Ocean Engineering 38 (2011) 1582–1599

Contents lists available at ScienceDirect

Ocean Engineering

0029-80

doi:10.1

� Corr

E-m

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

Numerical simulations of a fully submerged propeller subject to ventilation

A. Califano, S. Steen �

Department of Marine Technology, Rolls-Royce University Technology Centre ‘‘Performance in a seaway’’, Norwegian University of Science and Technology,

N-7491 Trondheim, Norway

a r t i c l e i n f o

Article history:

Received 31 January 2011

Accepted 10 July 2011

Editor-in-Chief: A.I. Incecikheavy sea states, experiencing continuous cycles in and out-of water. This leads to sudden thrust losses

and violent impact loads, which can damage shaft bearings and gears of azimuth and tunnel thrusters.

Available online 30 August 2011

Keywords:

Ventilation

Marine propeller

Dynamic loads

CFD

RANS

Tip vortex

18/$ - see front matter & 2011 Elsevier Ltd. A

016/j.oceaneng.2011.07.010

esponding author. Tel.: þ47 73 59 58 61; fax

ail address: sverre.steen@ntnu.no (S. Steen).

a b s t r a c t

Numerical simulations aimed at modeling the phenomenon of ventilation on a fully submerged

propeller were performed. Ventilation occurs on thruster propellers operating at high loadings and

Damages in rough seas were reported also during transit operations.

In the simulated configuration the propeller is fully submerged (h/R¼1.4) and working at high

loading (J¼0.1), where the blade becomes surface-piercing only after ventilation occurs.

The dynamic loads computed with the numerical model are in satisfactory agreement with the

experimental data at the upright position where the blade is piercing the free-surface, whereas thrust is

over-estimated at all the other angular positions. A thorough analysis of the causes of this deviation

was performed, identifying the inability of the numerical simulation to properly resolve the tip vortex

at some distance from the propeller blades as the most likely responsible factor. Unlike ventilation of

surface-piercing propellers with super-cavitating profile, it was found that the tip vortex plays an

important role in ventilation of conventional propellers, which is the object of the present study.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In the last 40 years, ship operations offshore have increasedfollowing an improved technology which allowed drilling in everdeeper waters. As a consequence, traditional position-keepingmethods such as jack-up barges and anchoring systems becameinadequate for those depths, leaving room to dynamic positioning(DP) systems. Propellers might be required to operate at very highloadings by the DP system in order to maintain a vessel’s positionand heading in heavy sea states, where thrusters can experiencecontinuous cycles of water exit and re-entry.

In these conditions, a number of accidents with damages to thelower bevel gear and propeller shaft bearings of azimuth and tunnelthrusters have been reported, causing service downtime and requir-ing costly repairs. Damages in rough seas were reported also duringtransit operations. Based on the analysis of the broken gear wheels,damages are identified as a Tooth Interior Fatigue Fracture (TIFF),which is a failure mode believed to be initiated as a fatigue crack inthe interior of the tooth of a gear (MackAldener and Olsson, 2000,2002). MackAldener and Olsson (2000) pointed out that the mechan-ical driving forces for the crack are twofold: (i) a constant residualtensile stress in the interior of the tooth due to case hardeningand (ii) alternating stresses due to the idler usage (gears with teeth

ll rights reserved.

: þ47 73 59 55 28.

loaded on both their flanks during each revolution). Although marinepropellers are not subject to idler usage (ii), alternating stresses canarise from excessive torsional vibrations, causing meshing gearsto loose contact and re-engage with considerable energy-impact(gear hammering).

Large torques and sudden variations of the load conditions canbe caused by intermittent ventilation, which occurs on thrustersexperiencing continuous cycles of water exit and re-entry duringsevere wave–vessel interactions. This leads to sudden thrustlosses and violent impact loads. Ventilation has been observedon fully submerged propellers operating at low advance speedand high loadings, thus the propeller does not necessarily requireto be surface-piercing for ventilation to occur (Koushan, 2006b;Califano and Steen, 2009; Kozlowska et al., 2009).

Thrust losses due to ventilation are traditionally accounted forusing semi-empirical methods, but there is little knowledge onhow to calculate the dynamic loads and the underpinningphysical phenomena. A better knowledge of the mechanismsleading to ventilation is needed in order to identify properoperational strategies and active control systems to reduce thedamaging load variations on the propellers.

1.1. Historical background

Propeller ventilation has been historically related to surface-piercing, partially submerged propellers, which were first employedon shallow draught ships, and in a second stage for high-speed craft,

Nomenclature

Acronyms

AKPA AK propulsor analysisBEM boundary element methodDP dynamic positioningHRIC high resolution interface capturingMRF multiple reference frameMARINTEK Norwegian Marine Technology Research InstituteRANS Reynolds-averaged Navier-Stokes equationsSM sliding meshSIMPLE semi-implicit method for pressure-linked equationsTIFF tooth interior fatigue fractureURF under-relaxation factorVOF volume of fluid

Greek letters

a under-relaxation factorb reduced thrust (KT=KT0

)Dp pressure jump across the propeller disc (Pa)Dp=p0 Propeller loading (%)Dt time-step size (s)Dx grid discretization (m)t propeller efficiency (J=2pKT=KQ )g circulation (m2/s)g heat-specific ratio (cp=cV )gair air-volume fractionl scale factor between lengths in model and full scale

(Ds=Dm)n kinematic viscosity (m2/s)r mass density (kg/m3)s standard deviation

Latin letters

A Coefficient for the discrete velocity in the SIMPLEalgorithm

a sound speed (m/s)c specific heat capacity (J/kg K)c0.7 chord length at 70% of the radius (m)Co Courant number (DtUc=Lc)D propeller diameter (m)Dhub Hub diameter (m)E bulk modulus (Pa)h shaft submergence (m)

h/R submergence-to-radius ratioJ advance ratio (U=nD)KQ torque coefficient (Q=rn2D5)KT thrust coefficient (T=rn2D4)KT0

nominal thrust coefficientL length (m)M Mach number (U/a)n shaft frequency (1/s)p pressure (Pa)p0 atmospheric pressure (Pa)Q torque (N m)R ideal gas constant (8.314 J/K mol)R radius (m)Re0.7 Reynolds number computed at 70% of the radius

(0:7pnD � c0:7=n)T absolute temperature (K)U free-stream velocity (m/s)x longitudinal coordinate (m)y lateral coordinate (m)z vertical coordinate (m)

Operators

@ partial derivative� vertical average (along index j)

Super–Subscript

0 correction variablen initial estimatea stagnation pointc cell indexF free vorticesg phase fractiong gaseous phasei index prescribing x directionj index prescribing y directionk current time level‘ index prescribing x, y and z directions (‘¼ 1,2,3)l liquid phaseP constant pressurep pressuretip tip of the bladeu velocityV constant volume

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–1599 1583

with super-cavitating-type profile. It is only recently that ventilationof conventional thrusters has gained much attention, due to theincreasing demand of offshore vessels and the new challengesencountered.

