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10thInternational Conference on HydrodynamicsOctober 1-4, 2012 St. Petersburg, Russia
Assessment of Simplified Propeller-Models for General
Purpose CFD Solvers
D.Durante1
, G.Dubbioso1*
, C.Testa1
1CNR-INSEAN
National Research Council-Italian Ship Model BasinVia di Vallerano 139, 00128 Rome, Italy
*Corresponding author: Dr. G. Dubbioso
e-mail: [email protected]
Abstract
CFD simulation of a whole hull-propeller
configuration requires high computational efforts tosolve accurately propeller hydrodynamics. This fact
may be unacceptable for preliminary design andoptimization applications. Thus, in order to overcome
this crucial aspect, the presence of the propeller
should be simulated by simplified models thatrepresent a good trade-off between accuracy and
computational costs. In this paper three differenthydrodynamic models for the analysis of propellers
working in hull-behind conditions are presented andcompared, from a numerical standpoint, in view of a
further implementation into a general purpose CFDsolvers. Their drawbacks and potentialities are
discussed to derive some guidelines on the use of fastand reliable algorithm suited to treat either design and
off-design conditions.
KEY WORDS: Unsteady propeller hydrodynamics;
Airfoil theories; Computational Fluid Dynamics.
INTRODUCTION
During the last decade, Computational Fluid
Dynamics (CFD) has been extensively verified andvalidated for marine hydrodynamic applications. Froma theoretical point of view, CFD yields a detailed
insight into the flow-field around the stern, accountingfor hydrodynamic effects induced by the propulsion
system (propellers and shaft-line appendages) andcontrol devices. As a matter of fact, computationswhere hull and propeller are solved jointly provide thewake onset-flow affecting propeller hydrodynamics
and, in turns, vessel maneuverability [1-2]. Although a
key-point for the success of ship-design lies on thecomputation of stern hydrodynamics, CFD analysis of
the whole vessel configuration (including the bare-hull, appendages and propeller system), may be
prohibitive in terms of computational effort, because a
very fine time-discretization may be required to
properly detect the unsteady behavior of propellerhydrodynamics. For this reason, the presence of
propeller-induced effects into CFD solvers aretypically modeled through the momentum theory
yielding a rough, quick prediction of the azimuthal-averaged propeller loads [3]. Despite the intrinsic
limits of such approach in describing hull-propeller
interactions, numerical results, in terms of averagedvelocity-field and propulsive performance in design
conditions, may be in satisfactory agreement withexperiments [4]. An enhancement of the
hydrodynamic simulation might be obtained bycoupling more accurate propeller hydrodynamic
models with the CFD solver.
Such consideration has inspired the present paper thatproposes a numerical comparison among three models
based on the potential-theory for incompressibleflows; in detail, the Nakatake hydrodynamic
formulation [5] is here compared with propeller
hydrodynamics predicted by Theodorsen and Searsunsteady sectional theories, respectively. The
Nakatake model has been successfully used in the pastfor the analysis of self-propulsion tests of full-
appended ships [6] and for steady maneuvers [2]. In
this model, blades shape is not described in terms ofgeometry, since the propeller is seen as an actuator-disk where the presence of the blades is accounted by
bound vortex sheet and free-vortices shed rearwards.To use this model, experimental input data in terms of
open-water propeller performance are necessary fortuning some coefficients related to both blade loads
computation and geometric features, otherwise not
described [5]. The need of an ad-hoc tuning procedurerepresents a drawback that encourages the
investigation of different hydrodynamic approaches.To this aim, a first attempt is addressed by the
Theodorsen theory where sectional unsteady loads areobtained from the knowledge of the upwash velocity
at the -chord and -chord points, respectively [7-8].
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Such an approach, widely used for the study offixed/rotary wing aeroservoelasticity (see for instance
[9-10]), is here applied to marine propeller in hull-behind conditions by observing that a spatially, non-
uniform flow produces periodical forces on the blades.
Next, since each section of the propeller blades,rotating within the non-uniform inflow coming from
the hull, behaves like an airfoil uniformly moving in amulti-harmonic gust, the Sears theory is proposed as
an enhanced bi-dimensional approach. Note that theSears theory has been applied in the past, for hydro-
acoustics purposes, to determine the unsteady pressuredistribution upon a marine blade operating in a wake-
field [11].
