owxtracttingusingvita-the

001

Francisco Pereirae-mail: f.pereira@insean.itPhone: 139 06 50299281

Francesco Salvatore

Fabio Di Felice

Istituto Nazionale per Studi ed Esperienze diArchitettura Navale,

Via di Vallerano, 139-00128 Rome, Italy

Measurement and Modeling ofPropeller Cavitation in UniformInflowAn experimental and theoretical investigation on a cavitating propeller in uniform inflis presented. Flow field investigations by advanced imaging techniques are used to equantitative information on the cavity extension. A refined map of the propeller cavitabehavior is established. Measurements are compared to numerical results obtainedan inviscid flow boundary element method for the analysis of blade partial sheet cation and supercavitation. The effect of the trailing wake vorticity on the prediction ofcavitation pattern is analyzed via a wake alignment technique.[DOI: 10.1115/1.1778716]

lf

a

y

-

ysar

e

n

a

hi

g9Ahipthefor

r ofn-thetheofallynd

,ns,avee of

m at isrmarymytions of-

ng-oth

nte-ive

od-lop-theen-ical

vy-alows.ter

IntroductionVisual observations and photographic recordings have been

are in many practical situations the common means to provid‘‘measure’’ of the cavitation pattern. With regard to the researchthe field, various methods are reported in the literature that almore accurate estimates of both the extension and volume ocavity shape. For instance, Dupont and Avellan@1# and Pereira@2#report precise measurements of the cavity length and thicknesisolated two-dimensional and three-dimensional hydrofoils, uscharge coupled devices~CCD’s! and laser light structured intoplane sheet. Pereira et al.@3# describe a fully three-dimensionamethod to measure the volume of the vapor structures as thegenerated by leading edge cavitation. In the case of cavitapropellers, the pin gauge method is the simplest and most imdiate approach, but highly intrusive as the pins also cavitate@4#.Lehman@5# seems to be the first to present nonintrusive quanttive measurements, using multiple orthographic views to buildestimate of the cavity volume on a model propeller. Using sreophotography, Sontvedt and Frivold@6# report volume measurements on model and full-scale propeller-hull configurations. Thfind large discrepancies, with an inaccuracy ranging from 30%100%. Working on model configurations, Ukon and Kurobe@7#devise a method where the spot of a laser beam scattered bcavity surface is detected and used to measure the thicknesextension of the cavity. Implementing laser beam scanningCCD technology, the same group@8# further develop this lasescattering technique, and perform the mapping of unsteady casurface on propellers behind complete ship models. Kodama e@9#, and later Tanibayashi et al.@10#, extend the principle usingtwo laser beams, applying it to full scale cavity thickness msurements. Another variant of this double-laser beam approacproposed by Stinzing@11#, with application to model scale tests inonuniform flow.

In the present work, new developments are introduced intoobservation and quantification of the cavitation pattern andapplied to a skewed four-blade model propeller in uniform infloUsing a novel cavitation analysis approach suitable for fieldplications, the area of the leading edge cavitation is accuraquantified throughout an exhaustive range of working conditio

A modified Wageningen type model propeller~INSEANE779A! was selected for the present research project, for itbeen thoroughly studied in non-cavitating flow conditions wpressure and noise measurements and state-of-the-art visualiz

Contributed by the Fluids Engineering Division for publication in the JOURNALOF FLUIDS ENGINEERING. Manuscript received by the Fluids Engineering DivisioJuly 4, 2003; revised manuscript received February 20, 2004. Associate EditoTsujimoto.

Copyright © 2Journal of Fluids Engineering

ande ain

owthe

s oning

lare

tingme-

ita-ante-

eyto

theandnd

vityt al.

a-h is

thearew.p-

telyns.

asthation

techniques, such as laser Doppler velocimetry@12# and particleimage velocimetry@13#. The present work extends the existindatabase to cavitating flow conditions. In addition, the E77geometry is very close to that of many existing twin-screw spropellers, thus enhancing the interest of the study. Finally,E779A propeller provides a complete and challenging test casethe validation of numerical codes.

The data collected in the present work covers a large numbeflow conditions. A rich cavitating flow database is being costructed, representing a powerful tool to investigate in detailrange of applicability of new theoretical models and to assessaccuracy of predictions. As a matter of fact, the validationnumerical predictions against experimental flow data is genernot possible, even though the numerical field is very active aeager for such information@14,15#. The lack of quantitative dataaccurate and covering a useful spectrum of flow conditioexplains the situation. Moreover, inadequate data formats hbeen used that are not suitable for a convenient exchanginformation.

Here, cavitating propeller measurements are used to perforthorough assessment of a computational prediction tool thavalid for sheet cavitation analysis of isolated propellers in unifoinviscid flows. The theoretical formulation is based on a boundintegral methodology for the velocity potential, derived froMorino et al. @16#. The trailing wake vorticity path is accurateldescribed through a wake alignment technique. The cavitamodel considered here is derived using standard procedureboundary element methods@17,18#. The model is based on a nonlinear free cavity length approach and is valid to study leadiedge attached cavities and supercavitating flow conditions. Bthe wake alignment technique and the cavitation model are igrated into the boundary integral methodology through iteratprocedures.