Shiba (1953) has carried out a comprehensive experimentalstudy of propeller ventilation, including sections with differentprofiles and analyzing the various parameters affecting thephenomenon. Later, during the 1970s, propulsion in a seawayand the related average loss of thrust and efficiency were studiedquite extensively in Germany (Gutsche, 1967; Fleischer, 1973)and in Norway (Faltinsen et al., 1981; Minsaas et al., 1983, 1987).The effort made in understanding ventilation led to modeling thetime-averaged reduced thrust b¼ KT=KT0

as a function of thesubmergence-to-radius ratio h=R, by means of the loss of discarea (Gutsche, 1967) and further including the losses due to the

Wagner effect (Minsaas et al., 1983). In the expression of b,KT ¼ T=rn2D4 is the measured thrust coefficient, whereas KT0

isthe non-ventilating, nominal value. More recently, Koushan per-formed experiments and measured the dynamic loads of a venti-lated propeller in open water (Koushan, 2006b) and withthe presence of a duct (Koushan, 2006a), taking into accountthe influence of factors normally encountered in a seaway, suchas waves and thruster azimuth angle (Koushan, 2006c, 2007a,b), inaddition to those commonly used (submergence and advance ratio).

Due to its nature being inherently non-linear and time depen-dent, the numerical modeling of ventilation is a difficult task. Thepresence of air cavities, spray and waves makes the mathematicalformulation of the phenomenon a real challenge. Since the begin-ning of the 1960s, modeling of thrust losses due to ventilation hasbeen attempted modifying ad hoc existing methods, such as blade

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–15991584

element method, lifting-line and lifting-surface theory. Morerecently, a three-dimensional boundary element method wasextended by Young and Kinnas (2004) to predict the unsteadyperformance of surface-piercing propellers during ventilation. Thismethod accounts for the exact cavity detachment location on thesuction side by means of an implemented search algorithm.

Although progresses were achieved toward the modeling ofpropellers piercing the free-surface, all methods present severalshortcomings related to the assumptions they are based on, limitingtheir validity to the global forces or the particular propeller object ofthe study. It can be seen that a more general purpose model isneeded to predict the dynamic loads occurring during ventilation, inall possible flow regimes. The first known work attempting themodeling of surface-piercing propellers using RANS was performedby Caponnetto (2003). He carried out numerical simulations of asurface-piercing propeller with super-cavitating profile, obtaining agood agreement with the experiments of Olofsson (1996), in termsof blade forces during a rotation cycle.

It should be mentioned here the main differences between thepresent study and those extensively performed on surface-pier-cing propellers with super-cavitating profile, e.g. Olofsson (1996):

Fig. 1. Propeller drawing.

(i) the propeller is fully submerged and may become surface-piercing only when ventilation occurs;

(ii)

the blade sections are not of the super-cavitating type (sharpleading edge and thick abrupt trailing edge), but designedwith a conventional lifting foil profile (blunt leading edge andsharp trailing edge);

(iii)

high propeller loadings were investigated.

As a consequence, the present study will have distinctive featuresnot present during ventilation of surface-piercing propellers withsuper-cavitating profiles:

(I)

ventilation must be triggered by some event (ventilationinception);

(II)

the blunt leading edge will not work as a sharp interfaceseparating the gaseous phase on the suction side from theliquid phase on the pressure side;

(III)

strong non-linearities are present in the tip region.

A first attempt to model ventilation of a fully submerged propellerby means of RANS methods has been performed by Califano andSteen (2009), obtaining a good agreement with the experimentalresults only for the most severe thrust losses. The present studyextends this previous analysis, focusing on the numerical imple-mentation of the physical phenomena underpinning propellerventilation, with special emphasis to the role played by the tipvortex.

2. Experiments

Model tests were performed in order to better understand thedynamic forces due to ventilation, and ultimately predict thecorresponding losses. The complete set of experiments performedis available in Califano (2010) while Califano and Steen (in press)focus on the relation between dynamic loads and ventilationregimes.

Tests were conducted at submergence ratios h/R rangingbetween 2.97 (identified as ‘‘infinite fluid case’’) and 1 (wherethe blade tip is touching the free-surface). For all the above waterdepths, the carriage speed U and the propeller shaft frequency n

were combined in order to obtain advance ratios J¼U/n D around0.1. The obtained results are presented in terms of the bladethrust. Statistics are computed over loads obtained at each

angular position during different revolutions, in terms of meanvalues and standard deviation s.

Only a summary of the obtained results will be given in thissection, needed to introduce the numerical simulations. Furtherdetails about the experimental set-up, the test matrix, and theprocedure used to analyse the data can be found in Califano andSteen (in press).

2.1. Propeller model

The propeller model has been extensively used for variouskinds of ventilation tests, with and without the presence of a duct.The model has thus a generic design, representing a typicalpropeller which can be used in different regimes. The propellerhas a diameter D of 0.25 m and a hub diameter Dhub of 0.06 m, andis right handed when mounted on a pulling thruster. Design pitchratio P/D is 1.1 and the blade area ratio EAR is 0.595. A propellerdrawing is depicted in Fig. 1 and the section characteristicspresented in Table 1, where c/D, t/D, s/D, P/D and f/D are,respectively, the chord, maximum thickness, skew, pitch andmaximum camber for each section, made dimensionless withthe propeller diameter. The rake is zero for all the sections.

A cartesian reference system is centered in the center of thepropeller, having the x-axis aligned along the propeller axis, thez-axis pointing upward towards the free-surface, and the y-axisfollowing a right-handed system, pointing on the portside (Fig. 2).

2.2. Results

Experimental results are summarized in Fig. 3, where the meanthrust ratio has been plotted as a function of the submergence ratio,having as parameters the advance ratio and propeller loadingDp=p0%, written in terms of percent of pressure jump across thepropeller disc Dp divided by the atmospheric pressure p0.

Three main regimes can be identified:

�

h=R42: at deep submergence all curves are overlappedand thrust losses occur only due to the proximity of thefree-surface. � h=Ro1:5: at low submergence curves are grouped solely by the

propeller loading. In this region, tip vortex is the dominating

Table 1Section characteristics.

r/R c/D t/D s/D P/D f/D

0.24 0.13 0.038 0.000 1.08 0.001

0.26 0.15 0.037 0.003 1.08 0.004

0.30 0.18 0.035 0.011 1.08 0.007

0.37 0.23 0.031 0.023 1.09 0.009

0.46 0.29 0.026 0.037 1.09 0.012

0.57 0.34 0.022 0.045 1.10 0.013

0.67 0.38 0.017 0.040 1.10 0.014

0.78 0.40 0.013 0.014 1.09 0.012

0.87 0.38 0.010 �0.030 1.06 0.010

0.94 0.32 0.008 �0.082 1.00 0.006

0.98 0.21 0.006 �0.125 0.95 0.003

1.00 0.03 0.006 �0.141 0.94 0.000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

Δ

0.133 1.70.117 1.70.100 1.80.114 2.40.100 2.40.086 2.40.100 3.10.088 3.20.075 3.2

2.52.42.32.22.121.91.81.71.61.51.41.31.21.1

Fig. 3. Mean thrust ratio as a function of the submergence ratio, having the

advance ratio and propeller loading as parameters.

Figsub

Fig. 2. Reference system.

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–1599 1585

ventilation mechanism, leading to very high thrust losses.Further reducing the submergence (h=Ro1), thrust would con-tinue to fall following the reduction of the submerged disc area.

� 1:5oh=Ro2: at intermediate submergence curves are still

grouped by the propeller loading, but the spreading is larger,due to the inherently unstable and random nature of thefree-surface vortex affecting ventilation in this range.

