Energy Conversion and Management: X 17 (2023) 100349
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Energy Conversion and Management: X
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Numerical analysis of propellers for electric boats using computational fluid
dynamics modelling
Oliver Lovibond a, Anas F.A. Elbarghthi a, b, Vaclav Dvorak b, Chuang Wen a, *
a
b
Faculty of Environment, Science and Economy, University of Exeter, Exeter EX4 4QF, UK
Department of Applied Mechanics, Faculty of Mechanical Engineering, Technical University of Liberec, Studentská 1402/2, 46117 Liberec, Czech Republic
A R T I C L E I N F O
A B S T R A C T
Keywords:
Wageningen B-series propeller
Electric Boat
Flow behaviour
Multiple Reference Frame
Computational Fluid Dynamics
In the maritime industry, propellers are the most commonly used form of propulsion and are core to the optimum
performance of a ship. Generally, the performance characteristics of a marine propeller are determined and
analysed by experiments like open water and self-propulsion scale model tests which are costly and timeconsuming at the initial design stage. In this study, the computational fluid dynamics (CFD) simulations were
performed to evaluate propeller performance. Three Wageningen B-series propellers with varying Expanded Area
Ratios (EAR) were modelled with respect to the design constraints, such as ship speed and rotational velocity.
The performance of the hydrodynamic coefficients, thrust, torque and open water efficiency are then analysed
using the CFD modelling. These characteristics are then validated against experimental data obtained from the
Netherlands Ship Model Basin open water test in Wageningen and used to investigate the flow behaviour. The
analysis considers the Multiple Reference Frame (MRF) model. This study provided a well-founded framework
for applying CFD in the analysis and selection of Wageningen B-series propellers, as well as investigated the
relationship between the EAR, flow behaviour, thrust coefficient, and torque coefficient for electric boats. The
results show that a lower thrust and torque coefficient can improve the flow behaviour with increasing the ef­
ficiency by up to 62%. Furthermore, the outcomes reveal that the lower expanded area ratio of 0.6 is more
suitable for electric boats, creating a larger pressure difference of 1.079 MPa and generating extra potential
thrust at the same advance ratio, which leads to greater open water efficiency.
1. Introduction
The shipping industry is one of the largest modes of transport for
global trade. Indeed 90 % of traded goods are carried by ships. An IMO
Global Greenhouse Gas (GHG) study estimated that for the period
2007–2012, the shipping industry accounted for 3.1 % of annual global
CO2 emissions [1–4]. According to more recent studies by CE Delft, the
shipping industry, if left unchecked, could represent 10 % of GHG
emissions by 2050 [5]. It can be seen that in the effort to keep global
temperature increase below 2 ◦ C, action must be taken in order to reduce
global shipping emissions. One way to reduce a vessel’s GHG emissions
is to improve the fuel efficiency of the vessel. This can be achieved not
only by improving the efficiency of the engine or motor, but also by
optimizing the vessel’s propulsion system, namely the propeller.
A modern screw propeller with an equivalent arrangement to a pair
of tandem propellers on a single shaft was patented in 1838 by James
Lowe. The propeller was designed comprising one or more blades where
each blade was a portion of a curve which, if continued, would produce a
screw [6]. This design was then improved upon by Brunel with respect to
contrarotating designs and in 1845 was implemented in the design of
Great Britain to much success, which forms the basis of the fixed pitch
propellers seen today. The fixed pitch propeller (FPP) is the most
commonly used propeller today due to its mechanical simplicity, resil­
ience and effectiveness. It consists of a main hub, connected to a drive
shaft, from which a number of blades are attached at a certain pitch
angle. The FFP comes in two forms; monoblock propellers which are
propellers cast as a single piece and built up propeller, whose blades are
cast separately from the boss and then bolted or fixed in some way after
machining [6]. In addition, the Wageningen B series propeller is a
standard series fixed pitch propeller that was designed and tested at the
Netherlands Ship Model Basin in Wageningen. The open water charac­
teristics of 120 propeller models of the B-series were tested at the N.S.M.
B and analysed with multiple regression analysis [7]. The hydrodynamic
characteristics obtained from this experimental work provides a good
foundation from which the calculated results can be validated.
