441
1 INTRODUCTION
Insomemodesofshipoperationormaintenancethe
propellershaftisimmobilisedandpropellerbecomes
locked. The case of fast ships fitted with multiple‐
engine and multiple‐propeller propulsion system
with three or four propellers is discussed by
Charchalis[1].Suchshipssometimesoperateathigh
speed but often sail at reduced or cruising speed.
Excessiveenginesarestoppedandpassivepropellers
are locked in order to prevent the gear and engine
fromwearorfailure.C
harchalisproposedthemethod
forevaluatingthedragandhydrodynamictorqueon
lockedpropeller.Themethodisbasedonthrustand
torque coefficients from four‐quadrant open water
charact
eristics of considered screw propeller. In
example calculations Charchalis refers to diagrams
forthree‐bladedscrewpropellerspresentedin[2].
Propellers may also stay locked when ship is
towedinemergencyorinthecourseofdeliveryfrom
shipyard to ship operator. Then the concern is to
protect the gear and engine, and despite the
additional drag during towi
ng the propeller shaft
maybeintentionallyimmobilized.
Allmodernsailboats,withexceptforthesmallest
ones, are equipped with mechanical propulsion.
Whensailing‐theengineisstoppedandsomesailors
decidetolocktheshaftandpropeller,sometimesdue
to the mista
ken belief that locked propeller is more
favourableforlowdrag.(Thequestionofwhetherthe
dragofthelockedpropellerislowerthanthedragof
the windmilling propeller was argued as late as in
2012.)Sometimesthe manufacturersof smallmarine
gearboxes specify tha
t the shaft should be locked
when the engine is stopped and the vessel is in
motion, because some gearboxes receive adequate
lubricationonlywhentheengineisrunning.
Althoughsometestresultsofhydrodynamicdrag
and torque acting on locked screw propeller are
availableinopenliteratureandonemayrefertothem
directly, it would be more convenient for ship
designers and operators to have a simple formula
ready to a
pply and provide the estimation of drag
and torque instantly. The present authors have
Drag and Torque on Locked Screw Propeller
T.Tabaczek
WrocławUniversityofTechnology,Wrocław,Poland
T.Bugalski
ShipDesignandResearchCentreCTOS.A,Gdańsk,Poland
ABSTRACT:Fewdataondragandtorqueonlockedpropellertowedinwaterareavailableinliterature.Those
data refer to propellers of specific geometry (number of blades,blade area, pitch and skew of blades). The
estimationofdragandtorqueofanarb
itrarypropellerconsideredinanalysisofshipresistanceorpropulsionis
laborious.Theauthorscollectedandreviewedtestdataavailableintheliterature.Basedoncollecteddatathere
were developed the empirical formulae for estimation of hydrodynamic drag and torque acting on locked
screwpropeller.SupplementaryCFDcomputationswerecarriedoutinordertoprovetheapplicabilit
yofthe
formulaetomodernmoderatelyskewedscrewpropellers.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 8
Number 3
September 2014
DOI:10.12716/1001.08.03.16
442
reviewedandcomparedtheavailabletestdata.Based
on a simple assumption of negligible effect of
interaction between blades in the case of locked
propeller they proposed the empirical formulae for
estimation of drag and torque on locked screw
propellerinopenwater.
The research was extended with CFD
computations
inordertoconfirmtheapplicabilityof
developed formulae to modern moderately skewed
propellers. The allowances to drag and torque were
proposedinsuchcases.
2 NON‐DIMENSIONALCOEFFICIENTS
The results of open water tests of marine screw
propellers are presented in the form of non‐
dimensionalcoefficients of speed,
thrustand torq ue.
Usually,instandardrangeofaheadspeedandahead
rotation,theadvancecoefficientJ,thrustcoefficientK
T
andtorquecoefficientK
Qaredefinedas
J=V
A/(nD)
K
T=T/(ρn
2
D
4
) (1)
K
Q=Q/(ρn
2
D
5
)
where:
V
A–advancespeedofpropeller
n–rotationalrate
D–propellerdiameter
T–propellerthrust
Q–torque
ρ–densityofwater.
