Laser Doppler Anemometry (LDA) and Com-

putational Fluid Dynamics (CFD) were used to acquire the efflux velocity from the two propellers with

different geometrical characteristics. The LDA and CFD results were compared in order to investigate

the equation derived from the axial momentum theory. The study confirmed the validation of the axial

momentum theory and its linear relationship between the efflux velocity and the multiplication of the

rotational speed, propeller diameter and the square root of thrust coefficient. The linear relationship of

these two terms is connected byan efflux coefficient and the value of this efflux coefficient reduced when

the blade number increased.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The investigations into predicting the velocity within the ship’s

propeller jet which can lead to seabed scouring are of particular

interest for the design of marine structures. In Whitehouse’s [2]

book “Scour at Marine Structures”, the potential damage made by

the propeller jet was highlighted. The action of the propeller jet

to the seabed scouring was also described in Sumer and Fredsøe

[3] book “The Mechanics of Scour in the Marine Environment” and

Gaythwaite [4] book “Design of Marine Facilities for the Berthing,

Mooring, and Repair of Vessels”. The jet impingement of a ro-ro

ship to the seabed is illustrated in Fig. 1.

The parameters to investigate the seabed scouring are pre-

sented in Fig. 2. The velocity field of a ship’s propeller jet was

preliminary investigated in order to determine the propeller

induced seabed scouring. The influences of the rudder [1,5], hull

and the berth geometry to the velocity within the jet would

normally be included after the unconfined propeller jet beingestablished. The impingement velocities were therefore used to

determine the erosion extent and erosion rate within the seabed

[6–9]. The influence ofthe bedmaterial, which resistedjet impinge-

ment, was considered in order to propose an effective remedial

∗ Corresponding author at: Department of Civil Engineering, Faculty of Engineer-

ing, University of Malaya, 50603Kuala Lumpur, Malaysia.

Tel.: +6 03 7967 7675/44028 9097 4006; fax: +6 03 7967 5318/44028 9097 4278.

E-mail addresses:[email protected] , [email protected] (W. Lam).1 Tel.: +44 0289097 4006; fax: +44 0289097 4006.

action [10]. The problem of theship’s induced scour was also inves-

tigated byusinga simplifiedround jet, which hasbeen documented

by Yeh et al. [11] and Yülsel et al. [12]. Lam et al. [13] attempted

to establish the velocity prediction at the efflux plane from a ship’s

propeller. Isbash [14] and PIANC [15] provided procedures for esti-

mating the contributions of numerous factors that determined the

seabed damage.

Previous researchers used the maximum velocity at the initial

stage as an input to investigate the ship’s propeller jet induced

scour. The velocity distribution within the entire propeller jet, the

bed velocity with or without rudder and the influence of the jet to

the sediment deposition were later predicted using these available

equations [16–22]. The accurate prediction of the efflux velocity

became important since it was an initial input in the entire pre-

dicting system.

The objectives of this paper are to investigate: (1) the validity of

the axial momentum theory used to predict the efflux velocity; (2)

linear relationship between the efflux velocity and the multiplica-tion of rotational speed, propeller diameter; (3) influences of the

propeller geometry to the efflux velocity prediction; (4) proposal

of efflux coefficients from different propellers. This paper presents

the objectives of the current research initially and then followed

by presenting the methodology used. The validity of the LDA and

CFD results is discussed in Section 3. Section 4 is the main part of

this paper with five sub-sections. The first subsection describes the

axial momentum theory and the second subsection describes the

limitation of theory. The third subsection describes the previous

works of the axial momentum equation by considering the geo-

metrical characteristics. The fourth and fifth subsections discuss

0141-1187/$ – see front matter © 2012 Elsevier Ltd. All rights reserved.

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W. Lam et al. / Applied Ocean Research 35 (2012) 14–24 15

Fig. 1. Jet impingement in the case of ro-ro ship with stern ramp [3].

the LDA and CFD results in terms of the efflux velocity and efflux

coefficient.

2. Methodology

A joint experimental and numerical approach was carried out

in order to acquire the time-averaged velocity within a rotating

ship’s propeller jet. Experimental measurements provided directdata within thediffusing jet andthe numericalmodelling gave data

at high rotational speeds that was hard to achieve in experiments.

The numerical modelling was able to provide additional blades to

the virtual model in order to investigate the blade influences to the

velocity.

2.1. Experimental measurements

The experiments are carried out in order to acquire the mean

velocitywithin a diffusingjet. A water tank which waslargeenough

toallowthe unhindered expansionand diffusionof thepropeller jet

wasusedin a seriesof experiments.The purpose-built test tank was

7.5m×4.4 m in plan by 1.0 m deep, as shown in Fig. 3. The water

tank was equipped with a drainage system from which the watercould be drained out for cleaning after the experiment ended. An

overflow spillway was also designed in order to prevent flooding.

The tankwasfilledto a heightof 880mm in1 h via a 50mmdiame-

terwaterpipe.The propellershaftwas located at almostmid-depth

in the tank. The effect of the tank bottom and water boundaries

was not found to influence the free expansion of the unconfined jet

under investigation.

The present study was undertaken using two propellers at ‘bol-

lard pull’ condition (zero advance speed), as shown in Table 1 and

Fig.4. Two propellers are termed as propeller-76 and propeller-131

Fig. 2. Parameters of the seabed scouring [10].