The paper describes, in a first part, the experimental methologies that are introduced to characterize the cavitation devement on a rotating propeller. In a second part, an outline ofnumerical approach is given. Finally, the results of the experimtal investigation are used to assess the validity of the numermethod.

Experimental Analysis

Setup. The experiments are carried out at the Italian Nacavitation tunnel facility~C.E.I.M.M.!. The tunnel is a closed water circuit with a 0.6 m30.6 m32.6 m square test section. Opticaccess to the section is possible through large Perspex windThe nozzle contraction ratio is 5.96:1 and the maximum wa

nr: Y.

004 by ASME JULY 2004, Vol. 126 Õ 671

Fig. 1 Experimental setup

i

-l

actc

rup.

fl

a

-r

p

hetask

n the

ingkeateer-toon-all

s theolethe

speed is 12 m•s21. The maximum free stream turbulence intensin the test section is 2%. The flow uniformity is within 1% for thaxial component and 3% for the vertical one.

The skewed four-blade model propeller has a radiusR5D/250.117 m, a pitch-to-diameter ratio of 1.1 and a forward raangle of 4.05 deg. The blockage ratio in the test section is ab10%.

In the following sections,J, KQ and KT are the advance, thetorque and the thrust coefficients, respectively.s0 is the cavitationnumber related to the reference pressurep0 measured at the propeller axis, andsn is the cavitation number related to the propelrotational speedn. The measurements described here have bperformed across an extensive range of working conditions, ws0 andJ being the control parameters.

The measurement configuration is pictured in Fig. 1. The blangular position is adjusted to allow a full view of the blade faA CCD camera is dedicated to the measurement of the paarea and is oriented at an angle with respect to the test sewindow. Looking at an angle through the thick window introducstrong aberrations, which are almost fully cancelled using a wafilled glass tank in the form of a wedge, and placed so thatcamera optical axis is normal to the wedge face. The propeshaft is equipped with a rotary encoder that supplies an electtrigger signal, used to pilot a pulse delay generator, which in tcontrols the phase between the propeller and the image caand synchronizes the camera with the illumination systemsingle high intensity flash lamp, with a 10ms light pulse, is usedto observe the cavitation pattern. For every flow condition, 1images are recorded, simultaneously with the correspondingparameters.

Cavitation Extension Measurement. We introduce a novelmethodology to determine the cavity area, designed to be higrobust and accurate. The common method of analyzing cavitaimages consists of enhancing the contrast between the cavitpattern and the rest of the image. This is usually done by threolding the image@19#. However, it is well known that this approach is very sensitive to variations in the illumination. Thegion of interest needs also to be easily distinguishable frombackground. In a complex environment such as a rotating proler, this approach is not reliable enough: tip vortex, scattering

672 Õ Vol. 126, JULY 2004

tye

keout

ereenith

dee.terntion

ester-thellericalrntureA

28ow

hlytiontionsh-

e-theel-of

the blade, fluctuations of the illumination intensity, objects in tfield of view, are among many other causes that make thedifficult and unreliable, especially if done automatically.

The approach used here, and depicted in Fig. 2, is based ocross-correlation between a template image~Fig. 2a!, i.e., theblade viewed in non-cavitating conditions, and the image beanalyzed~Fig. 2b!. The cross-correlation is a robust tool to mathe comparison between images: a high correlation would indicthat the template and the image are similar, while a slight diffence would result in a lower correlation coefficient. In orderlocalize the transition regions between the cavitating and the ncavitating situations, the cross-correlation is performed on smimage regions, represented in Figs. 2a and 2b. Figure 2c showresult. The convolution operation is performed across the whimage to produce a correlation image, see Fig. 2d, where

Fig. 2 Image cross-correlation procedure: „a… template image;„b… cavitation pattern image; „c… local cross-correlation; „d… cor-relation image

Transactions of the ASME

y

etricndis

resom-erae-

ns,is-

, wethe-

rdi-

ast-dis-

ith

ique

twooids

inple

thebe-

vi-re-

thelingage

rlym-

sti-the

a

sedss-dn-by

tialed

-

city

n-

differences between the template and the image being analare clearly identified as cavitation features: cavitating vorticfrom the blade tips, contour of the attached cavity.

The correct quantitative evaluation of the cavitation areaAc isonly possible if done in a known coordinate system. The regtered image is a perspective projection of the three-dimensio

Fig. 3 Warping procedure: original „top … and warped „bottom …

images

Fig. 4 Cavitation extension

Journal of Fluids Engineering

zedes

is-nal

blade surface and pattern. A back projection using basic geomoptics would report this image information into a plan view, athis would correct for the variable magnification. However, thisonly valid if the system is free of optical aberrations and requiexact knowledge of the optical parameters and of the blade geetry. In fact, the complex lens system composed of the camobjective, the prism, the test section window and the water mdium, introduces focusing aberrations and optical distortiowhich are the source of nonlinear magnification. The image dtortions can be corrected by calibration procedures. To do soapply here a technique known in the image processing field aswarping transformation@20#, with a second-order perspective operator to account for the nonlinear effects. Ifxi ( i 51,2) are theoriginal image coordinates of a point, the warped image coonatesxi8 are then defined by

xi85ai ,0x11ai ,1x21ai ,21ai ,3x1

21ai ,4x221ai ,5x1x2

bi ,0x11bi ,1x21bi ,21bi ,3x121bi ,4x2

21bi ,5x1x2

(1)

whereai , j and bi , j are the coefficients of the transform.ai , j andbi , j are unknowns that are determined by a Nelder-Mead lesquare minimization method, using reference points on thetorted image.

