First International Symposium on Marine Propulsorssmp09, Trondheim, Norway, June 2009

Coupled HydrodynamicsHydroacoustics BEM Modelling of MarinePropellers Operating in a Wakefield

Francesco Salvatore1, Claudio Testa1, Luca Greco1

1Italian Ship Model Basin (INSEAN), Rome, Italy

ABSTRACTCombined hydrodynamic/hydroacoustic formulations forthe analysis of marine propellers operating in a non-homogeneous flow are presented. The hydrodynamicmodel is based on a Boundary Element Method for invis-cid flows and is applied here to study pressure fluctuationsinduced by a propeller to solid boundaries and noise radi-ated to the open flow. Two hydroacoustic models basedon a standard Bernoulli equation model for incompressibleflows and on a wave propagation model are compared. Nu-merical applications are presented to analyse the capabil-ity of these two methodologies to describe the disturbancesgenerated by non cavitating and cavitating propellers in op-erating conditions.KeywordsMarine Propellers, Cavitation, Pressure Fluctuations,Ffowcs-Williams and Hawkings equation, Boundary Ele-ment Methods1 INTRODUCTIONThe design of high performance marine propellers is typ-ically the result of a trade-off among opposing factors.Modelling blade shapes to achieve efficiency gains is usu-ally hindered by increased levels of cavitation characteriz-ing the optimised configurations. The evaluation of a givendesign requires then a careful assessment of the nuisanceintroduced when additional cavitation is allowed. It followsthat the full exploitation of modern design and optimizationtechniques implies that reliable predictions of propeller in-duced pressures are available.Consistent with classical inviscid-flow propeller hydrody-namics models, computational approaches are commonlyused in which the pressure field induced by a propelleris described through the solution of the Bernoulli equa-tion for incompressible flows. With the advent of vis-cous flow models, induced pressure disturbances followfrom the numerical solution of Reynolds Averaged Navier-Stokes Equations (RANSE) in which incompressible flowassumptions are still present.Only recently, physically consistent models in which pres-sure disturbances are modelled by equations describingwave propagation through a compressible medium have

been introduced for marine propeller applications. In thiscontext, theoretical and computational studies addressinghydroacoustic models is addressed in Seol et al. (2002).An early attempt to apply a wave propagation model basedon the numerical solution of the Ffowcs-Williams andHawkings (FW-H) equation is proposed by Salvatore andIanniello (2003). A straightforward approach to describethe case of a cavitating propeller is derived in which a cav-itating blade is replaced by a solid body whose thickness ismodified to account for the vaporised region on its surface.The model is valid only to describe sheet cavities attachedto the blade surface.

An alternative approach to describe propeller cavitation ef-fects on noise emission and radiation is discussed by Testa(2008) where an original interpretation of the porous FW-H formulation is introduced. The methodology referredto as the Transpiration Velocity Model (TVM) is based onthe assumption that the perturbation induced by a cavityattached or lipping a solid surface may be approximatelyrepresented by a suitable velocity distribution normal to thesurface and thus violating the impermeability condition onthat surface.

Aim of this paper is to present numerical applications bythe FW-H/TVM model and to compare results with thoseobtained by a standard approach based on the Bernoulliequation. As an extension of previous work in Testa et al.(2005) and Testa et al. (2008) dealing with non-cavitatingpropellers in uniform flow conditions, the emphasis here ison the capability of FW-H/TVM and Bernoulli methods todescribe the effects of transient cavitation occurring on apropeller operating in a non-homogeneous wakefield.

Propeller flow predictions are obtained through a BoundaryElement Method (BEM) for potential flows combined to anunsteady sheet cavitation model. Propeller induced noise isstudied considering a single propeller in unbounded flow.Propeller induced pressure fluctuations on a solid boundaryare studied by considering a notional propeller/hull-plateconfiguration. Comparisons between isolated propeller andpropeller-plate configurations are made through the solidboundary factor model (Huse (1996)).

2 THEORETICAL MODELIn this section propeller hydrodynamics and hydroacousticsmodels are described.The hydrodynamic model used for propeller performancepredictions also provides the input for the two hydroacous-tic models considered. Propeller hydrodynamics is stud-ied via a Boundary Element Method (BEM) for inviscidflows around lifting/thrusting bodies. The methodologyis recalled here to clarify the coupling with hydroacousticmodels addressed in the following sections. Details of thishydrodynamic model are given in Salvatore et al. (2003)and Greco et al. (2004).2.1 Propeller Hydrodynamic modelAssuming the onset flow is incompressible and inviscid,perturbation velocity induced by fluid-body interactionsmay be represented as the gradient of a scalar potential asv = , where denotes the velocity potential. Continu-ity equation is recast into the Laplace equation 2 = 0which, following a classical derivation (see, e.g., Morino(1993)) yields a very general integral expression for atan arbitrary point x

E(x)(x) =SB

(

nG G

n

)dS(y)

SW

G

ndS(y), (1)

This general expression holds for the velocity potentialfield surrounding a solid body of surface S

Barbitrarily

moving with respect to a fluid with a prescribed onset flow.Quantity G = 1/4pix y is the Greens function inan unbounded three-dimensional domain, and n is the out-ward unit normal to S

B.