Fig. 3. Schematic of experimental tank layout (plan view).

Table 1

Propeller characteristics.

Propeller-76 Propeller-131

Propeller diameter,D p 76 mm 131 mm

Hub diameter,Dh 15.2 mm 35.0 mm

Blade number,N 3 6

Rake angle, 0◦ 0◦

Blade arearatio, ˇ 0.473 0.922

Thrust coefficient, C t 0.4 0.56

Mean pitch ratio (P ) 1 1.14

according to their diameters in millimetre. The propeller was fitted

to a stainless steel shaft on a stationary rig to allow the rotation

at the zero advance speeds. A central nut was used to lock the

propeller to the shaft. The propeller rig was 0.47m×0.57 m in

plan by 1.57m high and was fabricated using the galvanised steel

of 50mm×50mm in section by 5mm in thickness. The propeller

rig was located along the centreline of the water tank about 1.5 m

from the rear. The circulation effect of water within the tank has

been shown to be insignificant to the jet expansion [20,21].

A direct current electric motor was employed to rotate thepropellers at constant speeds. The motor was made by GEC Elec-

tromotors Limited. It was rated at 0.55kW and the design speed is

800 rpm. The design speed is the speed for maximum efficiency of

a motor. This motor used in this experiment has performed well by

gearing the output to a desire speed. In this experiment, the motor

was fitted at the upper part of the rig at a safe distance of 0.5 m

Fig. 4. Geometryof a six-bladed propellerand a three-bladed propeller: (a)aft view

and (b)side view.

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16 W.Lam et al. / Applied Ocean Research 35 (2012) 14–24

Fig. 5. Front view andside view of themeasurement grid.

above water surface. The rotational speed was adjusted using the

speed meter which was attached at the wall. Once the motor was

switched on, the motor transferred the torque force to rotate the

propeller.

Laser Doppler Anemometry (LDA) measurements were under-

taken to acquire data at the initial plane of a ship’s propeller jet

for efflux velocity study and CFD validation. LDA is a non-intrusive

optical technique used to measure time-averaged axial, tangential

and radial components of velocity along with turbulence levels.

In this experiment laser beams from the probe were focused onto

the measurement volume where the beams intersected. The mea-

surement volume was a few millimetres long, and the beam’s

intersection region was termed as the fringe. The Doppler fre-

quency was calculatedby the analysis of lightscattered fromseeded

particles. Flow velocity was calculated from the Doppler frequency

and the fringe distance [23].

The seeding particles were aluminium passivated wet powder

(E3064AR) from Silberline Ltd. The aluminium powder was silver

coloured with a 36m particle size and density of 1.46g/cm3. The

high number of recorded particle showed the particlesfollowed the

fluid well. The particle was able to scatter light as a burst to LDA

system. The average recorded burst count due to seeding was 6898

particles with a maximum 14,939 particles and a minimum 3982

particles for 40s. The measurement ateachpoint was taken for 40s

to obtain an accurate time-averaged velocity.

2.1.1. Measurement grid

In the current tests, one vertical and one horizontal line were

considered in order to measure the three components of velocity.

The vertical line was able to measure the axial and radial compo-

nents of velocity. The horizontal line was able to measure the axial

and tangential components of velocity. A 5 mm measurement step

was chosen to measure the velocity field, which were 0.13R p for

propeller-76 and 0.08R p for propeller-131. The measurement gridat the initial plane is illustrated in Fig. 5. The current measurement

step was two-time finer compared to the 10mm measurement

step chosen by Hamill [20]. The use of same spatial sampling of

the velocity field was recommended in the future study, which

may have a better choice for comparison. The ideal location of the

measurement was the plane immediate downstream after the pro-

peller. However, a 10mm gap between the propeller face and the

measurement point existed as a space for the intersection of laser

beams forming a measurement volume.

2.2. Numerical simulations

Fluent CFD code was used to predict the velocity field from a

ship’s propeller rotating at various speeds [24,25,32]. Seil etal. [33],

WS-Atkins-Consultants et al. [34] and Dargahi [8] carried out CFD

investigations on the flow field from a ship’s propeller.

Seiletal. [33] fromRolls-Royleinvestigatedtheflowfieldaround

the propeller in order to increase the efficiency of a ship’s pro-

peller. Seil et al. [33] investigated the problem using Fluent CFD

code by constructing a structured mesh with hexahedral cells. Seil

et al. [33] believed that the structured mesh would be able to con-

trol the aspect ratio of the downstream adjacent cells effectively.

Despite the unstructured mesh being able to save time on the grid

generation and cope with problems associated with the complex

geometry better, the structured mesh was still considered the bet-

ter method to predict the flow field around the propeller. Seil et al.

[33] implemented the RNG k–ε turbulence model derived from theBoussinesq’s approaches in this simulation due to its swirl modifi-

cation.

WS-Atkins-Consultants et al. [34] provided guidelines to sim-

ulate the hydrodynamic performance of a ship’s propeller. This

example demonstrates themanner in which the predictions for the

flow around a marine propeller can be achieved. The commercial

code CFX-TASCflow was used in this simulation. A structured mesh

with hexahedral grid was used in the model due to the restric-

tion of the CFD package. The standard k–ε turbulence model was

the choice in this case and the simulation was performed with the

second order discretisation scheme.