To make this correspondence, a grid is printed on a blade, wlines drawn spanwise at variousr /R, and radial lines regularlyspaced chordwise, see Fig. 3. Instead of determining one untransformation for the whole image, as commonly done, Eq.~1! iscalculated for each polygon defined by the intersection of thesets of lines. This approach refines the transformation and avthe side-effects inherent to the use of nonlinear polynomialsregions where calibration points are not available. Three samdistorted polygons are shown in the top image of Fig. 3 withcorresponding locally warped image displayed in the picturelow. For clarity, we show only four vertices for each polygon.

The warping procedure is applied to the correlation image~Fig.2d!, followed by standard threshold techniques to track the catation pattern, now possible without the drawbacks outlined pviously. Morphological operators may then be used to filtermain cavitation pattern from undesired features, such as travebubbles. This sequence of operations results in a pattern im~Fig. 4!, where the cavitation region over the blade is cleaoutlined, measurable and suitable for comparison with the coputations described in the next sections. The cavitation areaAc isexpressed in percentage of the blade face areaA0 obtained forr /R>0.3.

The error that arises from the warping transformation is emated by comparing the calculated and the actual areas ofsurface polygonal elements on the calibration blade~Fig. 3!. Amean error of 0.64% is found, which could be improved usingfiner calibration grid.

Theoretical ModelThe theoretical hydrodynamic model considered here is ba

on a boundary integral formulation for the analysis of incompreible inviscid flows around lifting bodies. Propeller load-inducevorticity shedding is described by means of a trailing wake aligment model, whereas cavity pattern predictions are obtainedusing a nonlinear sheet cavity model that is valid for both parand supercavitation. The theoretical formulation is briefly outlinhere. A detailed derivation is given by Salvatore and Testa@21#.

Governing Equations and BEM. Assuming that the unperturbed flow is inviscid and irrotational, a scalar potentialw isintroduced to express the propeller-induced perturbation veloas v5¹w. In a frame of reference (Oxyz) fixed to the propellerwith the x-axis parallel to the propeller axis and pointing dowstream, the incoming flow velocity readsvi5vA1V3x, wherevA5(v0,0,0) is the axial inflow to the propeller,V5(2pn,0,0) isthe angular velocity of the propeller, andx5(x,y,z). The present

JULY 2004, Vol. 126 Õ 673

r

m

t

e

u

e

e

u

e

-

v

dgeto

e

o-ry

ism-otex-

e

byer-

d

in-nts,ele-

ntd by

the

e.

alhe

hin

Ato theamicsed

ac-

derivation is valid for unsteady flow analysis of an isolated ppeller in which the inflowv0 is uniform whereas bothv0 and nmay be time-dependent. The velocity field in the propeller fraof reference isq5vi1¹w. Under incompressible flow assumptions, the continuity equation reduces to the Laplace equa¹2w50.

The pressurep is given by Bernoulli’s theorem that, in thframe of reference fixed to the propeller, reads

]w

]t1

1

2q21

p

r1gz05

1

2v l

21p0

r, (2)

whereq5iqi , v l5ivi i , g is the gravity acceleration andz0 de-notes depth.

Flowfield predictions are obtained by solving the Laplace eqtion for w with suitable boundary conditions. The potential-flodomainVP is delimited by the propeller surfaceSB , the trailingwake surfaceSW and the cavity surfaceSC .

On the cavitation-free portion of the propeller surfaceSWB , theimpermeability condition yieldsq"n50. Recalling q5vi1¹w,this condition reads

]w

]n52vi "n on SWB , (3)

wheren is the outward unit normal to the surface.The trailing wake surfaceSW represents a zero-thickness lay

over which the vorticity generated on the body is shed dowstream. In the potential field theory,SW denotes a discontinuitysurface for the velocity potential. Applying the mass and momtum conservation laws under non cavitating flow conditions, aletting D~ ! be the discontinuity across the two sides of the wasurface, we haveD(]w/]n)50 andDp50. Thus, from the Ber-noulli equation~2!, Dw is found to be constant as one follows thwake particles, i.e.