A key for present applications of Eq. (1) to hydrodynam-ics and hydroacoustics problems is function E(x). Thisquantity is defined in the whole space and equals 0, 1/2, 1,respectively, if x is inside, on, or outside the fluid/solid in-terface S

B. Introducing the field function E(x), Eq. (1)

is formally valid to evaluate the velocity potential over thesurface of a solid body or inside the fluid region surround-ing it. In the present study, S

Bdenotes the surface of a

propeller and of a solid plate above it, as shown in Fig. 1,where the rotating frame of reference fixed to propellerblades is also sketched. Following potential flow theory forlifting/thrusting bodies, S

Wdenotes the trailing wake sur-

face where vorticity generated on propeller blades is sheddownstream. Quantity represents the potential discon-tinuity on S

Wdirectly related to the intensity of vorticity

distributed along the wake (see Batchelor (1967)).Equation (1) with x S

Band E(x) = 1/2 provides a

boundary integral equation that is solved imposing imper-meability conditions on S

Band vorticity convection with-

out pressure jump at blade trailing edge for SW

(see Morino(1993), for details). Bondary conditions to account for cav-itating regions on blade surfaces are addressed later.

Figure 1: Notional propeller, rudder and hull-plate config-uration and definition of rotating frame of reference.

Once velocity potential is determined from Eq. (1), pres-sure is evaluated from the Bernoulli theorem which, in therotating frame of reference reads

t+

12q2 +

p

+ gz0 =

12v2I+

p0, (2)

where vI

is the velocity of flow incoming to the propeller,q = v

I+, whereas p0 is the freestream pressure, is

the fluid density, gz0 is the local hydrostatic head.The Bernoulli equation allows to determine the pressuredistribution over the solid surfaces (propeller and hullplate). By integrating pressure stress normal to the solidsurface, the inviscid-part of propeller loads is evaluated. Toaccount for viscosity induced tangential stress, the presentBEM can be coupled with a boundary-layer model as de-scribed in Salvatore et al. (2003). Although fundamentalfor a correct description of loads, viscosity effects play aminor role in noise emission and propagation aspects ad-dressed in the present paper. Thus, skin friction contribu-tions to thrust and torque are simply estimated here throughan approximated flat plate model.In the present study, Eq. (2) is also the basis for one ofthe two models addressed in the paper to evaluate pressureradiated in the flowfield, as described later.

2.2 Propeller cavitation modelThe inviscid-flow formulation outlined above is combinedto a cavitation model that is valid to address sheet cavi-ties forming at the leading edge of a lifting surface. Themethodology derived by Kinnas and Fine (1992) is de-scribed in Salvatore et al. (2003).Vaporization is directly related to local pressure droppingto vapor pressure pv . Then, the cavitating flow region isdetermined as the flow region attached to the body surfaceand limited by a surface S

Cwhere the following dynamic

condition derived from the Bernoulli Eq. (2) holds

qc = [(nDP )2n 2(/t + gz0) + v2I ]

12 , (3)

where n = (p0 pv)/ 12(nDP )2 denotes the cavita-tion number referred to the propeller rotational speed n,whereas D

Pis the propeller diameter. The above expres-

sion is manipulated to obtain a Dirichlettype condition

(, ) = (CLE

, ) + CLE

F () d on SC, (4)

where CLE

is the cavity leading edge abscissa in chord-wise direction , the abscissa in spanwise direction is .FunctionF() is derived from Eq. (3) and provides the linkbetween velocity potential and local pressure under cavi-tating flow conditions. Suitable conditions are imposed atcavity trailing edge as described in Salvatore et al. (2003).An expression of the cavity thickness hc is obtained by im-posing a nonpenetration condition on S

C. By combining

the constantpressure and the nonpenetration conditions,it follows that S

Cis a material surface. Denoting by S

CB

the cavitating portion of the body surface, and by S

thesurface gradient acting on it, one has

hct

+Shc q = c on SCB , (5)

where c = /n + vI n. Equation (5) provides apartial differential equation for hc that may be solved oncethe potential field is known.The resulting formulation for both non cavitating and cav-itating flows is numerically studied via a nonlinear BEM.Non cavitating potential flow is determined by solving aNeumann problem for in which quantity /n on S