In Dargahi’s [8] research, the three-dimensional computational

model was generated by using a structured grid. The structured

grid comprised hexahedral elements. Dargahi’s [8] work has been

solved using both the standard k–ε and the RNG k–ε turbulencemodels. The standard k–ε turbulence model is a robust turbulencemodel, whereas RNGk–ε turbulence model was the enhanced k–ε

model to predict the swirling flow.

In this investigation, the CFD simulation has been used in order

to predict the velocity in a wider range of rotational speeds. The

current numerical simulation is to obtain the efflux velocity at the

rotational speedsin which themeasurementis difficult to be taken.

The further documentation of the grid generation and mathemati-

cal model will be presented in the following sections.

2.2.1. Grid generation

The helicoidal surfaces of the propeller blades resulted in a

geometry difficult to create. The creation of the propeller geometry

was based on the geometrical data from the technical drawing of

the propellers. By substituting the geometrical datainto Eqs.(1)–(3)

[26], the points lying at the leading edge and trailing edges can be

obtained.

˚ = tan−1 P

2R p(1)

sin ˚ = z

L (2)

cos ˚ =r

L (3)

where ˚ is the pitch angle, P is the pitch distance, R p is the radius

of propeller, L is the distance of either the trailing edge or leading

edge tothe propellercentre dependingon theedgepointof interest.

The r , ,and z are the radial coordinate, angular coordinateand axialcoordinate from the central of propeller, respectively.

An unstructured grid was generated by using Gambit modeller

[36], obeying the Delaunay’s law, once the propeller domains were

created. The gridindependence wastestedto ensurethatthe virtual

model used an optimum grid number. The computational geome-

tries for propeller-76 and propeller-131 are shown in Fig. 6(a) and

(b). In addition, the computational geometry of propeller-76 has

been modified to produce a four-bladed propellerand a five-bladed

propeller by adding one and two blades, as shown in Fig. 6(c) and

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Fig. 6. Computational geometryfor CFD simulation: (a) three-bladed propeller; (b)

six-bladed propeller; (c) four-bladed propeller; (d) five-bladed propeller.

(d). Simulation tests were undertaken to determine the sensitivity

of the unstructured grid.Three unstructured gridswith 77,360 cells,

99,243 cells and 242,121 cells were tested. The grid with 242,121

cells is sufficient to capture the mean velocity within the jet.

2.2.2. Mathematical models

A rotating reference frame approach [32] was used to induce a

rotating propeller jet. The rotating reference frame is less compu-

tational consumption compared to the moving mesh method [32].

The standard k–ω turbulence model [37] was used in this sim-ulation. The standard k–ω model is designed to incorporate the

modification of the low-Reynolds number effects and the spread-

ing of the shear flow in predicting free shear flow. Fluent tutorialused a standardk–ω model to predict the propeller jet from a navalarchitect’s view [35].

The simulations were initially solved by using a first order

scheme in order to produce a preliminary result, and the sec-

ond order scheme was performed later to obtain a more accurate

result. In addition, an upwind scheme was used to predict the flow,

in which direction played a significant role. The SIMPLE scheme

was used to discretise the pressure–velocity coupling [32]. A seg-

regated solution algorithm was used to the Reynolds-Averaging

Navier–Stokes (RANS) equations [27] sequentially by updating the

property values until the convergence criterion has been satisfieds

[32].

A small under relaxation factor [32] was used in this simula-

tion, which non-dimensionlised value of 0.01 for both the pressure(default 0.3) and momentum (default 0.7) discretisations. After

the solution reached convergence using a first order discretisation

scheme, the iterative process was continued with a second order

process. The convergence of a second order scheme was achieved

at 34,000iterations. Forthe propeller jet simulation, a 2.2GHz Pen-

tium 4 personal computer with 512 Mb RAM needs 32h to reach a

converged solution.

3. Validation

The experimental measurements were validated by compar-

ing the velocity profile at various rotational speeds. The velocity

profiles of the axial component acquired from the horizontal

and vertical lines were compared to ensure the repeatability and

-2.0

-1.5

-1.0

-0.5

0.0

0.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Lateral section, r/R

a

b

p

V e l o c i t y ,

V o

( m / s )

76mm x-plane

76mm z-plane

-2.0

-1.5

-1.0

-0.5

0.0

0.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Lateral section, r/R p

V e l o c i t y ,

V o

( m / s )

76mm x-plane

76mm z-plane

Fig. 7. Comparison of the measurementsfrom horizontal line ( x-plane)and vertical

line ( y-plane) from propeller 76mm at efflux plane (10 mm from propeller) with

differentrotational speeds: (a)750rpm and (b)1250rpm.

validity of the experimental results. Hamill [20], Stewart [21],

Hashmi [22] and McGarvey [38] suggested that the velocity

profile of the efflux plane was a two-peaked-ridge profile. The

experimental measurements showed the velocity distribution has

a two-peaked-ridge profile for all tested speeds in a range of

750–1500rpm for propeller-76 and 350–1000rpm for propeller-

131. All the tested results obeyed the widely accepted velocity

profile proposed by the previous researchers.