Dw~x,t !5Dw~xTE ,t2t! on SW , (4)

wherexTE is a wake point at the blade trailing edge andt is theconvection time between wake pointsx and xTE . A further con-dition on w is required in order to assure that no finite pressjump may exist at the body trailing edge~Kutta condition!. Fol-lowing Morino et al.@16#, this is equivalent to imposeDw(xTE)5wTE

u 2wTEl , wherewTE

u and wTEl are evaluated, respectively, a

the body trailing edge suction and pressure sides.The boundary conditions on the cavity surfaceSC are imposed

assuming that the cavity is a fluid region where the pressurconstant and equal to the vapor pressurepv . Thus, requiring thatp5pv on SC , and using the Bernoulli equation~2!, it follows

q5F ~nD!2sn22S ]w

]t1gz0D1vi G1/2

on SC , (5)

where sn5(p02pv)/0.5r0(nD)2 is the cavitation number referred to the propeller rotational speedn and diameterD. Equation~5! is used to obtain a Dirichlet-type condition forw on SC interms ofsn . Introducing onSC a curvilinear coordinate systemwith s andu being the arclengths in the chordwise and spanwdirections, respectively, and recallingq5vi1¹w, one obtains anonlinear condition forw of the type

w~s,u!5w~sCLE ,u!1EsCLE

s

F S q,]w

]u,]w

]n,vi Dds on SC ,

(6)

wheresCLE is the cavity leading edge abscissa. A detailed derition of Eq. ~6! and the expression of the functionF is given bySalvatore and Testa@21#.

At the cavity trailing edge, the Dirichlet-type condition giveby Eq.~6! and the Neumann-type condition Eq.~3! should be bothimposed. However, this causes a non-physical discontinuityboth the velocity and pressure fields. To overcome the singula

674 Õ Vol. 126, JULY 2004

o-

e-ion

a-w

rn-

n-ndke

e

re

t

is

ise

a-

n

inrity,

we create a small transitional region across the cavity trailing eand apply a cavity closure condition: the pressure is forcedsmoothly vary frompv to the wetted flow values downstream thcavity. Inside the transitional region, the quantitysn in Eq. ~5! isreplaced by

sn* 5snH1~j!2~Cp!TDH2~j!2~Cp8!TDH4~j!, (7)

wherej spans between21 and11, andHi(j) are Hermite poly-nomials. (Cp)TD and (Cp8)

TD denote respectively the pressure cefficient and itss-wise derivative at the downstream bounda~TD! of the transitional region. This pressure-based conditionclosely related to the velocity-based condition proposed by Leonnier and Rowe@22#. However, the present approach does nrequire the introduction of any arbitrary parameters, once thetension of the transitional region is set~10% of the cavity lengthin our case!.

An expression of the cavity thicknesshc is obtained with anon-penetration condition onSC . By combining the constant-pressure and the non-penetration conditions, it follows thatSC is amaterial surface. IfSCB andSCW are the cavitating areas on thbody and on the wake surfaces, respectively, and if¹s is the sur-face gradient acting on those surfaces, then if follows

]hc

]t1¹shc•q5xc on SCBøSCW , (8)

wherexc5]w/]n1vi "n on SCB andxc5D(]w/]n) on SCW .The Laplace equation for the velocity potential is solved

means of a boundary integral formulation. Assuming that the pturbation vanishes at infinity, the third Green identity yields

w~x!5 RSB1C

S ]w

]nG2w

]G

]n DdS~y!

2ESW

Dw]G

]ndS~y! xPnP \]nP , (9)

whereSB1C5SWBøSC delimits both the body and the cavity, anG521/4pix2yi is the unit source.

Equation~9! is solved using a boundary element approachthe limit as the distance ofx from SB1C tends to zero. The boundary surfaces are divided into hyperboloidal quadrilateral elemeand the flow quantities are assumed to be constant on eachment. Equation~9!, in discretized form, is enforced at the elemecentroids and is solved with the boundary conditions expresseEqs. ~3! to ~6!. The sourceG and the dipole]G/]n influenceterms are integrated using an analytical approach@16#. Inviscid-flow hydrodynamic loads are computed after integration ofpressure onSB from Eq. ~2!. The viscosity contribution to thepropeller torque is estimated using flat plate friction line formula

Numerical Solution Procedure. The cavity surfaceSC andthe wake surfaceSW used in Eq.~9! are not known a priori. Inorder to limit the computational effort and to avoid numericinstabilities from the regridding of the cavity surface during tcavitation model iterative procedure, Eq.~9! is numerically solvedwith singularities distributed on the surfaceSCBøSCW rather thanon the cavity edgeSC . This simplification is consistent with thepresent sheet cavitation model that is limited to analyze tvapor-filled regions attached to body and wake surfaces.

The wake surface may be either prescribed~prescribed wakeapproach! or determined as a part of the flowfield solution~freewake approach!. In the prescribed wake approach,SW is approxi-mated by a helical surface with a prescribed pitch distribution.the trailing edge, the wake surface is assumed to be tangent tblade suction side, whereas an average between the hydrodynpitch of the unperturbed inflow and the blade pitch angle is udownstream the propeller.

In the free wake approach, a physically consistent shape ofSWis determined upon requirement that the wake points move

Transactions of the ASME

d

t

o

e

r

ooed.durepo-

ce

isekennat-

ridureller

cording to the local velocity field. A boundary integral represetation of the perturbation velocity field¹w is obtained taking thegradient from both sides of Eq.~9!. Starting with an initialprescribed-pitch guess forSW , the computed velocity field is useto update the location of each wake node. OnceSW has beenmodified, the potential field is re-computed through Eq.~9!. Theprocess is repeated until convergence of the aligned wake shDetails of this technique are given by Giordani et al.@23#.