The main goal of this work is to develop an unsteadyhydrodynamic tool that offers a good trade-off
between accuracy and required computational costs soas to be suited for a further integration into CFD-tools.In order to validate the proposed formulations,hydrodynamic results from the present approaches are
compared with those obtained through the fully three-
dimensional, time marching BEM hydrodynamicssolver [12-13], extensively validated for marine
propellers in uniform and non-uniform incoming, noncavitating flows. Fast and physically consistent
propeller hydrodynamic solvers may be very attractivefor ship maneuverability applications where the
reliable prediction of propeller hub-loads, deeply
affecting vessel dynamic response, is mandatory.
Unsteady Propeller Hydrodynamics
In this section, the prediction of the unsteady loads
arising when propeller blades operate within an inflowdue to the moving hull, is addressed. Although marine
propellers work in a very complex hydrodynamicenvironment, other contributions to the non-uniform
inflow, like perturbations induced by the propeller-wake vorticity and/or interferences with other bodies
(if present), are here neglected. Furthermore, only the
axial component of the onset flow is considered. Theinteraction between propeller blades and the hull-
wake determines periodic hydroloads; the moreirregular is the onset flow, the more important areblade-loads pulses. In the following, thehydrodynamic models used for the prediction of such
unsteady loads are briefly outlined (without loss ofgenerality, only propeller thrust is considered). More
details are found in the literature references.
The Nakatake Model
The propeller model proposed by Nakatake [5] is
represented by a continuum disk that takes intoaccount for thechordradial distribution and effective
pitch of the original propeller blades through apreliminary tuning process based on open-water
propeller performance given by experiments or highlyaccurate computations. A set of bound vortex sheets,
localized at the propeller-planep
S along with free-
vortices shed from rearwards, allow to describe thepropeller induced velocity-field through a potentialtheory [1-2]. For a propeller moving with constantadvancing ratio J=V/nD (V denotes the hull velocity
along thexdirection orthogonal to the propeller plane,
nthe propeller rate of revolution and Dthe propeller
diameter) the velocity-potentialwherever in the
flow-field (except onp
S and at the rearwards wake) is
given by [14-16]
=R
rboss
PdrdrzyxGrzyx
2
0
),,,,(),(41),,( (1)
where ,r identify the position of points on the disk,
zyx ,, are the coordinate of a point in the flow-field,
bossr and R denote the boss and blade radius,
respectively, pG represents the singularity function
defined in [16] whereas (r,)refers to the circulation
obtained from the solution of the boundary-conditionthat defines the conservation of mass through the
propeller-plane [5]. From the knowledge of both
and, the velocity field on pS may be written as
r
V
rnrV
P
V
xuV
x
22
2
+=
+
+=
(2)
where ),( ru defines the ahead axial-velocity field
from the hull-wake, Pis the nominal pitch associated
with the helical vortices shed rearwards, denotes thePrandtl tip correction factor [5], whilst the third term
at the right-hand-side of (2) represents the velocitycontribution due to both bounded and shed vorticity.The global thrust may be expressed as
=R
rboss
drrVrVT ),(),()( (3)
where is the fluid density. The nondimensionalthrust coefficient is finally defined by
42Dn
TK
T
= (4)
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Hydrodynamic Loads From Airfoil Theories
Blade hydroloads are here obtained as radialintegration of the loads given by sectional
hydrodynamic models. These are typically derived
from airfoil unsteady hydrodynamic theories. Let acoordinate axes be fixed to a thin, straight airfoil with
thexaxis parallel to the flow at infinity, and with theorigin at the mid-chord. For an airfoil of chord-length
c = 2b, moving in an incompressible flow with Vvelocity, subjected to a harmonic vertical translation
h(t)and to a pitch rate (t)imposed about an axis at asemi-chord from the mid-chord (t denotes time), the
Theodorsen theory yields the unsteady aerodynamicforce acting on it by combining the non-circulatory lift
Lnc, orthogonal to the chord, with the circulatory liftLcdirected along the normal to the relative wind (see, for
instance, [17]). Specifically, the non-circulatory lift isexpressed as
2/
2
cnc wb=(t)L & (5)
where 2/cw& represents the time derivative of the
normal component of the relative wind (upwash)