From the experimentalmeasurements, the regionof high veloc-ity gradient showed the average variation between the horizontal

andverticallines at 750 rpmwas only 3%,whereas the rotation axis

showed a maximum variation of 8%, as shown in Fig. 7. The region

of high velocity gradient was the area in between r /R p =−0.66 and

r /R p = 0.66. In a propeller jet, the main flow was contributed by the

rotation of blades and the secondary flow was a more complicated

flow with hub and tip vortices. A higher variation at the rotation

axis was expected as the hub vortex occurred at the rotation axis.

The hub vortex was more random in nature. However, the hub

vortex was not part of this study and was therefore treated with

limited concern. The velocity close to the jet boundary was low

and therefore the maximum variation reached 200%.

The horizontal and vertical measurements are not much differ-

ent as shown in Fig. 7. In a perfect experimentalsetup, thevariation

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-2.0

-1.5

-1.0

-0.5

0.0

0.5

2.01.51.00.50.0-0.5-1.0-1.5-2.0

Lateral section, r/R p

V e l o c i t y

, V o

( m / s )

76mm x-plane

76mm z-plane

CFD

Fig.8. Comparison of measurementsfrom horizontal line( x-plane)and verticalline

( y-plane) of propeller-76 at 1000rpm.

between the horizontal and vertical lines should be equal to zero.

The current variation is mainly due to the minor fluctuation of the

rotationalspeedfromthe power train which wasobservedfromthe

speed meter rather than the performance of the LDAmeasurement

system.

The validation of CFD results was undertaken by comparing the

LDAmeasurementswith theCFD predictions,as shown inFig.8. The

CFD results showed agreement with the LDA measurements with

variation of 13%in the high gradient region in between r /R p =−0.66

and r /R p = 0.66. Themaximum variationof 65%occurred at therota-

tion axis. The experimental measurements showed a peak due to

hubeffect,but it didnot happenin theCFD simulation. Thehub vor-tex was a low-pressure core started from the hub and the strength

of the vortex reduced gradually when moving downstream. Hub

vortex caused the thrust deduction by pulling the propeller boss

and subsequently reduced propeller performance to about 5% [28].

In this research, a low velocity core within the propeller jet was

expected when a coarse grid was designed to capture the time-

averaged velocity with a low computational consumption.

The validation showed that the experimental measurements

acquired from a LDA system have a high repeatability. There is no

much difference for the data acquired using vertical or horizon-

tal lines. The CFD was able to predict the velocity of the propeller

jet except the hub vortex at the rotation axis. The CFD did not

capture the complicated hub vortex that was seen in the experi-

mental measurements. The deficiency of the flow calculated fromCFD was limited by a low grid resolution designed to capture the

time-averaged velocity with a low computational consumption.

The current grid resolution was only able to capture the main flow,

but was not capable to capture the hub vortex. All the LDA and CFD

comparison showed a large variation at the rotation axis due to the

deficiency of CFD on predicting this highly turbulent region. LDA

showed a peak, but CFD showed a dip at the rotation axis. How-

ever, the efflux velocity described in this paper was the one with a

distance from the rotation axis, which is 0.67(R p−Rh) according to

Berger et al. [18]. The variation of efflux velocity between LDA and

CFD was in a range of 2–21% for different propellers. The difficulty

of CFD to capture the hub vortex gave insignificant influences to

the predictions of time-averaged velocity according to the previous

CFD experiences.

4. Results and discussion

After the validation, the maximum velocity at the initial plane is

acquired to investigate the theoretical equation. Efflux coefficient

is the dimensionless value obtained by dividing the efflux veloc-

ity with the rotational speed (rpm), propeller diameter (m) and

square root of thrust coefficient according to the axial momentum

theory. This linear equation was widely accepted by researchers

and the efforts were undertaken to testify the linear relation-

ship of the equation. Some researchers [20–22] suggested different

efflux coefficients based on the experimental investigation using

limited propeller models. The efflux coefficient of 1.59 is widely

accepted due to the strong theoretical support from axial momen-

tum theory. However, the coefficient of 1.59 was derived from a

one-dimensional assumption. The coefficient of 1.59 may be suit-

able for the plain water jet, but not for a ship’s propeller jet. The

two-peaked-ridge profile was found at the initial plane of a ship’s

propeller jet.

4.1. Axial momentum theory

The axial momentum theory is a widely accepted theory used

to predict the efflux velocity within a ship’s propeller jet. Blaauw

and van de Kaa [17], Berger et al. [18], Verhey [19] and Hamill [20]

used the axial momentum theory to describe the characteristics of

a ship’s propeller jet. The axial momentum theory was proposed

by Froude with reference to Rankine’s investigations in the 19th

century. The axial momentum theory made six assumptions:

(1) The propeller is represented by an ideal actuator disc of equiv-

alent diameter.

(2) The disc consists of an infinite number of rotating blades, rotat-

ing at an infinite speed.

(3) There is negligible thickness of the disc in the axial direction.

(4) The disc is submerged in an ideal fluid (inviscid fluid) without

disturbances.

(5) All elements of fluid passing through the disc undergo an equal

increase of pressure.(6) The energy supplied to the disc is, in turn, supplied to the fluid

without any rotational effects being induced.