When screw propellers are tested in extended
range of operating conditions, and when rotational
sped equals zero or is close to zero, the values of
conventionalcoefficientsK
TandKQapproachinfinity,
and non‐dimensional coefficients must be defined
otherwiseinordertopresentthefiniteperformance.
Nordstrom [3] and Miniovich [2] applied the
conventional coefficients (1) around bollard
conditionsandmodifiedcoefficients:
J’=nD/V
A=1/J
K
T’=T/(ρVA
2
D
2
) (2)
K
Q’=Q/(ρVA
2
D
3
)
around locked conditions (n=0). (Different symbols
are used across the literature for modified
coefficients.) The open water four‐quadrant
hydrodynamiccharacteristicsofeachscrewpropeller
werepresentedin[2]and[3]usingatthesametime
two systems of coefficients with overlapping ranges
of advance coefficient far from bollard
and locked
conditions. The definition of advance coefficient J
implies that it does not define the operating
conditions uniquely, and additional information is
necessary do distinguish the ahead (V
A>0; n>0) and
astern(V
A<0;n<0),aswellasthecrashback(VA>0;n<0)
andcrashahead(V
A<0;n>0)conditions.
The problem of two different systems of
coefficients has been overcome by application of
advance angle β and non‐dimensional coefficients
based on speed of blade section at radius r=0.7R
acrossthewaterV
r=[VA
2
+(0.7πnD)
2
]
1/2
[4]:
β=arctan(V
A/0.7πnD)
C
T*=T/{0.5ρ[VA
2
+(0.7πnD)
2
]π/4D
2
} (3)
C
Q*=Q/{0.5ρ[VA
2
+(0.7πnD)
2
]π/4D
3
}
(Similar idea by Lavrentiev, based on reference
speed(V
A
2
+n
2
D
2
)
1/2
,wasnoticedbutnotusedin[2].)
Performance characteristics C
T*(β) and CQ*(β) are
single‐valued, continuous and periodical, and more
suitable for theoreticalinvestigation of ship
manoeuvres.
In the following analysis only the thrust and
torqueatn=0areconsideredandtheauthorsusethe
coefficientsC
T*andCQ*atβ=π/2.Themodifiedthrust
andtorquecoefficientsK
T’andKQ’neededconversion
accordingtothefollowingcorrespondence(validonly
atn=0):
C
T*=KT’/(π/8)
C
Q*=KQ’/(π/8)
3 AVAILABLETESTDATA
In1948 Nordstrom[3] describedmodel testscarried
outinordertodetermineopenwaterperformanceof
screw propellers in all regimes ofoperation (infour
quadrantsofoperatingconditions).Testswerecarried
out with a series of four‐bladed propellers of blade
arearatioequalto0.45.Theonlyparameterdiffering
propellersintheserieswasthepitchratiothatvaried
fromzero to 1.6. Nordstrom presented theresultsin
the form of non‐dimensional coefficients plotted
against advance ratio. He applied two systems of
coefficients:theconventionalonebasedonrotational
speed n (Eq. (1)) and the modified one based on
advance speed V
A (Eq. (2)). (Nordstrom applied
differentsymbolsbutthesamedefinitionsasusedin
this paper.) Values of coefficients at n=0 were
convertedandshowninFig.1.
443
-0,5
-0,45
-0,4
-0,35
-0,3
-0,25
-0,2
-0,15
-0,1
-0,05
0
0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8
P/D
CT*
CT*
linear
-0,07
-0,06
-0,05
-0,04
-0,03
-0,02
-0,01
0
0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8
P/D
CQ*
CQ*
polynom.
Figure1. Thrust and torque coefficients of four‐bladed
propellerstestedbyNordstrom[3],n=0
In 1954 through 1956 Miniovich [2] carried out
model tests with a series of three‐bladed screw
propellers. The seriesincluded propellerswith three
valuesofblade arearatioA
E/A0:0.5,0.8and1.1.For
eachva lueof A
E/A0the pitchratioP/Dvariedinthe
range0.6≤P/D≤1.6.In[2]theresultswerepresented
in diagrams, using two systems of coordinates
definedbyformulae(1)and(2).Valuesofcoefficients
atn=0 wereconverted andare shownin Fig.2. The
resultsforA
E/A0=1.1readoutfromthetangleoflines
intheoriginal diagrams [2]are highlyirregular and
wereomittedinthefollowinganalysis.