In the proposed methodology, the cavitation model andwake alignment technique are combined under the assumpthat the trailing wake location is not affected by the caviinduced flowfield perturbation@24#. Within this context, the wakealignment procedure is performed under non-cavitating flow cditions. Once the actual shape of the trailing wake is determinan initial guessSCB

(0) for the cavity planform is imposed, Eq.~9! issolved under cavitating flow conditions, and the cavity thicknis evaluated by Eq.~8!. The leading edge cavity detachmentalso set, this being supported by experimental observations.cavity trailing edge is determined by imposinghc50 at each sur-face strip in the chordwise direction. If this condition is not ve

Fig. 5 Computational grid used for flowfield analysis of theE779A propeller, and wake surface evaluated by the flow-alignment technique „MBÄ36, MB ÕNBÄ2, MW ÕNBÄ5, NH

totÄ252…

Fig. 6 Effect of grid refinement on the non cavitating thrustcoefficient K T at JÄ0.71

Journal of Fluids Engineering

n-

ape.

thetiony-

n-ed,

ssisThe

i-

fied within SCB(0) , it is assumed that the guessed planform is t

small and a cavity planform extrapolation procedure is usHence, an updated cavity planform is obtained and the proceis repeated until convergence of the cavity shape and of thetential field.

A representative computational grid on the propeller surfaand on the wake is depicted in Fig. 5. On each blade side,MB536 andNB518 elements are used in chordwise and spanwdirections, respectively.MW590 elements are used on each waturn in streamwise direction, andNH

tot5252 elements are used oeach hub surface sector. For the sake of clarity, the wake emaing from only one blade is depicted in Fig. 5~free-wake calcula-tion result!.

The numerical solution sensitivity to the computational grefinement is addressed in Fig. 6. Specifically, the top pictshows the effect of grid refinement on the calculated propethrust coefficientKT under non cavitating flow conditions atJ50.71. In particular, numerical results using grids withMB

5@12,18, . . . ,60#, MB /NB5@1,2,3#, MW /NB55, and NHtot

Fig. 7 Effect of grid refinement on the cavity area fractionAC ÕA0 at JÄ0.71, snÄ1.515

Fig. 8 Cavity volume histories using different initial guessesof the cavity extension „JÄ0.71, snÄ1.515…

JULY 2004, Vol. 126 Õ 675

6

Fig. 9 E779A model propeller cavitation chart; pressure coefficient distributions on the blade suction and pressure sidesand associated images, at two characteristic conditions: J Ä0.65, snÄ0.528; JÄ0.77, snÄ2.082

o

hpd

o

s

b

h

e

apketnti-ion/ing

areoti-

the

onvi-kenyffi-

thetoffully

del,ethelcu-tionsof

ex.

5@132, . . .,372# are compared. The following observations aalso applicable to the propeller torque coefficientKQ calculations~not shown here!.

The ratio between the blade discretization elements in chwise and spanwise directions does affect the rate of convergeHowever, the extrapolated values as the grid refinement param1/(MB3NB) tends to zero are barely influenced by this grid prameter. Indeed, we find a convergedKT value of 0.246, while themeasured one is 0.255, hence the discrepancy is 3.5% circa.error can be seen as global, i.e., the combined result of the untainties that originate from the numerical simulation, from tmeasurement of the experimental flow parameters, from thepeller actual geometry and from the experimental setup andanalysis.

The effect of grid refinement on the computed cavity areashown in Fig. 7 for cavitating flow conditions (J50.71, sn51.515). The discretization errors on the cavity area predictitend to decrease as 1/(MB3NB) approaches zero.

The results above indicate that discretization errors are reaably small if computational grids havingMB>36 are used. Inparticular, all the numerical results presented hereafter haveobtained using the computational grid depicted in Fig. 5.

The convergence of the iterative procedure used to solvenonlinear algorithms~cavitation model and wake alignment, in thpresent methodology! is also a fundamental aspect of numericscheme verification. With regard to the cavitation model, a repsentative result is provided in Fig. 8, which shows different cavvolume histories for the caseJ50.71,sn51.515, obtained usingdifferent initial guesses of the cavity planform. Specifically, tchordwise locationxtec of the cavity trailing edge ranges from 7%to 74% of the local chord, the leading edge cavity detachmbeing preset. As one would expect, the converged value ofcavity volume is not affected by the choice of the initial guess

Flow Field InvestigationsAnalyzing the individual cavitation images across the compl

measurement chart, the E779A model propeller can be descr

76 Õ Vol. 126, JULY 2004

re

rd-nce.etera-

Thiscer-ero-ata

is

ns

on-

een

theealre-ity

e

entthe.

teibed

in terms of distinct cavitation patterns. Figure 9 represents a mof the cavitating behavior of the propeller, also known as a bucdiagram, where the main figures of cavitation have been idefied. The propeller presents five main patterns: bubble cavitattraveling cavitation, supercavitation, sheet cavitation, leadedge~LE! and tip vortex cavitation. A curvesn

min(J) is also plottedto delimit flow conditions characterized bysn,sn

min(J), wherethe cavity pattern predictions by the present theoretical modelnot reliable ~we recall that the current cavitation model is nintended for bubble cavitation!. In such cases, the iterative cavtating flow procedure usually fails to converge.