The change of the momentum due to the energy supplied to

the system through the presence of the actuator disc results in a

net thrust on the fluid. This thrust can be related to the Bernoulli’s

equation in order to develop an equation for the efflux velocity and

is balanced with the dimensional analysis of the thrust [29]. This

equation can be presented as,

1

2AP (V

2down − V

2up) = C t n

2D4 p (4)

where is the density of fluid, A p is the area of actuator disc, V downand V

up are velocity at far downstream and far upstream respec-

tively, C t is the thrust coefficient of the propeller, n is the speed of

rotation of the propeller in revolutions per second and D p is the

propeller diameter in metres. The thrust coefficient (C t ) is a non-

dimensional coefficient determined from measured performance

characteristics.

As the advance speed (inflow velocity to the propeller) V A = 0

and the area A p =D p2/4, the efflux velocity (V o), which is the max-imum velocity at the face of the propeller [16], becomes the widely

accepted theoretical equation used to predict the efflux velocity

V o = 1.59nD p

C t (5)

This equation is the fundamental equation from the axial

momentumtheory usedto predict the effluxvelocitywithin a ship’s

propeller jet.

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4.2. Limitations of the axial momentum theory

The axial momentum theoryonly considers the velocity charac-

teristicsof theaxialcomponent withina submergedjet. A propeller

jet has two other components of velocity (tangential and radial

components) in addition to the axial component of velocity. The

assumption of simulating a rotating propeller jetusing a plane sub-

merged jet is therefore not entirely satisfactory, due to the absence

of the tangential and radial components of velocity in a plane

jet [30]. Therefore assumption six in the axial momentum theory

“energy supplied . . . to the fluid without any rotational effects . . .”is invalid.

Hamillet al. [31] f ound the assumptions of theaxial momentum

theory are not applicable to true propeller jets. Thus, the equa-

tions derived from axial momentum theory are inherently flawed.

Assumption two in axial momentum theory “disc consists of an

infinite number of rotating blades, rotating at an infinite speed” is

clearly void since the true propeller cannot consist of an infinite

number of rotating blades and rotate at an infinite speed. The pro-

pelleris normallycarefully designedwiththreeto sixblades andthe

speed of rotation is carefully chosen to provide the maximum effi-

ciency when in service, and tends to be in hundreds of revolutions

per minute.

Assumption three that the disc (and hence propeller) has negli-

gible thickness in the axial direction is not practical as for efficient

operation, the blades of the propeller must have pitch (travel dis-

tance of a point in the longitudinal direction of the jet after one

rotation) in this plane. The resulting conclusion that the veloc-

ity on either side of the disc ( propeller) is approximately the

same is incorrect, with sizable differences by Hamill et al. [31].

As the blades on a ship’s propeller change in both pitch and area

within a three dimensional space, assumption five “equal increase

of pressure” is also invalid as there are significant differences in

pressure changes across the blade from the root at hub to the

tip.

These shortfalls have led to modification of the theoretical

equations in an attempt to take account of the geometrical char-

acteristics. Hamill et al. [31] f ound the assumptions in the axialmomentum theory to be inadequate to describe the process

involved in the formation of the propeller jet. Fuehrer et al. [7]

indicated that Eq. (5) wasfoundto bein error as much as ±20%.

4.3. Semi-empirical equations for efflux velocity

Thebehaviour of an actualship’s propellerjet contradictedmost

of the assumptions made in the derivation of the axial momentum

theory. However several researchers, such as Fuehrer and Römisch

[16], Berger et al. [18], Verhey [19] and Hamill [20] have developed

equations to predict the efflux velocity based on the axial momen-

tum theory. The maximum velocity taken from a time-averaged

velocity distribution along the initial propeller plane was termedthe efflux velocity denoted as V o [1].

Hamill [20] refined the theoretical equation of axial momentum

theory for the efflux velocity through an experimental investiga-

tion of a rotating ship’s propeller jet instead of a plain water jet.

He proposed a lower coefficient value, producing a semi-empirical

equation based on the detailed measurements on two propellers

V o = 1.33nD p

C t (6)

Stewart [21] performed similar experiments on two other pro-

pellers and proposed an equation for the efflux velocity where the

coefficient was based on geometrical characteristics of the pro-

pellers. Stewart [21] reported the coefficient used in the existing

equation to predict efflux velocity was not a constant but was

Table 2

Efflux velocity from propeller-76 at horizontal line.

Rotational speed (rpm)

750 1000 1250 1500

V o (1) 1.04 1.34 1.69 2.03

V o ( 2) 1.00 1.39 1.72 2.09

dependent on the propeller characteristics. The following equation

was subsequently suggested:

V o = ςnD p

C t (7)

where the efflux coefficient, is equal to:

ς = D p−0.0686P 1

.519ˇ−0.323 (8)

where ˇ is the blade area ratio (projected area of all blades relatedto the propeller disc area) and P is the pitch ratio of the propeller

(quotient of a pitch and the propeller diameter). The efflux coef-

ficient in Eq. (8) used a dimensional term (D p) in an otherwisenon-dimensional relationship. Hashmi [22] refined this equation

by non-dimensioning the propeller diameter (D p), by dividing by

the hub diameter (Dh

)

V o = E onD p

C t (9)

where the efflux coefficient E o, is equal to:

E o =

D pDh

−0.403C t

−1.79ˇ0.744 (10)

4.4. LDA measurements

The time-averaged velocities were measured to acquire the

efflux velocity at various rotational speeds for propeller-76 and

propeller-131. The changes of the efflux velocity with various

rotational speeds were later plotted in order to obtain the efflux

coefficient to compensate the aforementioned weakness of the

axial momentum theory.