evaluated at the airfoil mid-point (positive upwards).The circulatory lift is given by
[ ]4/3c1 ~2 wC(k)bV=(t)Lc
(6)
where denotes Fourier transformation,
][~ 4/3c4/3c w=w with 4/3cw representing the upwash
at the airfoil -chord point whereas C(k)indicates thelift deficiency function defined by Theodorsen [17] in
terms of the reduced frequency k = b/V ( is the
frequency of the harmonic motions). The applicationof the Theodorsen theory to marine propellers
operating in a non-uniform axial-inflow may be doneassuming that the periodic upwash on the airfoil is thecombination of multi-harmonics plunge and pitch
motionstjeh=h(t) 0 and
tje=(t) 0 with
constant phases (0
h and0
represents complex
amplitudes). Thus, the upwash w is described as thesuperposition of chordwise linear distribution ofvelocity amplitudes with constant phases such that
tjeabxh=t)w(x, )]([ 00 + (7)
The total lift L=Lc+Lnc is directly obtained by theintegration of the pressure-jump long the airfoil
b
b
dxtxp=tL ),()( (8)
with p~ ,for each harmonic component of the onset
flow, given by [8]
)(++
tgaV=(p 2sin2asin2a
22),~
210
(9)
where the variable [0,] is such that = cosbx
whereas coefficients 210 a,a,a are expressed as
)ww(jk
=a
wjk
+ww=a
w+wwC(k)=a
c
cc
c
2/4/3c2
2/2/4/3c1
2/4/3c4/3c0
~~
4
~
2
~~
~~~
(10)
An alternative approach to predict the unsteady loads
on an airfoil passing through a spatially non-uniformflow is that proposed by Sears [8] where the upwash
distribution, at any point of the airfoil and for a given
frequency of the multi-harmonic onset flow, may beexpressed as
( )VxtjWe=t)w(x, / (11)
stating that the sinusoidal gust pattern, with constant
amplitude W, moves past the airfoil with the speed V.Assuming that the wave-length of the gust is l,
lV/2 denotes the frequencywith which the wavepasses the airfoil. Following [8], the unsteady lift maybe written as
(k)cVWe=L tj (12)
where )(k denotes the Sears function representing
the frequency response of the lift to the gust. The
lifting force Lmay be expressed as in (8) where, foreach frequency of the wake-field, the pressure-
jump along the chord is
( )2
2),(~ 10
+= tgAAVWp (13)
with coefficients 0A and 1A given by
[ ]
)(
)()()(
11
100
kjJA
kjJkJkCA
=
= (14)
being (k)Jn the Bessel functions of the first kind. As
shown from (11), the Sears theory represents theupwash velocity-field on the airfoil as a combination
of chordwise constant distribution of velocityamplitudes with linear phases. Thus, differently from
the Theodorsen model, it is able to model (linearly)the inflow phase shift between points on the airfoil-
chord when a non-uniform flow is encountered. Notethat, in terms of jump-pressure, the blade response tothe uniform mean axial-velocity is obtained directly
from (9) or (13) by imposing k=0: as expected, for the
steady-state condition, both the Theodorsen and Searsformulations reduce to the Glauert theory. In thefollowing, Theodorsen and Sears models are applied
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by dividing the mean-surface of the propeller bladeinto a discrete number of sections and applying (9) or
(13) to each section, for each harmonic component ofthe inflow velocity. Further, in the time-domain, the
global thrust recasts
drdxtxpN=tTR
r
b
b
b
boss
))(,()( in (15)
where n is the outward unit normal to the meansurface, i denotes the unit vector aligned to thedirection orthogonal to the disk-plane
whereasb
N represents the number of propeller blades.
NUMERICAL RESULTS
In this section the hydrodynamic performance of a
marine propeller operating in a multi-harmonicswake-field is examined in order to validate thepresented solvers. Specifically, the hydrodynamic
predictions obtained through the proposed approachesare compared with those given by the fully three-dimensional, time marching, BEM hydrodynamicsolver. To this aim, the PROP 3714 four-bladed
propeller developed at the David Taylor ShipResearch and Development Center (DTRS) working
in a study-case non-uniform inflow characterized by a
high-frequency content, is considered. The diameterof the propeller isD=0.254m; the operating conditions
are defined by a rotational speed n=50rps, a free-stream speed V=12.7m/sand an advance ratioJ=1. A
detailed geometric description of the DTRC 3714model propeller is presented in [18]. Figure 1 showsthe azimuthal variation of the study-case onset flow uat different radial positions on the propeller disk. A
Fourier analysis (not shown here) shows that thespectrum of this wake-field contains up to 90
harmonics; moreover, the spatial distribution is suchthat both the velocity peaks and harmonic content
increase toward the end of the disk.
