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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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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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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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