-0,8
-0,7
-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
0
0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8
P/D
CT*
AE/A0=0.5
AE/A0=0.8
linear
linear
-0,14
-0,12
-0,1
-0,08
-0,06
-0,04
-0,02
0
00,20,40,60,811,21,41,61,8
P/D
CQ*
AE/A0=0.5
AE/A0=0.8
polynom.
polynom.
Figure 2. Thrust and torque coefficients of three‐bladed
propellerstestedbyMiniovich[2],n=0
In 1969 van Lammeren et al. [4] presented four-
quadrant hydrodynamic characteristics of 14 screw
propellers selected from the well-known B-series. The
selection was made so that the effect of three basic pa-
rameters (namely P/D, A
E
/A
0
and the number of blades
z) on performance in all regimes of operation was re-
vealed. Basic parameters of individual propellers are
given in Table 1. The measured thrust and torque were
converted into non-dimensional coefficients C
T
* and
C
Q
* according to the definition (3) and presented in
diagrams. Van Lammeren et al. [4] proposed also the
approximation of characteristics with the Fourier se-
ries of 20 terms, convenient for ship manoeuvring
study. However, some coefficients of Fourier series
given in [4] do not allow reproducing the correspond-
ing curves presented in diagrams. In order to deter-
mine the values of thrust and torque coefficients at
n=0 (β=π/2) the present authors used the approxima-
tion of characteristics with Fourier series of 30 terms
reprinted in report [5]. Calculated values of coeffi-
cients are listed in Table 1 and shown in Fig. 3.
Table1.PropellersfromB‐seriestestedinfourquadrants[3]
_______________________________________________
Propeller z AE/A0P/D CT* CQ*
_______________________________________________
B4‐70 4 0.70 0.5‐0.7818 ‐0.0575
B4‐70 4 0.70 0.6‐0.7935 ‐0.0690
B4‐70 4 0.70 0.8‐0.7958 ‐0.0954
B4‐70 4 0.70 1.0‐0.7719 ‐0.1155
B4‐70 4 0.70 1.2‐0.7092 ‐0.1237
B4‐70 4 0.70 1.4‐0.6545 ‐0.1364
B4‐40 4 0.40 1.0‐0.3550 ‐0.0442
B4‐55
 4 0.55 1.0‐0.5795 ‐0.0733
B4‐85 4 0.85 1.0‐0.9626 ‐0.1438
B4‐100 4 1.00 1.0‐1.0363 ‐0.1510
B3‐65 3 0.65 1.0‐0.6660 ‐0.0962
B5‐75 5 0.75 1.0‐0.8539 ‐0.1255
B6‐80 6 0.80 1.0‐0.8903 ‐0.1264
B7‐85 7 0.85 1.0‐0.9234 ‐0.1264
_______________________________________________
B4-70
-0,9
-0,8
-0,7
-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
0
0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6
P/D
CT*
CT*
linear
B4-70
-0,16
-0,14
-0,12
-0,1
-0,08
-0,06
-0,04
-0,02
0
0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6
P/D
CQ*
CQ*
polynom.
444
B4-40, B4-55, B4-70, B4-85, B4-100
-1,2
-1
-0,8
-0,6
-0,4
-0,2
0
0,00 0,20 0,40 0,60 0,80 1,00 1,20
AE/A0
CT*
CT*
linear
B4-40, B4-55, B4-70, B4-85, B4-100
-0,18
-0,16
-0,14
-0,12
-0,1
-0,08
-0,06
-0,04
-0,02
0
0,00 0,20 0,40 0,60 0,80 1,00 1,20
AE/A0
CQ*
CQ*
linear
B3-65, B4-70, B5-75, B6-80, B7-85
-0,25
-0,2
-0,15
-0,1
-0,05
0
0 0,05 0,1 0,15 0,2 0,25
(AE/A0)/Z
CT*/Z
CT*/Z
polynom.