4.4.1. Efflux velocity from LDA measurement

When the propeller is rotating, the propeller draws the

upstream water into the slipstream and discharges this water

downstream as a high velocity jet. The velocity within the jet

changed when the rotational speeds changed. The velocity distri-

bution immediate downstream of the propeller, which was termed

as efflux plane, varied along the propeller blade. The blades con-

tributed to the motion of the jet and therefore a low velocity core

should be expected close to the hub. The secondary flow and the

hub vortex were treated as an additional disturbance to the jet in

this study.

The efflux plane has two velocity peaks separated by the rota-

tion axis. Two peaks were measured due to the axis symmetricalcharacteristics of the jet as suggested by Hamill [20]. The efflux

velocities of propeller-76were acquired at four different speeds by

using both the horizontal and vertical lines. Two efflux velocities

across the horizontal section of propeller-76 are termed as V o (1)

on the left hand side and V o ( 2) on the right hand side (Table 2).

Two efflux velocities across the vertical section of propeller-76 are

termed as V o (3) on the top and V o ( 4) at the bottom (Table 3).

For the propeller-131, the same notation system was applied to V o(5) for the efflux velocity on the left hand side, V o (6) for the right

hand side, V o (7) on the top and V o (8) at the bottom, as shown in

Tables 4 and 5 respectively.

The hydrodynamics of the underwater propeller jet is com-

plicated associated with the swirling features. The hydrodynamic

behaviour at the efflux plane is therefore an area of interest for

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Table 3

Efflux velocity from propeller-76 at vertical line.


750 1000 1250 1500

V o (3) 1.03 1.33 1.70 2.05

V o (4) 1.07 1.35 1.72 2.07

Table 4

Efflux velocity from propeller-131 at horizontal line.


350 500 750 1000

V o (5) 0.91 1.25 1.94 2.53

V o (6) 0.91 1.29 1.99 2.59

Table 5

Efflux velocity from propeller-131 at vertical line.


350 500 750 1000

V o (7) 0.93 1.32 2.05 2.67V o (8) 0.92 1.32 1.98 2.77

hydrodynamist. The efflux velocity increased from the high rota-

tional speeds of 750rpm up to 1500rpm. The outcome surprisingly

showed that the rotational speed was the only variable influenc-

ing the magnitude of the efflux velocity. The efflux velocity can be

predictedby multiplying thetermsof D p

C t and efflux coefficient.

Thisrelationshipwas further validatedby usingthe results acquired

from the horizontal and vertical lines for propeller-76. The linear

relationship between the efflux velocity and the rotational speed

was also validated by using a larger propeller-131 at the lower

rotational speeds, as shown in Tables 2–5. The study found that

the propeller diameter and thrust coefficient are two fixed num-bers from the propeller design of naval architect which cannot be

changed. The efflux coefficient is the critical constant other than

the rotational speed.

4.4.2. Efflux coefficient from LDAmeasurements

The efflux velocities were acquired at various rotational speeds

to investigate the efflux coefficient proposals by the previous

researchers. The graph of efflux velocity V o versus nD p

C t wasplotted to investigate the coefficient of 1.59 derived from the axial

momentum theory. The LDA measurements of set V o (1)–V o (8)

are plotted as shown in Table 6 and Fig. 9. The linear lines of all

the sets were plotted to obtain the coefficients from different pro-

pellers.

The efflux coefficients of propeller-76 were in a range of 1.6901–1.7244. The study found that the efflux coefficient is

Table 6

Efflux coefficient from LDA measurements.

Set of

measurement

Propeller Efflux

coefficient

R2

V o (1) Propeller-76 1.6901 0.9985

V o (2) Propeller-76 1.7244 0.9973

V o (3) Propeller-76 1.6959 0.9982

V o (4) Propeller-76 1.7203 0.9963

V o (5) Propeller-131 1.5589 0.9987

V o (6) Propeller-131 1.5961 0.9990

V o (7) Propeller-131 1.6426 0.9988

V o (8) Propeller-131 1.6565 0.9957

sensitive to the rotational speeds and their locations of data acqui-

sition. The efflux coefficients were the values of 1.6901 on the left

acquisition, 1.7244 on the rightacquisition,1.6959on the topacqui-

sitionand 1.7203 at thebottom acquisition even thesamepropeller

used in themeasurements.The sensitivityof therotationalspeedto

the efflux coefficient was inherited when applying to a larger pro-

peller. Theeffluxcoefficientsof thepropeller-131werein a range of

1.5589–1.6565. The average efflux coefficients for the propeller-76

and propeller-131 were the values of 1.7077 and 1.6135 respec-

tively. The high efflux coefficient of the propeller-76 may be due

to the higher rotational speed rather than the impacts from the

propeller geometry. A more flexible study by using the CFD mod-

elling was therefore implemented for a wider range of rotational

speeds.