The average thrust ratio as a function of the blade angularposition for the case at J¼0.1 and n¼14 Hz is shown in Fig. 4,having the submergence ratio as parameter. The effect ofsubmergence and the characteristics of different regimes can bebetter observed through the envelopes of the thrust ratio as afunction of the blade angular position, normalized with respect tothe non-ventilating value:

�

.m

Free-surface vortex at deep submergence (Fig. 5)characterized by severe and discontinuous thrust lossesoccurring when a free-surface vortex reaches the blade’ssurface; the amplitude during a ventilation event candeviate significantly from the mean value, which isslightly lower than the nominal one.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 π2 π 3

2 π 2π

Blade angular position [rad]

h/R

Intermediate

Free-surface vortex

Tip vortex

1.801.721.64

1.56

1.481.401.321.241.00

4. Thrust ratio as a function of the blade angular position, having the

ergence ratio as a parameter (J¼0.1, n¼14 Hz).

Fig. 5. Thrust ratio (h/R¼1.8; J¼0.1, Dp=p0 ¼ 2:4%).

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–15991586

�

Tip vortex at moderate submergence (Fig. 6)characterized by uniform thrust losses during the completerevolution; the thrust encompasses a narrow amplitude

Fig. 6. Thrust ratio (h/R¼1.4; J¼0.1, Dp=p0 ¼ 2:4%).

Fig. 7. Thrust ratio (h/R¼1.56; J¼0.1, Dp=p0 ¼ 2:4%).

Fig. 8. Numerical domain. (a) Ensemble view, (b) clo

range around the mean value, which is in turn significantlylower than the nominal one.

�

Intermediate (Fig. 7)where both types of ventilation coexist, one dominated byfree-surface vortex, and the other by the tip vortex; thethrust encompasses a broad and uniform amplitude rangeand the mean value is somewhere in between those foundin the previous two regimes.

In the same figures the mean curve and the curves obtainedadding and subtracting two standard deviations 2s have alsobeen plotted, giving an idea of the upper and lower envelope ofthe measured data.

3. Numerical method

The commercial RANS code Fluent has been used to solve theviscous, incompressible, two-phase (air and water) flow. Thissolver is used in a wide variety of CFD applications, for whichvalidation cases are documented (Fluent, 2006). The momentumequation is solved with a second-order upwind scheme. Thepressure–velocity coupling is achieved using a Semi-ImplicitMethod for Pressure-Linked Equations (SIMPLE) algorithm. Thefree-surface evolution is handled using an implicit formulation ofthe Volume Of Fluid (VOF) and the transport equation of thevolume of fraction is solved with a modified High ResolutionInterface Capturing (HRIC) discretization scheme (Muzaferijaet al., 1999). An explicit formulation of the VOF method was alsoattempted to assess the influence of the integration time-step.For this explicit formulation, a second-order temporal discreti-zation was not available in the used solver, and a more diffusivefirst-order implicit scheme was chosen in order to use a consis-tent approach for all the simulations. Surface tension is neglected,but its effect will be verified while discussing the obtainedresults.

Further details about the solver can be found in the Fluent(2006) manual (Fluent, 2006).

Grid: The grid is fully unstructured in the rotating domain,with a superimposed prismatic layer close to the walls, inorder to better capture the boundary layer (Rhee and Joshi,2006). Prisms are extruded upstream and downstream the rotat-ing domain, whereas the remaining cells are fully structured.A total of about 2.35 million cells was used, most of themlocated around the propeller and across the interface betweenthe two phases. A typical size for the cell at the free-surface is5 mm, 2% of the propeller diameter (Fig. 8(b)). An ensemble viewof the grid used on the domain’s boundaries is shown in Fig. 8(a).

se up around the free-surface.

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–1599 1587

A closer view of the mesh topology on the blade can be seen inFig. 9.

Boundary conditions: The undisturbed free-surface elevation isassigned both at the inlet and outlet boundaries. At the inlet thefree-stream velocity is also specified. A zero flux of all quantitiesis enforced across the top and bottom boundaries. A no-slipcondition is set on the walls.

Propeller rotation: The propeller geometry is embedded in acylindrical domain, as shown in Fig. 10. The rotation of thisdomain was achieved both with a Multiple Reference Frame(MRF) model and using Sliding Mesh (SM).

In the MRF model the propeller is fixed, while its rotation istaken into account using a local reference frame rotating at thedesired propeller rate. The corresponding equations of motion aremodified to incorporate the additional acceleration terms arisingfrom the use of a rotating reference frame. This approach is mostsuitable when the interaction between stationary and movingparts are quasi-steady.

For the present study, where the unsteadiness of the afore-mentioned interaction becomes important, a SM model has to beadopted, accounting for the relative motion of stationary androtating components. The increased accuracy is achieved at theexpense of a higher computational time.

Fig. 9. Mesh topology on the blade wall.

Fig. 10. Rotating domain.

4. Open water

Fig. 11 shows the results in terms of propeller non-dimensionalcharacteristics: thrust KT, torque KQ and efficiency Z:

KT ¼T

rn2D4, KQ ¼

Q

rn2D5, Z¼ J

2pKT

KQð1Þ

Present RANS results were obtained on a single blade boundedby two periodic boundaries, such that the flow through one ofthe boundary planes is computed using the flow conditions atthe fluid cell adjacent to the corresponding periodic boundary.Periodic boundaries are generally a good assumption when theflow is rather steady and uniform, and recirculating regionsare mainly directed as the propeller rotation. The SST k�omodel (Menter, 1994) is used for the turbulence closure, with alow-Reynolds treatment of the boundary layer.

The obtained results are compared with the available experi-ments and Boundary Element Method (BEM) computations.Experiments were carried out in the large cavitation tunnel atthe Marine Technology Center in Trondheim, Norway. The testsection diameter is 1.2 m and the precision error of the testresults is found to be smaller than 1% using a 95% confidenceinterval. The code AK-Propulsor Analysis (AKPA) was used forBEM computations. AKPA is a velocity-based source BEM withmodified trailing edge (Achkinadze and Krasilnikov, 2001) in useat the Norwegian Marine Technology Research Institute (MAR-INTEK) for the analysis of marine propulsors.

The thrust coefficient computed with RANS is in satisfactoryagreement with the available experiments for the whole range ofadvance ratios. For high propeller loadings, i.e. for low advanceratios, the deviation is within the precision limit, about 1% for J¼0.1,the case which has been further investigated including thefree-surface in the next section.

The torque coefficient shows a systematic under-estimation ofthe experimental data, which is also seen in the BEM. An error inthe torque coefficient has been widely documented in otherRANS simulations (Bulten and Oprea, 2005; Rhee and Joshi, 2006;Berchiche and Janson, 2008), where an over-prediction was encoun-tered instead. The Reynolds number computed at 70% of the radiusRe0:7 ¼ 0:7pnD � c0:7=n is 1.2�105, where the skin friction coefficientmay vary significantly, depending on whether the flow is laminar,turbulent or transition occurs along the chord-line, being thelatter case the most common. The under-estimation of the torquecoefficient could be explained by the fact that the solver, even for

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

J

RANSBEM

experiment

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Fig. 11. Dimensionless thrust, torque and efficiency in open water: comparison

between experiments (—), RANS (–�–) and BEM (–n–).

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–15991588

a turbulent incoming flow, considers the flow on the walls as laminarbelow a certain Re, having a lower skin friction coefficient and thusleading to a lower profile-drag and lower propeller-torque.

The obtained contours of the pressure coefficient are shown inFig. 12(a) and (b), for the suction and pressure side, respectively.Three levels of grid refinement in the external domain—

obtained from the total domain subtracting the prismatic bound-ary layer—and one level in the boundary layer were appliedwithout producing significant changes in the global forces andmoments. The pressure contours for the three levels of refinementobtained for J¼0.1 are plotted in Figs. 13 and 14 for thesuction and pressure side, respectively. A sharper capture of thepressure difference can be observed decreasing the mesh size.This sharper contours are particularly visible on the leading edgeand on the suction side, where a finer grid better captures thestrong gradients existing at the leading edge and in the tip-vortexregion.