Two cases of interest are displayed on this figure, withassociatedCp distributions~both suction and pressure sides! andpattern image:J50.65, sn50.528; J50.77, sn52.082. Thesetwo cavitating situations are discussed hereafter.

Bubble cavitation is present at low values of the cavitatinumbersn and is coexisting in most of the cases with supercatation. This situation can be observed on the left photograph taat J50.65, sn50.528, at mid-span of the profile, and is fullconsistent with the numerical predictions of the pressure coecient Cp using a non-cavitating flow model~this flow case isbeyond the convergence limitsn

min(J)). At r /R50.4, the calcu-lated Cp on the suction side has a very smooth evolution inchordwise direction, which favors the explosion of nuclei inbubbles. Closer to the blade tip~e.g., atr /R50.8), the presence oa strong pressure gradient next to the leading edge starts adeveloped sheet cavitation extending past the trailing edge.

The second case~right part of Fig. 9! corresponds toJ50.77,sn52.082. Close to the tip~here atr /R50.88), theCp distribu-tion on the suction side, calculated using the cavitating moclearly shows the region atCp52sn and the recompression zonin the closure of the cavity. TheCp then recovers back to the we~non-cavitating! Cp levels, which are also represented to show tconsistency between the non-cavitating and the cavitating calations. The attached sheet cavitation observed in these condimixes with the tip vortex structure and, for the highest valuessn , transforms into the so-called leading edge cavitating vort

Transactions of the ASME

J

Fig. 10 Planform view of the cavitating blade and predicted cavitation area at J Ä0.71 „top row …, 0.77 „center … and 0.83„bottom …: free wake model „solid line …, prescribed wake model „dashed line …

d

l

rw

s

Fig.effi-n the

thealy-

thef thenhedatanand

ys-nge

theart,

The propeller also presents a form of streak cavitation, moately present forJ50.65,sn50.528. This pattern is characterizeby two or three vapor streaks detaching at precise locations athe leading edge. This situation is thought to be caused by surirregularities of the model blade geometry~i.e., local roughness!that, combined with the flatCp distribution, become the place oearly and very localized cavitation spots.

Images of the cavitation pattern for nine selected flow contions are shown in Fig. 10. These images have been obtausing the warp procedure described earlier. The cavity areasculated using both the free wake model~solid line! and the pre-scribed wake model~dashed line! are also shown for comparisonThe predicted extension of the cavitating area is clearly affecby the shape of the trailing wake used in the calculations.particular, if a correct location of the trailing vorticity is providevia the wake alignment technique~free wake model!, the numeri-cal results~solid curves! are in closer agreement with the expements as compared to those obtained using a prescribedmodel ~dashed curves!.

A quantitative comparison between the observed and the calated cavity extensions is made possible using the area mea

ournal of Fluids Engineering

er-dongface

f

di-inedcal-

.tedIn

d

i-ake

lcu-ure-

ment technique described in this work. Results are shown in11, where the data referred to five values of the advance cocient are shown. The area is expressed as the ratio betweemeasured cavitation areaAc and the blade planform areaA0 forr /R>0.3. The experimental mean values are represented withcorresponding standard deviation. We recall that the image ansis error was estimated to less than 1%.

The cavitation area fluctuations are in general very small, inrange of 1% to 3%, except for the cases at the lowest values ocavitation numbersn , where the occurrence of bubble cavitatiois the source of important variations of the pattern figure. Tagreement between numerical predictions and experimentalis generally very good. Some discrepancies are observed whesnapproaches the model convergence limit displayed in Fig. 9,when lowJ values are considered~e.g., curves atJ50.65 andJ50.71). We observe that the numerical simulations tend to stematically overpredict the measurements, and differences rafrom 1%–2% at highsn and highJ, to 10%–15% at lowsn andlow J.

The tendency of the present theoretical model to overpredictextension of the cavitating flow region may be due, at least in p

JULY 2004, Vol. 126 Õ 677

o

o

err

the

aaveeth-

ris-d in

daryper-ofnt

thatdict

s andodndrly

eri-jectsticri-ure-icalsis

lop-try,

rke-

to both the numerical discretization errors and the inviscid flassumptions. For instance, the caseJ50.71,sn51.515 shown inFig. 11 reports a discrepancy between the measured and predvalues of about 26%, half of which are due to discretizationrors, as also visible in Fig. 6. With regard to the inviscid floassumptions, Salvatore and Esposito@25# show that the inclusionof viscous flow effects induces a non-negligible reduction of bthe cavity extension and thickness as compared to inviscid flcalculations.

The propeller thrust and torque coefficients for three valuesthe advance coefficient are shown in Figs. 12 and 13. A commfeature is that bothKT andKQ increase as the cavitation numberincreased. At lowsn , this trend is more evident and is associatwith the presence of extensive cavitation on the blades. Numecalculations are also given for comparison. A satisfactory agment between numerical predictions and measurements is

Fig. 11 Effect of parameters J and sn on the cavity area A c.Comparison between measurements and numerical results„free wake model ….