4.5. CFD predictions

As describedin Section 2, the CFDmodels were created to obtain

the efflux velocities and subsequently the efflux coefficients from

a wider range of rotational speeds for further investigation.

4.5.1. Efflux velocity fromCFD results

The CFD models are able to provide a cheaper solution with

less time consuming. The CFD investigated the efflux velocity in

a wider range of rotational speeds (250rpm and 1750rpm with a

step of 250rpm), as shown in Table 7. The efflux velocities from

three additional rotational speeds were investigated in order to

provide a wider range of data. The comparison of CFD and LDA

at 750 rpm, 1000rpm, 1250rpm and 1500rpm was 0%, 3%, 9% and

19% respectively. The CFD predictions showed high accuracy for

the predictions of the low rotational speeds and their accuracy

decreased with the increase of the rotational speeds. The jet at

the high rotational speed is more complicated and therefore the

CFD may not be able to capture the efflux velocity as accurate

as those from the low rotational speed by using a personal com-

puter.

The efflux velocities of propeller-76 were predicted at variousrotational speeds by using the axial momentum theory, as shown

in Table 7. The CFD predictions have a closer agreement with the

axial momentum theory at a low rotational speed rather than a

high rotational speed as LDA. A more complicated flow occurred

at a high rotational speed and this flow is more difficult to predict

using CFD. For propeller-131, seven simulations were carried out

at various rotational speeds, as shown in Table 8.

The CFD prediction provided more data to investigate the efflux

velocity at the lower and higher ranges of rotational speeds. From

the CFD predictions, the efflux velocity remained sensitive to the

rotational speeds regardless the change of rotational speeds. The

efflux coefficient is further investigated in the next section.

4.5.2. Efflux coefficient from CFD results

The efflux coefficients obtained from various data sets are

denoted as V o (i)–V o (viii), as shown in Table 9. For propeller-76,

both the LDA measurements (points from horizontal and vertical

lines) and CFD predictions formed the linear relationship between

V o andnD p

C t witha coefficientof 1.7077 and1.4598respectively,

asshown in Fig. 10(a). The correlation coefficients(R2) for the efflux

velocity at various rotational speeds betweenV o and nD p

C t werehigh for both LDA measurement and CFD prediction, as shown in

Table 9. High correlation meant thehigh accuracy of thelinear rela-

tionship. For propeller-131, both the LDA measurements and CFD

predictions showed also the linear relationship between V o and

nD p

C t , asshownin Fig. 10(b). The CFDcoefficient waslower than

the LDA coefficient since the predicted CFD values were low.

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y = 1.6901x

R2 = 0.9985

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5

n Dp

a b

c d

e f

V o

( m / s )

t C

y = 1.7244x

R2 = 0.9973

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5

n Dp

V o

( m / s )

t C

y = 1.6959x

R2 = 0.9982

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5

n Dp

V

o ( m / s )

t C

y = 1.7203x

R2 = 0.9963

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5

n Dp

V

o ( m / s )

t C

y = 1.5589x

R2 = 0.9987

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

1.5

2.0

n Dp

V o ( m

/ s )

t C

y = 1.5961x

R2 = 0.999

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

1.5

2.0

n Dp

V o ( m

/ s )

t C

y = 1.6426x

R2 = 0.9988

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

n Dp

V o

( m / s )

t C

y = 1.6565x

R2 = 0.9957

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

n Dp

V o

( m / s )

t C

Fig. 9. Relationship of V o and nD p

C t from various measurement setsV o (1)–V o (8) correspondingto (a)–(h).

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Table 7

Efflux velocity from axial momentum theoryand CFDfor propeller-76.


250 500 750 1000 1250 1500 1750

Axial momentum theory (m/s) 0.32 0.64 0.96 1.27 1.59 1.91 2.23

CFD (m/s) 0.31 0.65 1.04 1.30 1.53 1.64 1.93

Variation 3% 2% 9% 2% 4% 14% 13%

Table 8

Efflux velocity from axial momentum theory and CFD for propeller-131.


250 350 500 750 1000 1250 1500

Axial momentum theory (m/s) 0.65 0.91 1.30 1.95 2.60 3.25 3.90

CFD (m/s) 0.58 0.79 1.03 1.72 2.31 2.89 3.46

Variation 11% 13% 21% 12% 11% 11% 11%

Table 9

Efflux coefficient from axial momentum theory, LDA measurements and CFD predictions.

Set Source of data Propeller Efflux coefficient R2

V o (i) LDA Propeller-76 1.7077 0.9963

V o (ii) CFD Propeller-76 1.4598 0.9596

V o (iii) LDA Propeller-131 1.6135 0.9936

V o (iv) CFD Propeller-131 1.4051 0.9980

V o (v) Axial momentum theory Actuator disc 1.5900 1.0000

V o (vi) LDA Both propeller-76 and propeller-131 1.6512 0.9868

V o (vii) CFD Both propeller-76 and propeller-131 1.4092 0.9790

V o (viii) CFD Modified four-bladed propeller-76 1.4812 0.9820

V o (ix) CFD Modified five-bladed propeller-76 0.8948 0.9287

The efflux coefficient from the axial momentum theory with-

out consideration of the geometrical characteristics was compared

to the LDA and CFD coefficients. The efflux coefficient was

plotted by using all the LDA efflux velocities at various rota-tional speeds regardless the propellers used. The efflux coefficient

of CFD results was also plotted by using all the CFD predic-

tions at various rotational speeds. The efflux coefficient obtained

from the LDA measurements regardless the propeller geome-

try was 1.6512, whereas the efflux coefficient of CFD results

regardless the propeller geometry was 1.4092, as shown in

Fig. 10(c) and Table 9. The efflux coefficient of 1.59 from the

axial momentum theory fell in between the LDA coefficient and

the CFD coefficient. The axial momentum theory has assumed

an actuator disc with infinite blade number. The results showed

that the efflux coefficient reduced when the blade number

increased.