B3-65, B4-70, B5-75, B6-80, B7-85
-0,035
-0,03
-0,025
-0,02
-0,015
-0,01
-0,005
0
0 0,05 0,1 0,15 0,2 0,25
(AE/A0)/Z
CQ*/
Z
CQ*/Z
polynom.
Figure3. Thrustand torque coefficientsofpropellers from
B‐series[4],n=0
Lurie and Taylor [6] investigated performance
characteristicsof2‐ and3‐bladed 13inch
commerciallyavailablepropellersdesignedforuseon
small and medium size sailboats. Besides the non‐
conventional screw propellers (folding or feathering
propellers) the four fixed‐blade propellers were
tested. The performance at forward speed was
measured including
propeller drag at n=0. Results
were presented in the form of drag plotted against
advance speed. Particulars of propellers along with
values ofthrustcoefficient (torquewas notreported
in original paper) at speed V
A above 3.0m/s are
summarizedinTable2.
Table2.Sailboatfixed‐bladepropellerstestedbyLurieand
Taylor[6]
_______________________________________________
Propeller
Campbell Campbell Michigan Michigan
Sailer Sailer Wheel Wheel
2‐bladed 3‐bladed 2‐bladed 3‐bladed
_______________________________________________
z2323
D[m]0.330 0.330 0.330  0.330
P/D 0.769  0.769 0.769 0.769
A
E/A0‐0.30 0.36 0.44
(A
E/A0)/z‐0.10 0.18 0.147
C
T* ‐0.212‐0.278‐0.238‐0.438
_______________________________________________
MacKenzie and Forrester [7] measured drag of
another three 12‐inch sailboat propellers.Results
were presented in the form of drag plotted against
speed.Particularsofpropellersalongwithcalculated
values ofthrustcoefficient (torquewas notreported
in original paper) at speed V
A of 3.09m/s are
summarizedinTable3.
Table3.Sailboatfixed‐bladepropellerstestedbyMacKenzie
andForrester[7]
_______________________________________________
Propeller
‘A’‘B’‘C’
_______________________________________________
z332
D[m]0.3050.3050.305
P/D0.50.50.667
A
E/A00.540.520.40
(A
E/A0)/z 0,180.1730.20
C
T*‐0.512‐0.463‐0.322
_______________________________________________
In 2013 Dang et al. [8] presented the outcomes
fromopenwatermodeltestsof4‐bladedcontrollable
pitch propellers denoted C4‐40, from the newly
developed Wageningen C‐series. There were 4
modern moderately skewed propellers designed
according the best design practice. The propellers
differed from each other with the
design pitch ratio
that was equal to 0.8, 1.0, 1.2 and 1.4. The
performance of propellers was measured in two
quadrantsofoperationincludingn=0.Testdatawere
presentedintheformofthrustandtorquecoefficients
C
T* and CQ* plotted in diagrams against advance
angleβ.
4 EMPIRICALFORMULAE
Forengineeringapplicationsthedragandtorqueofa
lockedpropellershouldbecalculatedusingonlybasic
parametersofpropellerthatareusuallyavailable,i.e.
propellerdiameter,numberofblades,bladearearatio
and pitch ratio. The available values
of thrust and
torque coefficients were used to determine the
relationships between drag/torque and basic
parametersofpropeller..
Therelations betweendrag/torqueandpitchratio
havebeendeterminedbasedondatafrom[2],[3]and
[4].Thevaluesofthrustandtorquecoefficients were
normalisedusingthevaluesforP
/D=1.0(seeFig.4).