In addition to the SST k�o, the standard k�o and the k�e(standard and realizable) turbulence models were tested without

Fig. 12. Pressure coefficient in open water (J

Fig. 13. Pressure contours on the suction side for three levels of refinement (J¼

Fig. 14. Pressure contours on the pressure side for three levels of refinement (J

finding significant differences with respect to the presented results.The turbulence intensity at the inlet is set to 1%, a rather low valuereproducing the undisturbed flow conditions of the cavitationtunnel. Simulations were also performed increasing the turbulenceintensity up to 10% and removing turbulence (viscosity is onlyaccounted for with its laminar component), obtaining, also in thesecases, no significant changes.

5. Fully ventilating propeller

The experimental results presented in Section 2 have shown alarge deviation among different revolutions within the same testcase, even for a fully ventilating propeller. Although the ventila-tion phenomenon is strongly unstable and time-dependent, somerecurring characteristics could be observed.

Numerical simulations were thus attempted with the aim toinvestigate these characteristics. A case of tip-vortex ventilationwas chosen (h/R¼1.4 and J¼0.1), where the resulting dynamic

¼0.1). (a) Suction side, (b) pressure side.

0.1). (a) Coarse (0.6 M cells), (b) base-line (1.5 M cells), (c) fine (3 M cells).

¼0.1). (a) Coarse (0.6 M cells), (b) base-line (1.5 M cells), (c) fine (3 M cells).

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–1599 1589

loads are more deterministic compared to those obtained duringventilation by free-surface vortex at deeper submergences.

5.1. Multiple reference frame

The free-surface deformation (gair ¼ 0:5) obtained using theMultiple Reference Frame (MRF) model is shown in Fig. 15, wherethe blade walls are colored with air-volume fraction. Although thephenomenon is inherently unsteady, the main features observedduring the experiments are captured by this model: air is suckeddown from the free-surface, covering the suction side of the blade tipat blade position angle p=4 rad with air. Due to the strongly non-linear recirculating velocity field existing on the tip at this highpropeller loadings (Greenberg, 1972), air is unable to escape down-stream and is convected from the propeller along its rotation.Residuals of air are visible on the following blades, at 3=4p and 5=4p.

The corresponding pressure coefficient contours are shown inFig. 16. The pressure on the suction (Fig. 16(a)) side is higher—withrespect to open water conditions (Fig. 12(a))—during the first halfrevolution, due to the presence of air. Differences in pressurebetween blades at different angular positions are barely visible onthe pressure side (Fig. 16(b)). The resulting thrust coefficient iswritten in Table 2, where the losses due to ventilation can be read.These are generally smaller than those found in the experiments,where thrust losses up to 60% of the infinite fluid case are obtained.

Fig. 15. Free-surface deformation, blades colored with air-volume fraction (h/R¼1.4;

MRF model).

Fig. 16. Pressure coefficients contours during ventilation (h/

5.2. Sliding mesh

The MRF model has shown a good representation of theventilation phenomenon, although losses are under-estimated.In order to handle the unsteady nature of the phenomenon, aSliding Mesh (SM) model was adopted. After the deformation ofthe free-surface, the simulation shows an increase of the residualerror for the solved equations. This behavior would after someiterations lead to divergence of the numerical solution, whichcould be stabilized by reducing the Under-Relaxation Factors(URFs) ap, au‘ , ag, for the update of, respectively, the pressure p,velocities u‘ (‘¼ 1,2,3) and volume fraction g used by the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) scheme.There is no theoretical underpinning to suggest specific values tobe assigned the coefficients as, but values between 0.5 and0.8 usually work and are thus widely used. Smaller values aresometimes needed in order to achieve a stable solution. Anoptimum relation between the URFs for velocity and pressureswas derived by Raithby (1979) and Ferziger and Peric (2002),based on the assumption that a steady solution is found iteratingfor an infinite time-step

ap ¼ 1�au‘ 𑼠1,2,3Þ ð2Þ

A stable solution for the present simulations was reached usingap ¼ 0:1, au‘ ¼ 0:2, ag ¼ 0:2, where, within each time-step, theresiduals of the numerical error are reduced by a factor of 20 inless than 40 inner iterations.

The obtained thrust ratio is shown in Fig. 17, for the singleblade (a) and the whole propeller (b). The propeller thrust lossesdue to ventilation are under-estimated by about 50% with respectto experimental data, but the unsteady approach applied with theSM model slightly improved the results obtained with the MRFmodel, shown in the same figure. The blade thrust is over-estimated at all angles, except around its upright position, wherethe agreement with the experimental results is satisfactory. Theimprovement obtained with respect to the MRF model can also beobserved around 0 rad.

R¼1.4; MRF model). (a) Suction side, (b) pressure side.

Table 2Blade’s thrust coefficient at four angular position and relative thrust loss with

respect to non-ventilating conditions (h/R¼1.4, MRF model).

Ang. position

(rad)

Pressure

side

D% Suction

side

D% Propeller D%

1=4p 0.15 �26.7 0.26 �32.0 0.41 �30.2

3=4p 0.13 �35.2 0.31 �21.3 0.43 �26.0

5=4p 0.17 �13.9 0.37 �5.8 0.54 �8.5

7=4p 0.19 �5.7 0.38 �1.9 0.57 �3.2

0

0.2

0.4

0.6

0.8

1

0 π2

π 32

π 2πangle [rad]

mean exp±2 exp

MRFSM

0

0.2

0.4

0.6

0.8

1

0 π2

π 32

π 2πangle [rad]

mean exp±2 exp

MRFSM

� �

Fig. 17. Thrust ratio averaged for each angular position (h/R¼1.4; ap ¼ 0:1, au‘ ¼ 0:2, ag ¼ 0:2).

Fig. 18. Pressure coefficient contours on the suction side at various angular positions (h/R¼1.4; ap ¼ 0:1, au‘ ¼ 0:2, ag ¼ 0:2). (a) 14p, (b) 0, (c) 7

4p.

Fig. 19. Pressure coefficient contours on the pressure side at various angular positions (h/R¼1.4; ap ¼ 0:1, au‘ ¼ 0:2, ag ¼ 0:2). (a) 74p, (b) 0, (c) 1

4p.

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–15991590

The effect of the presence of air is visible in Fig. 18, where thepressure coefficients on the suction side of the blade are shownfor the positions closer to the free-surface, in steps of p=4 rad. Airabove the free-surface on the propeller plane is visible with ashaded area above the blade. At these locations the tip of theblade shows contours of higher pressure, indicating the occur-rence of ventilation. The lowest levels are achieved at p=4, wherethe tip vortex can better entrain air from the free-surface(Califano and Steen, in press). The corresponding pressure coeffi-cients on the pressure side are shown in Fig. 19, where the bladetip at 0 and p=4 shows contours of lower pressure due toventilation.

Fig. 20 shows the pressure coefficient along the chord-line onthree radial stations—0.5, 0.7 and 0.9 of the radius—and along theradius on the blade’s axis line (Fig. 21). The plotted lines are splineinterpolations between the given points. All the stations present ageneral reduction of the pressure (in absolute value) with respectto the open water results, but only for the blades at 0 and p=4 radthis behavior appears remarkable. This reduction is stronger onthe suction side and most localized around the tip region, above

50% of the radius. At 90% of the radius (Fig. 20(c)) the blade at0 rad is subject to a pressure drop extending from the leadingedge to the mid-chord line, whereas at p=4 rad pressure hasdropped along the entire chord-line.