4.5.3. Influences of the propeller geometry to efflux velocityThe geometrical characteristics of propeller-76 and propeller-

131 were different in terms of propeller diameter, blade number,

blade area ratio, thrust coefficient and mean pitch ratio. Influ-

ences of the propeller geometry to the velocity field within a

ship’s propeller jet remained an interesting unknown in this area.

LDA measurements showed the efflux coefficient reduced when

the blade number increased. The efflux coefficients of actuator

disc, six-bladed propeller and three-bladed propeller increased

from 1.59 to 1.6135 and then to 1.7077, as shown in Table 9.

The blade number may give significant influence to the values of

efflux coefficient. The CFD was therefore implemented to inves-

tigate the influences of blade number to the efflux coefficient

by adding one and two additional blades on the three-bladed

propeller-76.

Stewart [21] and Hashmi [22] conducted studies to investigate

the influence of the propeller geometry to the efflux coefficients.

The three-bladed propeller has a LDA efflux coefficient of 1.71,

whereas the six-bladed propeller and the actuator disc of axialmomentum theory have efflux coefficients of 1.61 and1.59 respec-

tively. The six-bladed propeller was closer to the actuator disc

condition compared to the three-bladed propeller. The efflux

coefficients for the four-bladed and five-bladed propellers were

predicted as 1.4812 and 0.8948 by using the CFD model, as shown

in Fig. 11.

The CFD results of the four-bladed and five-bladed virtual pro-

pellers gave important references of the blade number influences

to the efflux coefficient. The efflux coefficient of the four-bladed

propeller was 1% higher than that of the three-bladed propeller,

whereas the efflux coefficient of the five-bladed propeller was 39%

lower than that of the three-bladed propeller. The geometrical

Table 10

Proposed equations based on propeller with different geometrical characteristics.

Propeller Source of

data

Proposed equation

Propeller-76 LDA V o = 1.71nD p

C t

Propeller-76 CFD V o = 1.46nD p

C t

Propeller-131 LDA V o = 1.61nD p

C t

Propeller-131 CFD V o = 1.41nD p

C t

Both propeller LDA V o = 1.65nD p

C t

Both propeller CFD V o = 1.41nD p

C t

Modifie df ou r- bladed pro pelle r- 76 CFD V o = 1.48nD p

C t

Modified five-bladed propeller-76 CFD V o = 0.89nD p C t

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y = 1.7077x

R2 = 0.9963

y = 1.4598x

R2 = 0.9596

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0 1.0 2.0 3.0

n Dp

V o

( m / s )

LDA

CFD

t C

y = 1.6135xR

2 = 0.9936

y = 1.4051x

R2 = 0.998

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0 1.0 2.0 3.0

n Dp

V o ( m / s )

LDA

CFD

t C

y = 1.4092x

R2 = 0.979

y= 1.6512x

R2 = 0.9868

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0 1.0 2.0 3.0

n Dp

V o ( m / s )

Axial Momentum Theory

LDA

CFD

t C

a

b

c

Fig. 10. Efflux coefficient of axial momentum theory, LDA measurement and CFD

prediction fromdatapointsof (a)propeller-76;(b) propeller-131; (c)bothpropeller-

76 and propeller-131.

characteristics of propeller-76 were designed with an optimum

blade number of three. The off-design condition of the four-bladed

and five-bladed propellers may influence the performance of the

propeller and subsequently the efflux coefficient. Blade area ratio

( AE / A0) becomes larger with a large number of blade areas. The

blade area becomes linearly larger with the number of blades.

y = 0.8948x

R2 = 0.9287

y = 1.4812x

R2 = 0.982

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 1.0 2.0 3.0

n Dp

V o ( m / s )

four-bladed

five-bladed

t C

Fig. 11. Efflux coefficient from CFD prediction by using two virtual propellers.

5. Conclusions

The applications of LDA and CFD to improve the efflux velocity

prediction have been demonstrated. The semi-empirical equations

are proposed in Table 10 and the findings are as follows:

(1) LDAand CFDresults confirmed thevalidityof theaxialmomen-

tumtheory to predict the efflux velocity from a ship’s propeller

by relating the efflux velocity,V o with nD p

C t linearly.(2) LDA and CFD results confirmed the linear line intercepting at

zero position with a strong correlation coefficient.

(3) LDA and CFD results showed the efflux velocity increased with

the rotational speeds.

( 4) The LDA measurement showed that the efflux coefficient

reduced when the blade number increased.

Acknowledgements

The current research was supported by SPUR studentship from

Queen’s UniversityBelfastand NationalNaturalScience Foundation

of China (grant no. 51006019).

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