* Corresponding author.
E-mail address: c.wen@exeter.ac.uk (C. Wen).
https://doi.org/10.1016/j.ecmx.2023.100349
Received 9 October 2022; Received in revised form 5 January 2023; Accepted 6 January 2023
Available online 7 January 2023
2590-1745/© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
O. Lovibond et al.
Energy Conversion and Management: X 17 (2023) 100349
Nomenclature
c
D
KT
KQ
J
n
r
T
Q
u
VA
Abbreviations
CFD
Computational Fluid Dynamics
Exp
Experimental
EAR
Expanded Area Ratio
FFP
Fixed Pitch Propeller
GHG
Green House Gas
IMO
International Maritime Organization
MRF
Multiple Reference Frame
N.S.M.B Netherlands Ship Model Basin
No.
Number
SST
Shear Stress Transport
RANS
Reynolds Average Navier-Stokes
Section chord length (m)
Diameter of propeller (m)
Thrust Coefficient (-)
Torque Coefficient (-)
Advance Coefficient (-)
Rotational speed (rps)
Propeller radius (m)
Thrust (N)
Torque (Nm)
Flow velocity (m/s)
Advance velocity (m/s)
Density of fluid (kg/m3)
Expanded area (m2)
Propeller disc area (m2)
Static pressure (Pa)
Number of blades (-)
ρ
AE
A0
p
Z
Units and Symbols
ŋ
Open Water Efficiency (-)
There are several studies using computational fluid dynamics (CFD)
simulations to analyse propeller hydrodynamic characteristics [8–11].
Triet et al. investigated the Wageningen B-Series propeller in open water
conditions using the k-epsilon turbulence model [12]. This research,
with others such as Tran Ngoc Tu [13] and Subhas et al. [14], utilised
the Multi-Reference Frame (MRF) method to model the rotational mo­
tion of the propeller and received satisfactory results. The method
consists of separating the model into two regions. A rotating region
encapsulates the propeller and a static region covers the rest of the
simulation domain. The dimensions of the two domains are comparable
to Kutty and Rajendran [15], Subhas et al. [14] and Tran Ngoc Tu [13]
and provide a good foundation from which the current model is con­
structed. Tran Ngoc Tu compared different approaches to calculating
propeller open water characteristics using the Reynolds Average NavierStokes (RANS) equations method. These approaches are sliding grid,
rotating reference frame and rotating domain. The calculations were
performed on a hexahedral grid and two-equation of SST k-omega tur­
bulence model was used in the model. The result revealed that rotating
reference frame is a suitable method for open water simulation and is
now commonly deemed the Multi-Reference Frame (MRF). An under­
standing was gained that the rotating domain must be relatively tight to
the propeller, width about 0.38D in order to get accurate results for both
the propeller and the wake field. Similarly the static domain should be
sufficiently large enough to not interfere with the wake field and also
small enough to keep the mesh elements and computational time down.
As a result, Triet et al. study provided the knowledge that the k-epsilon
model is suitable for design purposes or coarse meshing and facilitates
fast simulation time. This is because the model uses wall functions to
calculate the near-wall region flows and theoretically requires a coarser
mesh at the boundary layer to decrease simulation time.
Huge consideration was given to other models. For instance, Guil­
mineau et al. simulated the wake field of a marine propeller using two
different RANS equation models; the k-ω SST of Menter and an aniso­
tropic two-equation Explicit Algebraic Reynolds Stress Model (EARSM)
[16]. Giancarlo investigated the RANS equations for turbulence
modeling [17]. The author concluded that turbulence modeling is an
attempt to devise a number of partial differential equations for
turbulent-flow calculation based on appropriate approximations of the
exact Navier–Stokes equations. This field is relatively challenging to be
investigated numerically, which supports visualizing the flow patterns
of the modeling.
The present study aims to verify the basic knowledge for the flow
simulations of a Wageningen B series propeller in electric boats using
computational fluid dynamics modelling. The CFD simulation with
detailed procedures was conducted to address and predict the impact of
the propeller characteristics and map its performance. The study in­
vestigates the relationship between the expanded area ratio of the pro­
peller, the thrust coefficient, the torque coefficient, and efficiency based
on the numerical results.