The details of the air-volume fraction around the tip regionand the leading edge are shown in Fig. 22, only for the bladeduring the first half revolution. Plotted lines are spline interpola-tions between the given points.

5.3. Surface tension

The contribution to the pressure due to surface tension isproportional to the curvature of the free-surface location and thusimplies the computation of a second-order derivative. This opera-tion performed on a tetrahedral mesh can lead to inaccuracy, andeventually solution instability. Fig. 23 shows a comparison of thethrust coefficient obtained with and without surface tension,where differences in terms of global loads between the twosimulations are very small. Its presence seems not to modify theobtained loads during ventilation, but it should be mentioned that

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

Trailing Edge

01/4 π1/2 π3/4 π

π5/4 π3/2 π7/4 π

open water

-3

-2

-1

0

1

2

3

4

5

6

7

8

90

1/4 π1/2 π3/4 π

π5/4 π3/2 π7/4 π

open water

Leading Edge mid-chord

Trailing EdgeLeading Edge mid-chord

Trailing EdgeLeading Edge mid-chord-3

-2

-1

0

1

2

3

4

5

6

7

8

90

1/4 π1/2 π3/4 π

π5/4 π3/2 π7/4 π

open water

Fig. 20. Pressure coefficients at various stations and angular positions (h/R¼1.4;

ap ¼ 0:05, au‘ ¼ 0:1, ag ¼ 0:1). (a) r/R¼0.5, (b) r/R¼0.7, (c) r/R¼0.9.

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

Tip

01/4 π1/2 π3/4 π

π5/4 π3/2 π7/4 π

open water

Hub 0.5 0.7 0.9

Fig. 21. Pressure coefficients along the blade axis (h/R¼1.4; ap ¼ 0:05, au‘ ¼ 0:1,

ag ¼ 0:1).

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

0.1

Suction Side

Pressure Side

01/4 π1/2 π3/4 π

π

0.2

0.1

0

0.1

0.2

Tip0.9

Suction Side

Pressure Side

01/4 π1/2 π3/4 ππ

Leading Edge

Fig. 22. Air-volume fraction at various stations and angular positions (h/R¼1.4;

ap ¼ 0:05, au‘ ¼ 0:1, ag ¼ 0:1). (a) r/R¼0.9, (b) axis.

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–1599 1591

simulations including surface tension are more unstable and tendpromptly to diverge. Its effect can be important in the formation/destruction mechanism of bubbles from the air-sheet sucked fromthe propeller and in the formation of the free-surface vortex.

5.4. Grid refinement

A systematic verification of the grid used to perform thecomputations could not be performed, due to very long computa-tional time, in the order of several weeks using a node with 16processors. The grid size was then chosen according to thesensitivity analysis performed in open water (Section 4), whichdoes not take into account the presence of the free-surface.

In order to assess the effect of the grid size on the free-surface,the grid was refined once only in that part of the domain wherethe free-surface was expected to be located, as shown in Fig. 24.With respect to a dynamic grid-refinement, this approach wasdeemed as more robust and less computationally expensive inthis case, where the location of the free-surface around the propeller

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–15991592

is rather constant in time. The choice of a dynamic grid-refinementwould have reduced the number of cells to refine, but certainlyincreased the operation of refinement and coarsening at each time-step. Fig. 25 examines the influence of the described local grid-refinement, comparing the obtained thrust coefficient with thedefault mesh. The solution obtained with the refined grid presentsa larger spreading, better approaching the experimental results. The

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time [s]

� = 0� = 0.07

Fig. 23. Effect of surface tension on the blade thrust ratio.

Fig. 24. Refined region.

0

0.2

0.4

0.6

0.8

1

0 π2

π 32 π 2π

angle[rad]

mean exp±2� exp

RANSrefinedRANS

Fig. 25. Thrust ratio averaged for each angular position, effect of local refin

improvements can be observed especially for the propeller thrust,indicating that the refined grid is able to better capture ventilationoccuring on the deeper submerged blades. However, overallimprovements are modest, and not such to fill the gap betweenthe numerics and the experimental results. Due to the numericaldifficulties connected to a simulation with a refined grid, in terms ofcomputational resources and tuning of the numerical parameters,this mesh sensitivity study is not fully comprehensive, and theindependence of the obtained solution cannot be ensured.

5.5. Integration time-step

The choice of the integration time-step can change thecomputational time drastically, thus care was taken to use thehighest value leading to a converged solution. The differencesobtained changing the time-step for the default, not-refined gridare plotted in Fig. 26, and a time-step of 1�10�4 s was chosen,corresponding to about half degree of rotation for n¼14 Hz. Thedifferences between the curves with Dt¼ 1� 10�4 and 5�10�5 sare barely visible and the first time instants computed furtherreducing the time-step seem also to follow the same trend. Theresults for those lower time-steps—Dt¼ 2� 10�5 and 1�10�6 s—are available for a shorter time interval due to the divergence ofthe numerical solution.

For the chosen time-step, there are only five cells exceedingthe cell Courant number of 40 required by the solver for a stablecalculation (Fluent, 2006). The Courant number is the ratio of thetime-step Dt to the characteristic convection time (Dx=u in 1D)which is required for a disturbance to be convected a distance Dx.For a generic 3D domain, the local cell Courant number is definedby the solver as

Co¼DtUc

Lcð3Þ

where Uc is the magnitude of the cell velocity and Lc is the minimumcell edge length. The Courant number computed in the interfaceregion between air and water is everywhere below 0.2, except in thetip-vortex region, where it increases up to values of 10. Co¼0.2 is anupper limit for the HRIC scheme in order to have a sharp repre-sentation of the free-surface (Muzaferija et al., 1999), thus asmearing of the interface in the tip-vortex region has to be expected.

The simulation with the lowest time-step was performedusing an explicit scheme for the VOF method, whereas all othertime-steps use an implicit scheme. For this explicit formulation, amaximum Co of 0.25 was imposed on the entire domain, limitingthe time-step to 1�10�6 s.

For the solution obtained using the refined grid, a time-step of5�10�5 s was chosen, and all the cells present a Co below 40.

0

0.2

0.4

0.6

0.8

1

0 π2

π 32 π 2π

angle[rad]

mean exp±2� exp

RANSrefinedRANS

ement (h/R¼1.4; ap ¼ 0:1, au‘ ¼ 0:2, ag ¼ 0:2). (a) Blade, (b) propeller.

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5time [s]

= 1 · 10−4s

= 5 · 10−5s

= 2 · 10−5s

explicit, = 1·10−6s

Fig. 26. Effect of the integration time-step on the blade thrust ratio.

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8time [s]

= 0.1, = 0.2, = 0.2= 0.05, = 0.05, = 0.1

= 0.1, = 0.05, = 0.2= 0.1, = 0.1, = 0.1

Fig. 27. Effect of the URFs on the blade thrust ratio.

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–1599 1593

Values of the time-step above 5�10�5 s are inadequate tocapture the characteristic convection time. For Dt¼ 1� 10�4 onlythree cells exceed Co¼40, but the number of cells with20oCoo40 is much larger with respect to the not-refined gridusing the same time-step size.

5.6. Under-relaxation factors

The default Under-Relaxation Factors (URF)s suggested by thesolver were reduced in order to stabilize the performed simula-tions. A further reduction was performed in order to test thesensitivity of the simulations to these parameters, in terms of

�

all the URFs to ap ¼ 0:05,au‘ ¼ 0:1,ag ¼ 0:1, � only the momentum equation URFs to au‘ ¼ 0:1, � only the volume fraction equation URF to ag ¼ 0:1.