Fig. 1 Axial onset flow velocity
Although a typical wake-field encountered by apropeller at each radial location has different features
from that herein used and, may be reasonably definedthrough the first eight/ten harmonics [19], the high
frequency content of the inflow shown in Fig.1 allows
to stress the behavior of the proposed hydrodynamicmodels respect to the flow nonuniformity. Hence, in
the following, the blade response to the uniform meanaxial-velocity is neglected. The capabilities of the
three methodologies in capturing the unsteady bladeloads is first investigated by analyzing the response of
a single-blade 3714 propeller to an axial-inflowderived from the study-case inflow using the first 4, 8,
12 and 16 harmonics, respectively. To this aim thethrust coefficient predicted by the Nakatake, Sears and
Theodorsen approaches is compared with thatevaluated by BEM hydrodynamics. The tuning
process of the Nakatake formulation has been donethrough the open-water propeller performance givenby BEM computations.For the 4-harmonics inflow, Fig. 2 shows that the
thrust signal predicted by Theodorsen almost perfectly
match that obtained using the Sears theory. Withrespect to BEM results, the trend of the wave-form is
well captured, especially in terms of signal-phase;relevant discrepancies, in the signal-peaks, are
highlighted at the first quarter and at third-quarter ofthe blade revolution. On the contrary, the agreement
with BEM computations is good about at half and at
the end of the blade revolution, corresponding to thoseazimuthal positions where the blade encounters largevariations in the incoming flow (see Fig.1).
In these regions, numerical results yielded by theNakatake model exhibit signal oscillations not shown
neither in BEM results nor in Theodorsen/Sears
outcomes; however, at the first quarter and at third-quarter of the blade revolution, the agreement with
BEM computations is slightly better than that obtainedthrough the Theodorsen/Sears models.
Fig. 2 Blade response to the first 4 harmonics inflow
This behavior is also confirmed by the analysis of Fig.
3, showing the blade response to the first 8-harmonics
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10thInternational Conference on HydrodynamicsOctober 1-4, 2012 St. Petersburg, Russia
inflow. By increasing the harmonic content of theincoming inflow, relevant discrepancies arise between
Theodorsen and Sears predictions. This is well shownin Figs. 4-5 depicting the blade response to the first 12
and 16-harmonics inflow, respectively. Such
discrepancies consist in oscillations of the signalpredicted by the Theodorsen model with respect to
that given by the Sears formulation; the greater is theharmonics content of the inflow, the greater are the
signal oscillations. Amplitude oscillations are larger athalf and at the end of the blade revolution, that is in
those regions where higher circumferential gradient ofthe inflow velocity are present. This behavior is due to
the fact that the Theodorsen theory describes theupwash velocity on an airfoil encountering a gust by
combining basic (harmonic) motions havingchordwise linear distribution of velocity amplitudes,
with constant phases; in this way, each point of theairfoil is forced to experience the same signalsimultaneously.
Fig. 3Blade response to first 8 harmonics inflow
Fig. 4Blade response to the first 12 harmonics inflow
Conversely, the Sears theory accounts for a chordwiselinear variation of the upwash phase, so that it models
(somehow) an airfoil progressively encountering aspatially, non-uniform flow. The two theories yield
comparable results until the sectional chord-length issmaller than, or comparable with, the minimum
wavelength associated with the incoming inflowharmonic components.
Fig. 5Blade response to the first 16 harmonics inflow
For a blade section located at r/R = 0.75, Fig. 6shows
the ratio between the minimum wavelengthassociated with the onset-flow and the local chord-
length, for the 4, 8, 12, 16-harmonics inflow,
respectively. As expected, is greater than one forthe 4 and 8-harmonics inflow whilst it becomes lessthan one for inflow velocity with higher frequency
content. This result states that the Theodorsen theorymay be used to study the unsteady response of anairfoil travelling into a non-uniform flow until thewavelength associated with the (relevant) higher
frequencies upwash components is greater, or at least,comparable to local chord-length (low reduced
frequency).
Fig. 6 Minimum Wavelength-chord ratio at r/R = 0.75,
for different harmonic contents of the onset flow
Furthermore, Figs.4-5 confirm the ability of theNakatake approach to capture, better than the Sears
model, the load-peaks at the first quarter and at third-quarter of the blade revolution, thus yielding a better
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agreement with BEM computations. However, byincreasing the harmonics content of the inflow-
velocity, a worsening of the results in terms of bothpeaks and phase signal is highlighted. An in-depth
analysis of the correlations among hydrodynamic
predictions is given in Fig.7 that depicts the spectrumof the thrust signal for the 16-harmonics inflow. As
shown, for the 1st and 2
nd harmonic, the Nakatake
model yields amplitudes that are closer to BEM
results than Theodorsen/Sears outcomes. For higherharmonics, the Sears formulation yields predictions
closer to BEM results almost throughout the wholespectrum. This fact confirms the difficulty of the
Nakatake theory in capturing local high-frequencychange of the onset flow. The load spectrum also
points out that the Theodorsen theory is reliable up tothe 3/rev frequency since for higher values the
amplitudes of the response tend to diverge. All theaspects discussed above, may be well observed inFig.8 where results from Theodorsen, Sears andNakatake approaches are compared with BEM
outcomes for the inflow velocity characterized by 90
harmonics.