2. Theory and methodology
The research aims to model the flow behaviour of a propeller in open
water conditions. To achieve this, it is important to consider the con­
servation of mass and momentum in the analysis, as this will allow
thrust and torque to be predicted, which is defined by the governing
equations. Using this principle, the standard k − ε model will then be
applied to model the hydrodynamic characteristics of the propeller as it
represents the time-averaged turbulent nature of the flow around the
propeller. Once this has been achieved, the coefficient formula will be
applied to the hydrodynamic characteristics to produce the hydrody­
namic coefficients and allow comparison and validation against exper­
imental data. Finally, the data will be used to investigate the
relationship between flow behaviour and thrust and torque coefficients
to ascertain the most appropriate expanded area ratio of the propeller
design for electric boats.
The flow of a viscous incompressible fluid with constant properties is
governed by the Navier–Stokes equations [17]:
δui
δ
δp
δ2 ui
+v
+ (ui uj ) =
δxi
δt δxj
δxj δxj
(1)
δui
=0
δxj
(2)
where ui is the fluid velocity, p is the pressure divided by the density
ρ, v is the fluid kinematic viscosity, and body forces do not appear
explicitly, where the convective term of equation (1) is expressed in
conservative form. The standard k − ε model is a semi-empirical model
based on model transport equations for the turbulence kinetic energy (k)
and its dissipation rate (ε) [18]. The model transport equation for k is
derived from the exact equation. In contrast, the model transport
equation for ε was obtained using physical reasoning and bears little
resemblance to its mathematically exact counterpart. In the derivation
of the k − ε model, the assumption is that the flow is fully turbulent, and
the effects of molecular viscosity are negligible [19]. The stand­
ard k − ε model is therefore valid only for fully turbulent flows.
The open-water propeller characteristics conventionally are repre­
sented in the form of the thrust and torque coefficients KT and KQ in
terms of the advance coefficient J where:
2
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Energy Conversion and Management: X 17 (2023) 100349
Fig. 4. Surface mesh on propeller, first mesh case (Isometric View).
Fig. 1. 3D flow domain, including the stationary and rotating domain.
Table 1
Mesh options & positioning.
Mesh options
Position
Element size (mm)
Face sizing
Body sizing
Body sizing
Propeller blade
Rotating domain
Stationary domain
8
50
100
Table 2
Global meshing details.
Fig. 2. Flow domain dimensions based on the propeller diameter D.
Physics preference
CFD
Solver preference
Element order
Element size
Max adaptive size
Growth rate
Mesh defeaturing size
Curvature min size
Curvature normal angle
Fluent
Linear
Default (0.30714 m)
Default (0.61428 m)
Default (1.2)
Default (1.5357 mm)
Default (1.5357 mm)
Default (18.0◦ )
Table 3
Mesh quality.
Physics
preference
CFD
Physics
preference
CFD
Solver preference
Element order
Element size
Fluent
Linear
Default (0.30714
m)
Default (0.61428
m)
Solver preference
Element order
Element size
Fluent
Linear
Default (0.30714
m)
Default (0.61428
m)
Max adaptive
size
KT =
Fig. 3. Unstructured mesh with tetrahedron elements, first mesh case (Sec­
tion View).
KQ =
3
T
ρn2 D4
Q
ρn2 D5
Max adaptive
size
(3)
(4)
O. Lovibond et al.
Energy Conversion and Management: X 17 (2023) 100349
Table 4
Mesh Sensitivity Analysis.
J=
Mesh
no.
No. of
elements
No. of elements
x105
KT
Error from KT
(%)
1
2
3
254,100
550,646
1,343,546
2.541
5.50646
13.43546
0.3364614
0.3281122
0.3281318
− 4.024
− 6.671
− 6.664
VA
nD
(5)
where, T is the propeller thrust, Q is the propeller torque, ρ is the
fluid density, n is the number of propeller revolutions per second, D is
the propeller diameter and VA is the speed of advance. The open water
efficiency of the propeller is:
η=
J KT
2π KQ
(6)
And the percentage difference in Experimental (Exp) and CFD results
is calculated using the following equations:
ΔKT (%) =
KTCFD − KTExp
*100
KTExp
(7)
ΔKQ (%) =
KQCFD − KQExp
*100
KQExp
(8)
Δη(%) =
ηCFD − ηExp
*100
ηExp
(9)
The Expanded Area Ratio (EAR) is the most commonly used pro­
peller outline used by designers. It converts the face of a propeller from
its helix to a flat plane. The expanded area is given by the relationship:
∫R
AE = Z
cdr
(10)
Fig. 5. Mesh convergence graph.
rn
where Z is the number of blades. To calculate this area, it is sufficient
for most purposes to use a Simpson’s rule procedure with 11 ordinates.