Differences in terms of thrust losses among different simulationsare barely visible (Fig. 27). Changes in the residuals are also notsignificant.

Both the URFs and the integration time-step have shown toaffect strongly the obtained solution. Their effect can be analyzedwithin the discrete form of the variable updates of the SIMPLEscheme (McDonough, 2007), written in 2D for the u-momentum

equation at the next time level kþ1 on the grid coordinates i and j:

ukþ1i,j ¼ un

i,jþauDt

Aui,jDxðp0i,j�p0iþ1,jÞ ð4Þ

where ‘‘n’’ denotes an initial estimate, and ‘‘0’’ represents a correc-tion. Au

i,j is an additional factor due to the discretization of theSIMPLE scheme, depending on Dx, Dt and the vertical-velocityaverages ~u and ~v at the time level k, expressed as

Aui,j ¼ 1þ4

nDt

Dxþ

Dt

2Dxð ~uk

i,j� ~uki�1,jþ ~v

ki,jþ1� ~v

ki,jÞ ð5Þ

The combination of the URFs, the time-step and the grid size willdetermine in a non-trivial manner how the term with the pressurecorrection will contribute to the new velocity field.

There is a strong similarity between the algebraic equationsresulting from the use of under-relaxation when solving steadyproblems and those resulting from implicit Euler scheme appliedto unsteady equations. The following relation between the under-relaxation factor a and time-step Dt can be derived by requiringthat the contributions be same in both cases (Ferziger and Peric,2002):

Dtpa

1�aLc

Ucð6Þ

The use of a constant under-relaxation factor is equivalent toapplying a different time-step to each control volume.

6. Discussion

Although the agreement of the CFD results with the experi-ments is qualitatively good, being able to capture the occurrenceof thrust losses during the propeller rotation, numerical resultsgenerally under-estimate the thrust loss relative to theexperiments.

In order to understand the causes of this discrepancy, theinstantaneous flow field of the fully ventilated flow for thenumerical simulation and the experiments will be compared,starting from a qualitative point of view. The presence of air is anexcellent marker describing ventilation, and its content can becompared in Fig. 28. The free-surface visualized in the CFD results(Fig. 28(a)) is the location where gair ¼ 0:5.

The fully ventilated flow observed during experiments(Fig. 28(b)) is characterized by a thin sheet cavity covering thesuction side of the blades, extending radially from the tip toroughly half of the blade radius. An additional air-flow character-ized by small bubbles is superimposed, extending radially asmuch as the sheet cavities, and covering the entire propeller disc,including the blade-to-blade passage. This secondary bubbly flowis quasi-steady, while the sheet cavities follow the rotation of thepropeller. Comparing the results obtained with the numericalsimulation (Fig. 28(a)), although the air pattern is very similar,substantial differences can be observed:

�

Extension of the cavities

J Angular: The air initially drawn from the free-surface isconvected from the blade rotation only for half a revolu-tion. Only a small amount of air is able to follow partiallythe remaining half revolution (residuals of air are visibleon the blade at 3=2p). Most of it is instead being trans-ported downstream by the axial flow.

J Radial: The blade is covered by air only in proximity ofthe tip.

�

Bubbles

The rapid formation of bubbles observed during theexperiment is not visualized in the contours of air-volumefraction.

Fig. 28. Instantaneous flow field. (a) CFD: Contours of air-volume fraction. (b) Experiments.

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–15991594

The ventilation phenomenon simulated numerically is a sim-plified model of the complex physical phenomenon, and is based

on several assumptions, the most restrictive being incompressi-bility, the absence of turbulence and cavitation. Furthermore, thespatial discretization of the domain might neglect important flowfeatures, such as bubbles and discrete vortices. The following listenumerates the approximations which might possibly invalidatethe present numerical results. The listed parameters will then beanalyzed in detail in the remaining part of this section.

�

Simulation time. � Turbulence. � Cavitation. � Air loss. � Bubbly flow mechanics. � Compressibility. � Pressure drop.

Simulation time: The simulated time—only 1 s for most simu-lations, corresponding to about 15 revolutions—is much shorterthan the duration of an experimental test, but long enough toreach oscillatory, quasi-steady forces. This assumption is corro-borated by the fact that a steady-state flow pattern is achieved inless than 10 revolutions during experiments. Recent experiments(Kozlowska et al., 2011) have shown different ventilation modesoccurring during the same test, where the blade loads are movingfrom one level of quasi-steady oscillatory forces to a completelydifferent level, without changing any parameter in the set-up.This change occurs after several seconds, for a duration muchlonger than the one reached with a numerical simulation.Although simulating different ventilation modes is out of thescope of this study, it is possible that a simulation of much longertime could capture other flow modes.

Turbulence: All the results presented in Section 5 wereobtained using a laminar flow, without introducing a turbulencemodeling. This assumption was based on the fact that the free-surface deformation due to an attached flow over a lifting surfaceis due to the pressure forces exerted from the body, whileviscosity plays a marginal damping role. However, turbulence isimportant for flow separation, and it is not known a priori whatrole it is playing when ventilation occurs. A simulation includingturbulence was thus attempted, showing very little changes withrespect to the laminar solution.

Cavitation: The pressure on the blades of the propeller mightfall below the vapor pressure leading to cavitation. The propelleris specifically designed not to cavitate when fully submerged.Nevertheless, Nishiyama (1986) has shown that the low pressure

achieved at the core of vortical systems—tip-vortex or free-surface vortex—might induce cavitation on the blade surface.For the present fully ventilated case, ventilation should alwaysoccur before cavitation, since the vapor pressure leading tocavitation is lower than atmospheric pressure. This is question-able for a partially ventilated case, where a link might existbetween ventilation and cavitation (Kozlowska et al., 2009).

The possible cavitation inception can trigger bubble formationthrough a nucleation process and interact with the ongoingventilation. If occurring, this phenomenon is not captured withthe present formulation, whereas a cavitation model should beintroduced in order to account for

�

the formation and transport of vapor bubbles; � the turbulent fluctuations of pressure and velocity; � the magnitude of noncondensable gases, which are dissolved

or ingested in the operating liquid.

Singhal et al. (2002) have derived the phase-change rate expressionsfrom a reduced form of Rayleigh-Plesset equation for bubbledynamics. These rates depend upon local flow conditions (pressure,velocities, turbulence) as well as fluid properties (saturation pres-sure, densities, and surface tension). The phase-change rate expres-sions employ two empirical constants, which have been calibratedwith experimental data covering a very wide range of flow condi-tions, and do not require adjustments for different problems.

Air loss: Air mass-flow is conserved through the sliding interface,i.e. the air flowing through the interface rotating with the propelleris equal to the air flowing through the corresponding fixed surface.However, air bubbles smaller than the cell size might still be filteredout at the sliding interface. The air mass-flow-rate is also conserved:the flow rate of air entering and leaving the sliding interface over acertain simulation time is equal to the rate of change of aircontained in the rotating cylinder delimited by the sliding interfaces.

Bubbly flow: Experiments show—from the very beginning,when ventilation occurs—a rather uniform mixture of air andwater around the propeller disc, rather than two distinct phaseswith a sharp separation between air and water, as seen from theCFD calculations. The tiny bubbles seen in the experiments,characterized by a diameter of Oð10�2

Þ m, are not captured bythe numerics with the used discretization. The buoyancy forceacting on a bubble of diameter D is OðD3Þ, while the drag force isOðD2Þ. The larger bubbles simulated with CFD will thus be subjectto higher buoyancy-to-drag force ratio—compared to the smallbubbles filling the same enveloped volume. As a result, thesimulated entrapped air will rise quicker and dissolve faster thanin reality.