Fig. 7 Blade response spectrum at the first 16 harmonics
inflow
Fig. 8Blade response to 90 harmonics inflow
As shown, results from the Theodorsen formulation
appear to be completely unreliable; Sears
computations well capture the unsteady thrust atazimuthal regions where the blade encounters largevariations in the incoming flow, albeit some lower
level of agreement is observed out of these regions;results from the Nakatake model, indeed, exhibits a
greater level of accuracy in those regions where theinflow velocity is more regular with lower gradients.
In the framework of the Sears approach, the more
accurate is the evaluation of the upwash and the morerealistic are the hydroloads predicted: computations
previously shown, do not take into account for theupwash induced by the blade-wake vorticity, in that
only the close shed vorticity, generated by theexamined section, is modeled in the airfoil theory
[7],[17]. However, these three-dimensional (3D)effects are relevant especially for low advance-ratio
where the wake is closer to the propeller disk. On thecontrary, the Nakatake model intrinsically accounts
for the presence the vorticity shed downstream thepropeller plane by a set of trailed-vortices. [5]. The
importance of these free vortices is highlighted inFig.9 that shows the comparison between the thrustpredicted by BEM hydrodynamics and the Nakatakeformulation, with and without 3D effects, for the 90
harmonics-inflow. As expected, the better agreement
with BEM results is obtained including 3D-effects.Further, Fig.10 shows that outcomes from the
Nakatake model without 3D effects are closer to thosepredicted by the Sears theory in terms of signal-peaks,
even if relevant discrepancies remain in those regionswhere the incoming flow is much irregular. In view of
these preliminary results, an enhanced propeller
modeling, suitable for treating both high-frequencyonset flow and three-dimensional induced effects,might be obtained from the Sears theory including a
more detailed description of the unsteady airfoilupwash.
Fig. 9 Behavior of Nakatake thrust prediction with and
without 3D effects
Finally, Fig.11 depicts the thrust coefficient, for thefour-bladed propeller subjected to the 90 harmonicsinflow, computed by BEM, Sears and Nakatakeformulations. The spectrum of the signal, limited to
the first 32 harmonics, is shown in Fig.12. Akin to the
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10thInternational Conference on HydrodynamicsOctober 1-4, 2012 St. Petersburg, Russia
wavelength is less than the local chord-length;differently, Theodorsen results are completely
unreliable. On the contrary, Sears results match quitewell BEM-computations in terms of signals trend,
especially in those azimuthal regions where the inflow
is much irregular. This behavior highlights thecapability of the Sears model to capture the
hydrodynamic blade response to higher frequencies.However, signal peaks are overpredicted respect to
BEM results. Nakatake computations denote theattitude of such a model to capture the hydrodynamic
response components at low-frequency better thanSears; however, a noticeable worsening of the results
arises at higher frequency. The strength of theNakatake model consists of its ability to accounts for
some 3D effects through the set of vortices shedrearwards; however, it requires an initial tuning
procedure based on open-water propeller performanceprovided by experiments or accurate computations.On the contrary, except for the contribution describedin the airfoil theory, Sears methodology is not able to
model the hydrodynamic effects from the blade-wake
shed rearwards; these information could be includedinto the analysis through advanced hydrodynamic
solvers yielding a more accurate description of theupwash field on the mean-blade surface. Both
approaches take few seconds to perform a calculationsso, in view of CFD applications, they are
computationally cheap. In this context the major
difference to be highlighted is the different body-forcedistribution that can be derived by the two simplifiedmethodologies: Nakatake computations would provide
an angular load distribution that does not accountproperly for the presence of a multi-bladed propeller,
as the Sears approach does. This aspect should be
considered in design or off-design maneuvers wherethe unsteadiness involved may have a high frequency
content so that the actuator-disk modeling may furnishinaccurate results. All these aspects will be addressed
in future investigations.
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