The expanded blade area ratio is simply the expanded blade area AE ,
divided by the propeller disc area A0 to give the relationship AE /A0 . This
ratio is significant in propeller design and directly affects many propeller
characteristics. Lee et al. reported that cavitation might be decreased by
increasing the propeller blade area, but efficiency is consequently
decreased [20]. This relationship has been emphasized and illustrated
the open water efficiency of an MAU standard propeller with respect to
an expanded area.
Table 5
Cell zone conditions.
Rotating Domain
Stationary domain
Motion
Frame motion
Relative to cell zone
Rotation-axis origin
Rotation axis direction
Rotational velocity
Motion
Absolute
(X, Y, Z) = (0,0,0)
(X, Y, Z) = (0,0,1)
Diff. velocity corresponding with n
Stationary
Table 6
Boundary conditions.
Pressure outlet
Outer enclosure wall
Propeller blade
3. Numerical model descriptions
Reference frame
Absolute
Velocity magnitude
Coordinate system
Turbulent intensity
Backflow reference frame
Backflow direction
Turbulent intensity
Wall motion
Wall condition
Wall motion
Wall condition
6.22 m/s
(X, Y, Z) = (0,0,1)
5%
Absolute
Normal to boundary
5%
Stationary
No slip
Stationary
No slip
3.1. Flow domain
The numerical predictions presented in this study were performed
using ANSYS FLUENT 2021R1 commercial CFD solver. As mentioned
previously, the MRF model approach was used to predict the flow
around the propeller numerically. The domain is defined and illustrated
in Figs. 1 and 2. The domain is split into a global stationary domain and a
subdivided rotating region, called the rotating domain. The rotating
domain contains the entire propeller specified with dimensions of 1.15D
in diameter and 0.4D in length.
It should be noted that if the rotating domain is too small, predicted
results may be inaccurate due to the effect of large vortices near the
propeller. On the other hand, if the domain is too large however it will
significantly increase the simulation time. The static domain must be
large enough to prevent the full development of the upstream and
downstream flow from affecting the results of the analysis. However, if
the domain is too large, the computational time increases. For that
reason, the inlet is located 2.5D upstream of the origin of the propeller
and the outlet is located 5.8D downstream of the propeller to allow the
turbulence to collapse freely before hitting a boundary wall. This is all
encapsulated by a 3.3D square. Proper selection of the flow domain
upstream and downstream distance is very important to prevent recir­
culation of the flow that will cause convergence problems [15].
Table 7
Simulation flow conditions.
Case
number
Rotational speed n
(rps)
Velocity of advance
(U) m/s
Advance
coefficient (J)
1
2
3
4
5
30
25
20
17.5
15
6.22
6.22
6.22
6.22
6.22
0.346
0.415
0.518
0.592
0.691
4
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Energy Conversion and Management: X 17 (2023) 100349
Table 8
Propeller dimensions.
Propeller
Diameter (m)
No of blades
Expanded area ratio
P/D ratio
Rake
Material
Elements
1
2
3
0.6
0.6
0.6
4
4
4
0.65
0.6
0.7
1.10627
1.10627
1.10627
15
15
15
Ni-Al Bronze
Ni-Al Bronze
Ni-Al Bronze
550,646
556,324
563,826
3.3. Mesh sensitivity and model validation
Table 9
Solver Settings.
Pressure Link
SIMPLE
Pressure
Velocity formulation
Gradient
Momentum
Turbulent kinetic energy
Turbulent dissipation rate
Turbulence model
Near wall treatment
Models
Solver
Standard
Absolute
Least squares cell based
First order upwind
First order upwind
First order upwind
Standard k-epsilon
Standard wall functions
Single phase
Steady
The mesh sensitivity analysis was performed at J = 0.415 with the komega model on Propeller 1 and compared to experimental data ob­
tained by Barnitsas et al. (KT = 0.035) [7]. The study was conducted
using three grids whose element number increases by the square rule.