Fig. 12 Effect of cavitation number sn on the propeller thrustcoefficient

678 Õ Vol. 126, JULY 2004

w

icteder-w

thow

ofon

isd

icalee-ob-

tained, except for flow conditions at lowsn where bothKT andKQ calculations are biased by the convergence limit and byoverestimation of the cavitation pattern.

Concluding RemarksAn experimental investigation on a cavitating propeller in

uniform inflow has been presented. The cavitation figures hbeen analyzed through novel, accurate and robust analysis mods. A complete map of the propeller hydrodynamic charactetics has been established and the cavitating behavior quantifieterms of cavitation extension.

We used the measurements to assess an inviscid flow bounelement method for the analysis of blade partial sheet and sucavitation. The methodology included a refined evaluationthe trailing wake vorticity path by means of a wake alignmetechnique.

Comparison with the experimental measurements showedthe present theoretical methodology was able to accurately prethe sheet cavitation area across a wide range of advance ratioof cavitation number values. The importance of having a godescription of the trailing wake vorticity has been highlighted athe range of applicability of the present prediction model cleaidentified.

The present work constitutes the first phase of a joint expmental and theoretical research project. Phase two of the proextends the current results to the hydrodynamic/hydroacouanalysis of propellers in non-uniform inflow conditions. Expemental effort is also ongoing to extend the cavity area measment principle to the volume measurement. From a theoretviewpoint, the inclusion of viscous flow modeling and the analyof tip-vortex cavitation are being addressed.

A database accessible to the public~http://crm.insean.it/E779A!is being regularly updated with the latest results and devements, complementing the already existing propeller geomepressure, noise, PIV, and LDV data.

AcknowledgmentsThe authors are grateful to the CEIMM personnel. The wo

was performed in the construct of the INSEAN 2000-2002 Rsearch Program supported by the ItalianMinistero dei Trasporti edella Navigazione.

Fig. 13 Effect of cavitation number sn on the propeller torquecoefficient

Transactions of the ASME

r

le

at-

ceding

u-

ss

itys of

t of

g

n-l-

vo-s

of

nelnd

ryns,’’

eT,

od

ble

s

i-

of a

of

aly-

Nomenclature

n 5 Boundary surface outward unit normalq 5 Total velocity:q5vi1¹wv 5 Perturbation velocityvi 5 Inflow velocity

SB 5 Body surfaceSCB 5 Cavitating area on body surfaceSCW 5 Cavitating area on wake surfaceSC 5 Cavity surface

SWB 5 Non-cavitating body surfaceSW 5 Trailing wake surfaceVP 5 Potential flow domainA0 5 Blade face area forr /R>0.3Ac 5 Cavity areaCp 5 Pressure coefficient, (p2p0)/0.5r(nD)2

D 5 Propeller diameter, 2Rhc 5 Cavity thicknessJ 5 Advance coefficient,v0 /nD

KQ 5 Torque coefficient,Q/rn2D5

KT 5 Thrust coefficient,T/rn2D4

MB 5 Chordwise number of blade face elementsMW 5 Streamwise number of wake-turn elements

n 5 Propeller angular velocity~rps!, normalNB 5 Spanwise number of blade face elementsNH

tot 5 Number of hub sector elementsp 5 Pressure

p0 5 Pressure at propeller axispv 5 Vapor pressureQ 5 TorqueR 5 Propeller radiuss 5 Surface arclength in chordwise directionT 5 Thrustu 5 Surface arclength in spanwise direction

v0 5 Upstream axial inflow velocityVc 5 Cavity volumer 5 Fluid density

s0 5 Cavitation number referred tov0 , (p02pv)/0.5rv02

sn 5 Cavitation number referred ton,(p02pv)/0.5r(nD)2

w 5 Perturbation velocity potential

References@1# Dupont, P., and Avellan, F., 1991, ‘‘Numerical Computation of a Leading Ed

Cavity,’’ Proc. of Int. Symposium on Cavitation and Multiphase Flow, vol.FED-116, ASME-JSME, Portland, Oregon~USA!, pp. 47–54.

@2# Pereira, F., 1997, ‘‘Pre´diction de l’Erosion de Cavitation: Approche E´nerget-ique,’’ Ph.D. thesis, E´ cole Polytechnique Fe´derale de Lausanne, Institut deMachines Hydrauliques et de Me´canique des Fluides~IMHEF-LMH !, no1592.

@3# Pereira, F., Avellan, F., and Dupont, P., 1998, ‘‘Prediction of Cavitation Esion: An Energy Approach,’’ J. Fluids Eng.,120~4!, pp. 719–727.

@4# Chiba, N., Sasajima, T., and Hoshino, T., 1980, ‘‘Prediction of PropelInduced Fluctuating Pressures and Correlation With Full Scale Data,’’Proc. of

Journal of Fluids Engineering

ge

o-

r-

the 13th Symposium on Naval Hydrodynamics, ONR, Tokyo, pp. 89–103.@5# Lehman, A. F., 1966, ‘‘Determination of Cavity Volumes Forming on a Rot

ing Blade,’’11th International Towing Tank Conference, ITTC, Tokyo ~Japan!,pp. 250–253.