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–1599 1595

Looking at Fig. 28(a), the display of the interface with gair ¼ 0:5does not give a complete picture of the presence of air in thenumerical domain. The visualizations of air-volume fractionsbetween 0 and 0.5 (Fig. 29) show a larger domain with smallerair content, giving a final picture compatible with the presenceof bubbles observed during the experiments (Fig. 28(b)). Thisqualitative similarity does not allow us to draw a final conclusionabout the effect of these bubbles on the forces on the blade. Thepresence of air with gair o0:5 can be the result of the numericaldiffusion introduced by the VOF method, which tends to spatiallydiffuse the originally sharp air–water interface while the time ismarching.

On the other hand, the domain with gair o0:5 can be due to thephysical nucleation of air bubbles and following coalescence atthe high pressures achieved in proximity of the propeller. Bubbledynamics—breakup, coalescence and interactions—is stronglydependent (Clift et al., 1978; Brennen, 2005) on the mutual actionof surface tension and turbulence.

6.1. Compressibility

Compressibility was neglected throughout the present work. Thisis an established assumption for water, but could be too stringent

Fig. 29. Instantaneous flow field (0:01rgair r0:5).

Fig. 30. Contours of air-volume fraction and static pressure on the x–y plane cutting the

(the free-surface is displayed with gair ¼ 0:5). (b) Contours of static pressure (Pa).

within the gaseous phase. The region containing air was cut with anx–y plane passing through the propeller axis (Fig. 30(a)) and thecorresponding contours of pressure—subtracted from the atmo-spheric pressure—were displayed (Fig. 30(b)). Assuming an isen-tropic process, i.e. adiabatic and reversible, of an ideal gas, pressureand density can be expressed by the isentropic relationship:

p

rg ¼ constant ð7Þ

where g¼ cp=cV is the heat capacity ratio, which is equal to 1.4 for adiatomic ideal gas (air). cP and cV are the specific heat capacities of thegas, suffix P and V referring to constant pressure and constant volumeconditions, respectively. According to Eq. (7), a volume expansionwould follow a pressure decrease of an air bubble drawn down fromthe free-surface. The constant in the isentropic relation (Eq. (7)) canbe referred to the initial state in air above the free-surface in order todetermine the expansion of the drawn bubble:

p

rg ¼p0

rgair

ð8Þ

The maximum pressure-difference computed with CFD in the regionof air (Fig. 30(b)) is about 7000 Pa, which is small with respect to theatmospheric pressure (101 325 Pa). According to Eq. (8), the corre-sponding density inside this region will not differ significantly fromthe air density, and the expansion the air-volume would be subjectto—including compressibility—would be negligible. It will later bestressed that the pressure difference computed by CFD can be under-estimated within the core of the tip vortex. A larger pressuredifference might invalidate the assumption of incompressibility,leading to a volume expansion which cannot be neglected.

Full-scale pressure-differences will also be larger, as computedfrom the equality of the pressure coefficients in model (m) and full(s) scale:

ps�p012rU2

s

�pm�p012rU2

m

) ps�p0 ¼Us

Um

� �2

ðpm�p0Þ ¼ lðpm�p0Þ ð9Þ

Using a typical full-scale diameter of 4 m, the pressure difference infull scale would be l¼Ds=Dm ¼ 16 times the corresponding model-scale value, about 16� (�7000)¼ �112 000 Pa, thus larger thanthe atmospheric pressure in absolute value. It is clear that for thisobtained vacuum pressure compressibility will matter and the airdrawn by the full-scale propeller will be subject to an expansionleading to a stronger ventilation.

region with air. (a) View of the cutting plane with contours of air-volume fraction

Fig. 31. Contours of air-volume fraction on the x–y plane cutting the region

with air.

0

500

1000

1500

10−7 10−6 10−5 10−4 10−3 10−2 10−1 10

500

1000

1500

0

20

40

60

80

100

0.9

Air volume fraction

Sou

ndsp

eed

[m/s

]

airwater 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Fig. 32. Sound speed for the mixture phase computed using Eq. (16) (Wood, 1930).

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–15991596

In order to assess the assumption of incompressibility, we startwriting the Bernoulli equation for an inviscid compressible flow:

U2

2þ

gg�1

p

r¼ constant ð10Þ

The sound speed a can be written for an isentropic process:

a2 ¼dp

dr¼ g p

rð11Þ

We now assume that the entropy level is constant not only in time(isentropic), but also in space, i.e. the process is homentropic.Eq. (10) can be re-written as

U2

2þ

gg�1

p

r ¼g

g�1

pa

ra

ð12Þ

with pa and ra computed at a stagnation point, i.e. where thevelocity is zero. Using Eq. (11), the previous formula becomes

g�1

2U2þa2 ¼ a2

a ð13Þ

Dividing by a2 and introducing the Mach number M¼U/a a differentexpression for the Bernoulli equation can be found:

a2a

a2¼ 1þ

g�1

2M2 ð14Þ

Using the isentropic relation (Eq. (7)) and the ideal gas lawp=r¼RT (R and T are the ideal gas constant and the absolutetemperature, respectively), an analogous expression for pressureand density can be found:

pa

p¼ 1þ

g�1

2M2

� �g=g�1

ð15aÞ

ra

r ¼ 1þg�1

2M2

� �1=g�1

ð15bÞ

According to Eqs. (15), considering the flow as incompressible is areasonable assumption for M-0. An acceptable upper limit isM¼0.3, where it can be shown that the error obtained from theincompressible Bernoulli equation is about 2.3% (it is about 1% forM¼0.2). The maximum velocity attained around the propeller incorresponding open water conditions is about 12 m/s. Using the valueof 340 m/s for the sound speed in air, the corresponding Machnumber is 0.04, thus well below the upper limit for incompressibilityjust obtained.

This consideration was drawn assuming that the gaseous phaseis only constituted of air. The contours of air-volume fraction on thex–y plane cutting the region with air (Fig. 31) identifies a mixture ofair and water, where the air content at the center of this region isabout 80%. This mixture is rapidly formed after air is drawn andmixed with water due to propeller rotation. For such a mixture, thespeed of sound will be different from those of the single phases. Anexpression was derived by Wood (1930):

amixture ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiElEg

½ð1�gairÞrlþgairrg �½ð1�gairÞEgþgairEl�

sð16Þ

where E is the bulk modulus and subscripts l and g are referred tothe liquid and gaseous phase, respectively. Eq. (16) is plotted inFig. 32. For a wide range of the air-filling-ratio between 0.1 and0.9 the mixture sound speed is below 40, reaching at 0.5 theminimum value, around 24 m/s. Using the maximum velocity foundin open water, the Mach number can increase up to 0.5 for a 50% air-volume fraction. The corresponding error obtained using the incom-pressible Bernoulli equation is still modest, about 6.4%, but mightinvalidate the assumption of incompressibility.

6.2. Pressure drop

The thin cavity-sheet visible on the blade wall (Fig. 28(b)) suggeststhat the suction side is completely covered with air. In theseconditions, the minimum thrust measured in the experiments isabout one third of that measured in deep water, and it corresponds tothe thrust exerted only by the pressure side when fully submerged.