The summarized results of the mesh sensitivity are illustrated in Table 4.
The aim of the mesh convergence study is to determine the mesh
density at which the difference of propeller open water characteristic KT
obtained from two subsequent meshes reaches a sufficiently low value
[13]. The study was conducted by modelling each mesh at J = 0.415 and
obtaining the predicted value of thrust (T). The thrust coefficient KT is
then calculated using equation (3) which is then compared against the
experimental thrust coefficient obtained from the N.S.M.B. [7]. The
difference between the CFD value and the Experimental (Exp) value is
the error. The percentage error from KT is then calculated and plotted
against element number, as shown in Fig. 5.
It can be seen from Fig. 5 that as the element number increases, the
error in results also increases. This is unusual as generally, the trend
should be that of which, with increased element number, the obtained
results have increased accuracy. However, the higher element mesh
results are in strong agreement with each other with a discrepancy of
0.00598 % despite a difference of 7.9 x105 elements. As a result, the
decision was made to use the secondary mesh (element No. 550646) as a
basis for the model as it receives greatly the same results as the more
detailed mesh number 3 but with a significant reduction in computa­
tional time. The mesh sensitivity results are shown in Table 4.
There are multiple reasons for the error in the results. One reason is
that the simulation is performed under the single-phase condition. This
3.2. Meshing method
The mesh was generated using the mesh tool in the ANSYS program.
The grid is fully tetrahedral unstructured in both the stationary and
rotating domain, as shown in Fig. 3. The selection is based on justifi­
cations that unstructured tetrahedral grids have the capabilities to dis­
cretize complex geometries fast and with minimum user intervention.
The maximum cell size of the propeller and the rotating domain is 8 mm
and 50 mm respectively. The propeller surface mesh can be seen in
Fig. 4. The mesh was designed so that the cell sizes near the blade wall
were small and steadily increased towards the stationary region. The
global and local mesh details are displayed in Tables 1 and 2. The mesh
quality is shown in Table 3.
Table 10
Propeller 1 Hydrodynamic Characteristics.
J
0.346
0.415
0.518
0.592
0.691
KT
KT (Exp)
ΔKT (%)
KQ
KQ (exp)
ΔKQ (%)
η
η(Exp)
Δη(%)
0.3477
0.3281
0.2962
0.2682
0.2270
0.3700
0.3600
0.3300
0.2900
0.2300
− 6.0367
− 8.8577
− 10.2300
− 7.5132
− 1.2965
0.0582
0.0555
0.0507
0.0468
0.0411
0.0750
0.0720
0.0630
0.0580
0.0440
–22.3375
–22.8804
− 19.5104
− 19.2424
− 6.5191
0.3300
0.3900
0.4819
0.5399
0.6071
− 0.5256
0.3900
0.4900
0.5500
0.6200
− 3.4513
− 0.0047
− 1.6446
− 1.8421
− 2.0823
Table 11
Propeller 2 Hydrodynamic Characteristics.
J
0.346
0.415
0.518
0.592
0.691
KT
KT (Exp)
ΔKT (%)
KQ
KQ (exp)
ΔKQ (%)
η
η (Exp)
Δη (%)
0.3482
0.3294
0.2972
0.2710
0.2322
0.3700
0.3600
0.3300
0.2900
0.2400
− 5.8981
− 8.4891
− 9.9441
− 6.5410
− 3.2420
0.0578
0.0554
0.0511
0.0476
0.0423
0.0750
0.0710
0.0630
0.0560
0.0480
–22.8730
–22.0337
− 18.9460
− 15.0734
− 11.8825
0.3310
0.3928
0.4801
0.5373
0.6039
0.3400
0.3900
0.4800
0.5400
0.6200
− 2.6375
0.7081
0.0227
− 0.5018
− 2.5973
Table 12
Propeller 3 Hydrodynamic Characteristics.