@6# Sontvedt, T., and Frivold, H., 1976, ‘‘Low Frequency Variation of the SurfaShape of Tip Region Cavitation of Marine Propeller Blades and CorresponDisturbances on Nearby Solid Boundaries,’’Proc. of the 11th Symposium onNaval Hydrodynamics, ONR, London, pp. 717–729.

@7# Ukon, Y., and Kurobe, Y., 1981, ‘‘Measurement of Cavity Thickness Distribtion on Marine Propellers by Laser Scattering Technique,’’Proc. of the 16thITTC, vol. 2, Leningrad~USSR!, pp. 241–245.

@8# Ukon, Y., Kudo, T., and Kurobe, Y., 1991, ‘‘Measurement of Cavity ThickneDistribution on the Blade of Propeller Models by Laser-CCD Method,’’Proc.of the 1st ASME-JSME Fluids Engineering Conference, vol. FED-116, Port-land, Oregon~USA!, pp. 99–104.

@9# Kodama, Y., Takei, Y., and Kakugawa, A., 1983, ‘‘Measurement of CavThickness on a Full Scale Ship Using Lasers and a TV Camera,’’ PaperShip Research Institute,73, pp. 1–42.

@10# Tanibayashi, H., Saito, Y., and Fujino, R., 1991, ‘‘Full-Scale MeasuremenCavity Over the Blades of Marine Propellers,’’Proc. of Int. Cavitation andMultiphase Flow Forum, vol. FED-109, 1st ASME/JSME Fluids EngineerinConference, Portland, Oregon~USA!, pp. 107–110.

@11# Stinzing, H. D., 1990, ‘‘Cavity Thickness on Rotating Propeller Blades,’’Proc.of the 18th Symposium on Naval Hydrodynamics, ONR, Ann Arbor, Michigan,pp. 75–85.

@12# Stella, A., Guj, G., Di Felice, F., and Elefante, M., 2000, ‘‘Experimental Ivestigation of Propeller Wake Evolution by Means of LDV and Flow Visuaizations,’’ Journal of Ship Research,44~3!, pp. 155–169.

@13# Di Felice, F., Romano, G. P., and Elefante, M., 2000, ‘‘Propeller Wake Elution by Means of PIV,’’Proc. 23rd Symposium on Naval Hydrodynamic,ONR, Val de Reuil~France!, pp. 62–76.

@14# Kim, Y. G., and Lee, C. S., 1997, ‘‘Prediction of Unsteady PerformanceMarine Propellers with Cavitation Using Surface-Panel Method,’’Proc. of the21st Symposium on Naval Hydrodynamics, Trondheim~Norway!, pp. 913–929.

@15# Mueller, A. C., and Kinnas, S. A., 1997, ‘‘Cavitation Predictions Using a PaMethod,’’ Proc. of the ASME Symposium on Marine Hydrodynamics aOcean Engineering, Dallas~USA!.

@16# Morino, L., Chen, L. T., and Suciu, E. O., 1975, ‘‘Steady and OscillatoSubsonic and Supersonic Aerodynamics Around Complex ConfiguratioAIAA J., 13, pp. 368–374.

@17# Lee, J. T., 1987,A Potential Based Panel Method for the Analysis of MarinPropellers in Steady Flow, Tech. Rep. 87-13, Dept. Ocean Engineering, MICambridge, Massachusetts~USA!.

@18# Kinnas, S. A., and Fine, N. E., 1992, ‘‘A Nonlinear Boundary Element Methfor the Analysis of Propeller Sheet Cavitation,’’Proc. of the 19th Symposiumon Naval Hydrodynamics, ONR, Seoul~Korea!, pp. 1–17.

@19# Pham, T. M., Larrarte, F., and Fruman, D. H., 1998, ‘‘Investigation of UnstaCloud Cavitation,’’Proc. of the 3rd Int. Symp. on Cavitation, vol. 1, pp. 215–220.

@20# Wolberg, G., 1996,Digital Image Warping, IEEE Computer Society Press, LoAlamitos, CA~USA!.

@21# Salvatore, F., and Testa, C., 2002,Theoretical Modelling of Marine PropellerCavitation in Unsteady High-Reynolds Number Flows, Tech. Rep. 2002-077,INSEAN, Rome~Italy!.

@22# Lemonnier, H., and Rowe, A., 1988, ‘‘Another Approach in Modelling Cavtating Flows,’’ J. Fluid Mech.,195, pp. 557–580.

@23# Giordani, A., Salvatore, F., and Esposito, P., 1999, ‘‘Free Wake AnalysisMarine Propeller in Uniform Flow,’’Proc. of the XXI Int. Conference onBoundary Element Methods, Oxford ~U.K.!.

@24# Arndt, R. E. A., Arakeri, V. H., and Higuchi, H., 1991, ‘‘Some ObservationsTip-Vortex Cavitation,’’ J. Fluid Mech.,229, pp. 269–289.

@25# Salvatore, F., and Esposito, P., 2001, ‘‘An Improved Boundary Element Ansis of Cavitating Three-Dimensional Hydrofoils,’’Proc. of the 4th Interna-tional Symposium on Cavitation, B1.006, Pasadena, CA~USA!.

JULY 2004, Vol. 126 Õ 679