The corresponding picture from the numerical simulations(Fig. 28(a)) shows a suction side covered with air only partially,and computes subsequently a thrust higher than in the experi-ments. Two mechanisms are introduced in order to explain theunder-estimation of air content on the surface of the blade:

(I)

the amount of air reaching the blade is not sufficient; (II) the amount of air reaching the blade, although being sufficient,

remains confined around the tip, not being able to propagate tothe remaining surface of the blade.

Both mechanisms assert that the pressure drop is not sufficient tosuck enough air, respectively:

(i)

along the air channel connecting the blade with the free-surface;

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–1599 1597

(ii)

locally, on those locations of the surface of the blade notreached by the air.

Fig. 33. Path-lines in open water colored with the radial position (m).

The capture of the pressure field inside the surface of the blade isnormally achieved accurately with present RANS methods. How-ever, an inappropriate modeling of turbulence could lead to localerrors in the pressure field (ii). The flow on the blade is assumedcompletely laminar or turbulent, depending on whether turbu-lence modeling is implemented. In reality, a laminar-to-turbulenttransition will occur somewhere on the suction side, in thepresence of adverse pressure gradients. The location of thetransition would in turn modify the local pressure field onthe walls.

The under-estimation of the pressure locally in the computa-tional domain can be related to the mesh refinement and to thenumerical accuracy achieved. The unsteady nature of the phe-nomenon required an unsteady simulation over a truncatedcomputational domain sufficiently large to avoid unphysicalreflections from the boundaries. Although run in parallel, thistype of simulations requires long-time simulations. In this frame-work, a consistent mesh refinement study could not be performedand the uncertainty of the obtained results (Roache, 1997) couldnot be assessed. Nevertheless, in order to establish the differenceswith respect to the base-line grid, all the cells encompassing theventilated region were refined once halving the size of thesegments forming a cell. A comparison of the thrust coefficientobtained with the base-line and locally-refined grids was shownin Fig. 25. The trend achieved with the refined grid is very similar,and the differences obtained cannot explain the substantialdiscrepancy of the numerical simulations with respect to theexperiments. However, the mesh refinement was probably notadequate to resolve some local flow features where strongpressure gradients exist, such as at the leading edge of the bladeand in the core of the tip vortex, connecting the blade to the free-surface through an air channel (i). Once ventilation has started,the amount of drawn air will depend on the sectional area of thechannel and on the achieved pressure drop.

6.2.1. Tip vortex

The under-pressure computed in the tip vortex connecting theblade with the atmospheric pressure might be insufficient to allowmore air-mass-flow entering the blade surface. Califano and Steen(in press) have described the inception of ventilation for a moder-ately submerged propeller by means of the visualization of the tipvortex ‘‘breaking’’ a hole in the free-surface. This tip vortex plays stillan important role after the blade has become surface-piercing,acting as a channel continuously supplying air to the blade’s surface.The air mass-flow-rate through this channel will depend on itsdiameter and on the minimum pressure at the blade tip, where thetip vortex originates from. Accurate predictions of the vortex flowphenomena require a very fine mesh in the vortex core. The exactlocation of the vortex core and the level of its under-pressuredepend strongly on the mesh refinement, due to the high velocitygradients present in the flow (Chen, 2000; Bulten and Oprea, 2006;Li et al., 2006).

An estimation of the radius of the tip vortex Rtip can beobtained assuming a solid body rotation distribution within thecore of the vortex, in terms of the pressure in its center pð0Þ andthe circulation gtip:

Rtip ¼gtip

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir

p0�pð0Þ

rð17Þ

The total circulation of the tip vortex gtip can be computedsumming up the circulation of all the free vortices gF ðrÞ rollingup to form a concentrated tip vortex. The observation of thepath-lines along the blade in open water (Fig. 33) can give a rough

estimation of the radial location above which free vortices mergeto form the tip vortex, about r¼0.1 m (r/R¼0.8).

The circulation of each free vortex shed at a certain radialposition r can be obtained assuming a span-wise circulation GðrÞalong the blade:

gF ðrÞ ¼�@GðrÞ@r

dr ð18Þ

Using the span-wise circulation GðrÞ obtained using the BEMsolver AKPA, gtip was found to be equal to 0.13 m2/s. Introducingthe pressure computed from CFD calculations p(0)¼83 000 Pa,Eq. (17) gives Rtip 5 mm. With this value, one would expect thatthe vortex is properly described close to the blade, where the cellsize is much smaller than 1 mm.

Since the distance traveled by the vortex from the tip to thefree-surface is very short, we will assume for simplicity thatthe radius would remain constant along this path, neglecting theeffect of diffusion. In proximity of the undisturbed free-surface—

where the cell size is about 5 mm—the diameter of the tip vortexis distributed over only two grid cells, which seem inadequate tocapture the strong gradients occurring within the vortex core.

There is a mutual interaction between the tip vortex and thedrawn region with air. Since the circulation must remain constant(Kelvin’s theorem), vorticity must increase after stretching of the tipvortex toward the free-surface, leading to increased velocities in thevortex plane (Fig. 34). Vorticity amplification by vortex stretching is awell known phenomenon within a single phase, but it is not clearwhether the presence of air itself has the effect of increasing thevelocity field, reducing the density of the mixture phase. The fact thatthe Courant number in the region of the tip vortex largely exceeds thelimit of 0.2 for the HRIC scheme to properly resolve the air–waterinterface (Section 5.5) is a likely reason for the insufficient represen-tation of the tip vortex ventilation in the numerical simulation.

7. Conclusions

By means of numerical simulations the present work aimed atmodeling the phenomenon of propeller ventilation widely studied inmodel tests (Califano and Steen, in press). In the simulated config-uration the propeller is fully submerged (h/R¼1.4) and working athigh loading (J¼0.1), where the blade becomes surface-piercing

Fig. 34. Velocity vectors colored with velocity magnitude (m/s) on the x–y plane. (a) View of the port and starboard side.

A. Califano, S. Steen / Ocean Engineering 38 (2011) 1582–15991598

only after ventilation occurs. This condition is characterized byuniform thrust losses during the complete revolution, and the thrustencompasses a narrow amplitude range around the mean value,which is in turn significantly lower than the nominal one. Thisparticular case was chosen because the resulting dynamic loads aremore deterministic compared to those obtained during ventilationby free-surface vortex at deep submergences (Califano and Steen,in press).

The dynamic loads computed with the numerical model are insatisfactory agreement with the experimental data at the uprightposition where the blade is piercing the free-surface, whereas thrustis over-estimated at all the other angular positions. A thoroughanalysis of the causes of this deviation was performed, identifyingthe inability of the numerical simulation to properly resolve the tipvortex at some distance from the propeller blades as the most likelyresponsible factor. Unlike ventilation of surface-piercing propellerswith super-cavitating profile, it was found that the tip vortex has adominant role in the ventilation of conventional propellers.

Among other causes, the assumption of incompressibility wasfound to be too stringent within the ventilated region, consistingof a mixture of air and water, characterized by a density lowerthan in water and a speed of sound lower than in air and in wateralone. This assumption becomes even weaker in full scale,characterized by much lower pressures. In corresponding full-scale conditions, ventilation is expected to occur faster, with moresevere thrust losses, possibly leading also to cavitation. A scenariowhere both cavitation and ventilation coexist, and compressibilityof air cannot be neglected, would lead to a physical phenomenonmore complex than the one described in the previous sections.

Acknowledgments

The authors gratefully acknowledge the Rolls-Royce UniversityTechnology Center in Trondheim for supporting the present research.The support of the Norwegian HPC project NOTUR that grantedaccess to CPU time is also acknowledged.

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