J
0.346
0.415
0.518
0.592
0.691
KT
KT (Exp)
ΔKT (%)
KQ
KQ (exp)
ΔKQ (%)
η
η (Exp)
Δη (%)
0.3504
0.3301
0.2956
0.2680
0.2274
0.3700
0.3600
0.3300
0.2900
0.2300
− 5.2923
− 8.3075
− 10.4205
− 7.5739
− 1.1109
0.0596
0.0567
0.0517
0.0478
0.0419
0.0750
0.0710
0.0650
0.0570
0.0480
− 20.5109
− 20.1717
− 20.4331
− 16.2078
− 12.6487
0.3233
0.3844
0.4715
0.5291
0.5967
0.3400
0.3800
0.4700
0.5300
0.6200
− 4.9226
1.1479
0.3248
− 0.1703
− 3.7630
5
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Energy Conversion and Management: X 17 (2023) 100349
Fig. 6. Propeller 1 hydrodynamic characteristics.
Fig. 7. Propeller 2 hydrodynamic characteristics.
Fig. 8. Propeller 3 hydrodynamic characteristics.
means that cavitation is not simulated; however, it would affect the
results in the open water test performed at Wageningen. Another reason
for the error is the lack of a propeller hub and shaft in the model’s
geometry. Both of these solid objects will affect the flow over the pro­
peller and alter the physical hydrodynamic characteristics of the pro­
peller and were present at N.S.M.B. Another reason is that the fluid used
6
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Energy Conversion and Management: X 17 (2023) 100349
Fig. 9. Propeller 2 static pressure contour for J = 0.415: front face left side and back face right side.
Fig. 10. Propeller 3 static pressure contour for J = 0.415: front face left side and back face right side.
Fig. 11. Pressure contour at J = 0.415: Propeller 3 left side and Propeller 2 right side.
in the model is pure liquid H2O whereas the fluid used at Wageningen is
impure sea water with a different density which can affect the experi­
mental result. The final reason is that the mesh quality could be
improved given other meshing software such as ANSA, which could
produce more accurate results [21]. Despite there being an error in re­
sults, the error is sufficiently low enough to validate the model against
the experimental data obtained from the ship model basin in
Wageningen with only 6.671% error.
3.4. Solver and boundary conditions
The domain continuum was chosen as a fluid and the properties of
pure liquid water were assigned to it with a density of 998.2 kg/m3 and a
viscosity of 0.001003 kg/(m.s). The cell zone and boundary conditions
7
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Energy Conversion and Management: X 17 (2023) 100349
(a) Propeller 1 Velocity Contours at J=0.415
Subhas et al. recommendation on CFD analysis of a Propeller Flow and
Cavitation [14]. However, in future studies, the 2nd order upwind
setting will be used to get more accurate results at the cost of greater
computational time. The detailed setting is shown in Table 9.
In the simulation, the numerical algorithm used an implicit method
for the pressure -Linked equation known as SIMPLE as the default
scheme. The problem was chosen to be a steady state where the mo­
mentum equation was solved to attain the velocity field and the pressure
gradient term determined based on the initial condition or previous
iteration using the pressure distribution. The solution gradient for the
simulation is calculated based on the Least-Squares Cell-Based approach
because it delivers the same range of accuracy with less intensive
computation. Since the iterative domain contains a huge number of cells
that require faster convergence, the First-Order Upwind with first-order
accuracy. The Standard k-epsilon turbulent model was selected using the
transport equation for the kinetic energy and the dissipation rate and fits
the current simulation study because it was assumed for fully turbulent
flows. In the calculations, the mesh should predict the velocity gradient
within the boundary layer. This essentially requires capturing the thin
viscous layer by the first cell adapted at the wall. However, if the first
cell is placed at the log-layer region, then the standard wall function
could be chosen, which can predict various ranges of Reynolds-number
covering y + higher than 30 [24]. In general, both standard and real­
izable k-epsilon model variants used wall functions. The only matter is
that the value of y + near the wall must not be below 30 as has been
checked in this simulation.”.