Using the least square approximation the relation
between the normalised thrust coefficient C
Tn* and
P/Dwas fittedwithlinear function,and therelation
between the normalised torque coefficient C
Qn* and
P/Dwithquadraticpolynomial:
445
C
Tn*=‐0,31P/D+1,31 (4)
C
Qn*=‐0,30(P/D)
2
+1,35P/D‐0,05
TherelationsbetweencoefficientsC
T*andCQ*and
P/Dbecomeasfollows:
C
T*=CT*(P/D=1.0)(‐0,31P/D+1,31) (5)
C
Q*=CQ*(P/D=1.0)(‐0,30(P/D)
2
+1,35P/D‐0,05)
0
0,2
0,4
0,6
0,8
1
1,2
1,4
0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8
P/D
CTn*
Nordstrom
Miniovich 0.5
Miniovich 0.8
B4-70
linear approx.
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8
P/D
CQn*
Nordstrom
Miniovich 0.5
Miniovich 0.8
B4-70
polynom.approx.
Figure4. Variation of the normalized thrust and torque
coefficientswithpitchratio
The assumption of negligibleinteractionbetween
bladesinthecaseofn=0allowedtoconsidertheforce
ortorqueonasinglebladeinsteadonentirepropeller.
Twobasicparametersi.e.bladearearatioandnumber
of blades were combined into a single parameter,
namelythearearatioofa
singleblade(AE/A0)/z.The
relationship was investigated using data for
propellers with P/D =1.0, an average value
encounteredindesignofmerchantships.Muchdata
were available directly without the need for
extrapolation from other values of pitch ratio. Only
thecoefficientsforsailboatpropellersfrom[6]and[7]
were extrapolated
using the relationship (5). The
valuesofthrustandtorquecoefficientsperoneblade
werecollectedandpresentedinFig.5.
-0,3
-0,25
-0,2
-0,15
-0,1
-0,05
0
0 0,05 0,1 0,15 0,2 0,25 0,3
(AE/A0)/Z
CT*/Z
Nordstrom
Miniovich
B-series
sailboat props
polynom.approx.
-0,04
-0,035
-0,03
-0,025
-0,02
-0,015
-0,01
-0,005
0
0 0,05 0,1 0,15 0,2 0,25 0,3
(AE/A0)/Z
CQ*/
Z
Nordstrom
Miniovich
B-series
polynom.approx.
Figure5.Relationbetweenthethrustandtorquecoefficients
per one blade (C
T*/z and CQ*/z) and unit blade area ratio
(A
E/A0)/z,atP/D=1.0
Because the data for two‐ and three‐bladed
propellersfrom[2],[6]and[7]donotalignwithother
data,theleastsquareapproximationwasfittedusing
only data for B‐series [4] and those given by
Nordstrom [3]. For P/D =1.0 the following formulae
areproposed:
C
T*(P/D=1.0)/z=4,28[(AE/A0)/z]
2
‐2,61(AE/A0)/z+0,13(6)
C
Q*(P/D=1.0)/z=0,796[(AE/A0)/z]
2
‐0,453(AE/A0)/z+0,026
Combining the approximations (6) and (5) the
empirical formulae for 4‐ to 7‐bladed propellers
becomesasfollows:
C
T*=z{4,28[(AE/A0)/z]
2
–
2,61(A
E/A0)/z+0,13}(‐0,31P/D+1,31) (7)
C
Q*=z{0,796[(AE/A0)/z]
2
‐
0,453(A
E/A0)/z+0,026}(‐0,30(P/D)
2
+1,35P/D‐0,05)
The fit of approximation (7) to empirical data is
illustratedinFig.6.
446
0
0,2
0,4
0,6
0,8
1
1,2
0 0,2 0,4 0,6 0,8 1 1,2
-CT* (test data)
-CT* (eq.(7))
Nordstroem
Miniovich
van Lammeren
sailboat props
CT* (7)=CT* (test)
0
0,04
0,08
0,12
0,16
0,2
0 0,04 0,08 0,12 0,16 0,2
-CQ* (test data)
-CQ* (eq.(7)
)
Nordstroem
Miniovich
van Lammeren
CQ* (7)=CQ* (test)
Figure6.Thefitofapproximation(7)toempiricaldata
5 CFDCOMPUTATIONS
In order to assess the applicability of empirical
formulaetomodernmoderatelyskewedpropellersa
number of RANSE‐CFD computations was carried
outusingreadygeometriesandgridspreviouslyused
for computations of propeller flow in forward
operating conditions [9], [10]. Main particulars of
propellersarecollectedin
Table4.