(b) Propeller 2 Velocity Contours at J=0.415
4. Results and discussion
4.1. Hydrodynamic characteristics
The simulation was run for each propeller at 5 different advanced
coefficients in order to find the hydrodynamic characteristics, KT , KQ
and η. The results of these characteristics are then compared to experi­
mental data (Exp) and the error in the results is calculated. These results
are displayed in Tables 10, 11 and 12. The advance coefficient J is
changed by varying the revolution of the propeller while the advance
velocity was kept constant at 6.22 m/s. The hydrodynamic data for all
three propellers is being modelled.
It can be seen from Figs. 6 -8 that the overall accuracy of results
increases with an increasing advance ratio (J). The CFD efficiency
almost exactly replicates the Experimental data with the greatest dif­
ference being − 4.92 % with Propeller 3 at J = 0.346. For the torque
coefficient, accuracy increases and converges at the high end of J.
Whereas with the thrust coefficient, the greatest inaccuracy is found in
the middle region of J, most notably at J = 0.518.
It is noticeable in Fig. 8 that propeller 3 experienced the lowest ef­
ficiency. This is in correlation with Chang Sup Lee’s prediction that
increasing the expanded area ratio will cause the efficiency to decrease
[14]. It can be seen from the Tables 10-12 that Propeller 3 consistently
has the greatest KT for all values of J. This suggests that the greatest EAR
of 0.7 can produce the largest thrust. Propeller 3 also has the largest KQ
indicating it delivers the most torque and subsequently power out of all
propellers. This does not mean it has the highest efficiency, however.
The greatest efficiency is reached by Propeller 3 at the low end of J.
Nevertheless, towards the higher end of J, Propeller 3′ s efficiency drops
off and both Propeller 1 and 2 experience greater efficiency. This sug­
gests that a lower EAR produces an improved efficiency at high advance
ratios. Yet a higher EAR produces an improved efficiency at low advance
ratios.
Interestingly, the highest accuracy results of KT and KQ which appear
at J = 0.691, correlate with the lowest accuracy efficiency seen in
Figs. 6-8.
(c) Propeller 3 Velocity Contours at J=0.415
Fig. 12. Propeller velocity contours.
are displayed in Tables 5 and 6, respectively.
CFD simulation was conducted over a range of propeller advance
ratios corresponding to the rotational velocity of the rotating domain, as
summarized in Table 7.
3.5. Propeller geometry and Solver setting
The geometry of the Wageningen B-series can be obtained from the
official Wageningen B-Series Propeller generator, which allows you to
create a propeller with B-series dimensions identical to those tested at
Wageningen and the expanded area ratio as a variable. In this paper,
three propellers that have identical dimensions but have three different
expanded area ratios: 0.6, 0.65 and 0.7 will be investigated. The pro­
peller geometry can be seen in Table 8 [25].
As mentioned previously, a MRF method is used to model the pro­
peller rotation. A number of hypotheses are used to simplify the case of
simulating propeller hydrodynamic characteristics. These include
steady-state flow, non-cavitation and incompressible flow, hence the
single phase model. The solver settings used in the k − ε model are dis­
played in Table 9. The first order upwind settings were used following
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Energy Conversion and Management: X 17 (2023) 100349
Fig. 13. Isometric view for velocity vectors at J = 0.415 for Propeller 2.
4.2. Pressure and velocity distributions
particles are accelerated to a higher velocity. Therefore, a greater force
must be applied to the fluid, henceforth a greater reaction force is
applied to the propeller culminating in improved thrust. The minimum
velocity with respect to the boundary conditions in all three figures is 0
m/s.
The isosurface velocity as three-dimensional analog of Propeller 2 is
displayed in Fig. 13. It used to link up all the point with same values
within a volume of space. The result showed that the fluid reaches its
maximum velocity at the propeller tips. It can also be noted that Pro­
peller 2 has a higher maximum velocity of 46.95 m/s. This suggests once
again that the lower EAR can produce a greater thrust, albeit by an
almost negligible improvement of 0.32 %.
In general, the propeller with the lowest EAR, as well as the lower
thrust and torque coefficients, generates the highest slipstream velocity,
improves the flow behaviour, and suggests suitability for optimisation
for electric motors, which can generate maximum torque instantly.
Fig. 9 shows the pressure distribution across the front and back face
of Propeller 2 operating at an advance ratio of J = 0.415. It can be seen
that the pressure at the back side of the propeller (the pressure face), is
slightly greater than the pressure at the front side of the propeller, (the
suction face). Indeed a maximum pressure of 435.9 kPa is reached on the
pressure face compared to a maximum pressure of 121.3 kPa on the
suction face. This difference in pressure between the front and back faces
of the propeller is the creating force in the forward direction known as
thrust. The greatest difference in pressure between the two faces is sit­
uated at the leading edge of the propeller and reaches a total pressure
difference of 1.079 MPa. When compared to Propeller 3 displayed in
Fig. 10 at J = 0.415, the maximum pressure experienced is 385.0 kPa,
which is less than Propeller 2 with the lower EAR and the maximum
pressure difference obtained is 1.042 MPa. This shows that a smaller
EAR creates a slightly larger pressure difference and has the potential to
create more thrust at the same advance ratio.
The pressure difference between the front and back faces of the
propeller can be clearly seen in Fig. 11. The maximum pressure differ­
ence appears between R/R0 = 0.8 and the tip of the blades. Here the
fluid pressure difference reaches a maximum of 586.4 kPa for Propeller
3 and 594.4 kPa for Propeller 2. This is where the thrust is generated,
and the results indicate that greater thrust can be produced with the
smaller EAR consistent with Propeller 2. It is important to note the area
of relatively low pressure situated aft of the propeller hub. This area is
likely to cavitate due to the local pressure dropping below the vapour
pressure of the fluid, and this effect is enhanced due to the vortical
nature of the propellers slipstream and the positive feedback loop
associated with cavitating vortices [22].
The velocity contours of each propeller are displayed in Fig. 12. All
propellers are used as a comparison in the velocity section. It can be seen
that at the same Advance Coefficient J = 0.415, Propeller 3 (Fig. 12.c)
induces the lowest maximum velocity of 45.39 m/s and that Propeller 1
and 2 which have a lower EAR induce a greater maximum velocity by
0.39 % and 0.31 % respectively. This means that a lower EAR has the
potential to create more thrust due to Newton’s third law [23]. The
5. Conclusion
This research was devoted to a CFD simulation to evaluate three
Wageningen B-series propellers’ performance. The study successfully
provides the framework needed to numerically analyse a propeller’s
open-water characteristics. The commercial CFD code FLUENT was
found to be reliable in providing good initial predictions and thus is a
viable alternative to open water tests at the design phase. The validation
revealed that the CFD simulations used in this study are in good
agreement with the experimental results recording an average error of
8.65 %. This diversion in the accuracy of the experiment can be
explained by the effect of some minor factors, such as using pure water
and the absence of the hub or the shaft in the validation.
Moreover, the results for the thrust coefficient showed a slight
underprediction for all advance ratios, with this underprediction max­
imising at J = 0.518 and minimising and converging for high advance
ratios, namely J = 0.691. The results for the torque coefficient showed a
much greater underprediction with a decrease in error towards the
higher advance ratios. The open water efficiency was very accurate
throughout the advance ratio, with a slight underprediction at the
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Energy Conversion and Management: X 17 (2023) 100349
extremities of the advance ratio. Overall results showed a reliable
capability to predict the performance of a Wageningen B-Series
propeller.
However, based on the pressure and fluid velocity analysis, the lower
expanded area ratio showed the greatest potential in creating the most
thrust. Concerning flow behaviour, the greatest thrust and torque co­
efficients correlate with the lowest propeller slipstream velocity, as seen
in Propeller 3. This suggests that lower thrust and torque coefficients,
which correspond with a lower expanded area ratio, as shown in Pro­
pellers 1 and 2, can generate an improved flow behaviour with a higher
particle velocity. It could be noticed that up to 62 % of the efficiency
increase could be gained with lower thrust and torque coefficients. In
addition, the outcome revealed that the lower expanded area ratio of 0.6
is more suitable for electric boats, creating a larger pressure difference of
1.079 MPa and generating extra potential thrust at the same advance
ratio, which leads to greater open water efficiency.
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Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Data availability
Data will be made available on request.
Acknowledgment
The second author, Anas F.A.Elbarghthi would like to acknowledge
the support of the Student Grant Competition of the Technical Univer­
sity of Liberec under project No. SGS-2023-5323.
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