Table4.PropellersusedinCFDcomputations
_______________________________________________
KP505 CP469 P0 P2 P9
(KCS) (Nawigator)
_______________________________________________
z5 44 4 5
D[mm] 250 226 233.33 247 3000
P
0.7/D0.997 0.942 1.00.920 0.973
A
E/A00.796 0.674 0.581 0.517 0.634
rake[mm] 0 05.83 ‐27.64 0
(A
E/A0)/z0.159 0.169 0.145 0.129 0.127
skew‐back 0.224 0.229 0.120 0.248 0,133
ratios(R)/D
skewangle 24.9 25.7 13.2 27.6 14.7
θ
S(R)[deg]
C
T*CFD‐1.042‐0.842‐0.572‐0.598‐0.766
C
T*eq.(7) ‐0.886‐0.767‐0.635‐0.557‐0.666
(‐15%) (‐9%)(+11%) (‐7%) (‐13%)
C
Q*CFD0.152 ‐0.112‐0.077‐0.076‐0.106
C
Q*eq.(7) 0.129 ‐0,106‐0.092‐0.072‐0.091
(‐15%) (‐5%)(+19%) (‐5%) (‐14%)
_______________________________________________
Computations were carried out using the
commercial CFD software CDadapco STAR CCM+.
High accuracy of CFD computations using that
software andappropriate grids has been proved for
computations of flow around screw propellers
operating in the conventional range of open water
characteristics(0<J<J(K
T=0)).TheCFDcomputations
have not been verified and validated at locked
conditions(atn=0).
The results of computations in the form of non‐
dimensional coefficients are included in Table 4, in
comparison to values calculated using the formulae
(7). The fit of approximation is illustrated in Fig. 7
where
approximated values are compared to results
of CFD computationsand totest datain thecase of
controllablepitchpropellersC4‐40[8].Approximated
values areunderestimated by approximately5 to15
percentinrelationtobothCFDandtestdata.
0
0,2
0,4
0,6
0,8
1
1,2
0 0,2 0,4 0,6 0,8 1 1,2
-CT* (CFD)
-CT* (eq.(7)
)
P602
CP469
P0
P2
P9
C4-40 (test data)
CT*(eq.(7))=CT*(CFD)
0
0,04
0,08
0,12
0,16
0,2
0 0,04 0,08 0,12 0,16 0,2
-CQ* (CFD)
-CQ* (eq.(7)
)
P602
CP469
P0
P2
P9
C4-40 (test data)
CQ*(eq.(7))=CQ*(CFD)
Figure7.Thefitofapproximation(7)toCFDandtestdata
6 CONCLUSIONS
Using the available test data for locked screw
propellers the formulae (7) for estimation of thrust
and torque coefficients C
T* and CQ* (defined by eq.
(3)) areproposed. The coefficients are relatedto the
basic parameters of propeller, namely to pitchratio,
bladearearatioandthenumberofblades.
Theformulaearevalidprincipallyfornon‐skewed
orlow‐skewedscrewpropellers,intherangeofpitch
ratio P/D from
0.6 to 1.6 and in the range of single
bladearearatio(A
E/A0)/zfrom0.10to0.25.
In application to 2‐ or 3‐ bladed propellers the
approximated values of drag and torque may be
overestimatedbyupto30%inrelationtoactualforces
(Fig.6). In the case of moderately skewed propellers
the approximated values may be underestimated.
Based on the outcomes
from CFD computations the
authorsproposetheallowanceof+16%.
Knownthedragandtorquecoefficients,thedrag
and torque on locked screw propeller are estimated
accordingtotheformulae:
447
D=‐T=‐0.125C
T*ρVA
2
πD
2

Q=0.125C
Q*ρVA
2
πD
3

using the relevant advance speed. It is proposed to
usetheaveragevelocityof flow innominal wake of
ship:
V
A=VS(1‐wn)
where V
S denotes ship speed, and wn‐the nominal
wakefraction.
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