Bis

known through the impermeability condition. When vapor-ization on the body surface is detected, the solution followsfrom a mixed Neumann-Dirichlet problem in which Eq. (4)provides a non-linear boundary condition for on the cav-itating portion of the body surface. Recalling that both thetrailing wake surface S

Wand the cavity surface S

Care not

known a priori, the resulting problem is non linear and isnumerically solved through an iterative procedure.In the present study, the problem of determining a flow-aligned wake shape (see, e.g., Greco et al. (2004)) is notaddressed and surface S

Wis prescribed as a helicoidal sur-

face with given distribution of local pitch.2.3 Radiated pressure by the Bernoulli equationIn the framework of inviscid-flow formulations, pressureradiated by a moving body can be evaluated through theBernoulli equation (2). Specifically, the perturbation in-duced at an arbitrary location xa (acoustic observer) in thefluid domain reads

p(xa, t) = p0 (at

+ v0ax

+12|a|2

), (6)

where a = (xa). The above expression is derived fromEq. (2) in the particular case of an acoustic observer travel-ling at speed v0 along a x-axis, see Fig. 1.Equation (1) is still valid if the fluid/solid interface S

Bin-

cludes propeller and hull plate surfaces. Combining Eq. (1)and the Bernoulli theorem, a coupled hydrodynamic andhydroacoustic model for incompressible flows based onBEM is formulated.If the acoustic observer lies on S

B, the velocity potential

and its gradient are determined by the numerical solu-tion of Eq. (1) used as a boundary integral equation (withE(x) = 1/2). Next, pressure follows from Eq. (2).If the acoustic observer is immersed into the fluid region,a two-step boundary integral problem is solved. First, thevelocity potential on S

Bis determined by solving Eq. (1)

as a boundary integral equation (same as above). Next, ve-locity potential a at acoustic observer is evaluated fromEq. (1) recast as a boundary integral representation (withE(xa) = 1). Finally, pressure pa at acoustic observer fol-lows from Eq. (6).2.4 Propeller Hydroacoustics: the Transpiration Ve-

locity ModellingIn this section the Ffowcs Williams and Hawkings Equa-tion (FWHE) is proposed as hydroacoustic solver for theprediction of noise generated by marine screw propellers.Although the formulation addressed is general, the empha-sis here is posed on the modelling of cavitation effects.Specifically, a hydroacoustic solver based on the Transpi-ration Velocity Model (TVM) introduced by Testa (2008)is used to describe noise emission and radiation due to oc-currence of a fluctuating vapor cavity on the blades of apropeller operating in a wakefield. Transient cavity emis-sions are accounted for through fictitious flow velocitiesdistributed on the cavitating region of the blade.Through an elegant manipulation of mass and momentumequations and using the generalised function theory ( Faras-sat (1994)), under assumptions of compressible flow with-out significant entropy changes, a non-homogeneous waveequation may be derived. Considering two-phase flows toaddress cavitation, the additional assumption of negligiblespatial gradients of local speed of sound and density has tobe imposed. Then denoting by f(x, t) = 0 a permeablesurface S moving with velocity v and enclosing both thenoise sources and solid surfaces, the permeable form of theFW-H equation (Ffowcs Williams and Hawkings (1969),Brentner (2000) and Di Francescantonio (1997)) reads

22p =

t{[0 v + (u v)] f (f)}

[P f (f)] [u (u v)f (f)]+ { [T H(f)]} x

represents the density perturbation and c0, 0 denoting, re-spectively, the speed of sound and the density of the undis-turbed medium. The bars denote generalized differentialoperators and 22 = (1/c20)(

2/t2)2 is the general-

ized wave operator. In addition, P = (p p0) I = p Iand T = u u + (p c02) I denote the compressivestress tensor and the Lighthill tensor, respectively, u is thefluid velocity, whereas H and are Heaviside and Diracdelta functions. These two operators point out the differentnature of the source terms in the right-hand side of Eq.(7).The Dirac operator yields surface terms directly related tothe effect of the surface f(x, t) = 0, whereas the Heavi-side operator introduces volume contributions accountingfor noise sources outside this surface (quadrupole term).Akin to noncavitating propellers, the quadrupole term canbe neglected in Eq. (7) in that a small thickness attachedcavity does not induce strong velocity perturbations in theflow field. Hence, assuming the nonlinear perturbation fieldterms to be negligible, and choosing f such that |f | = 1,the boundary integral representation of the acoustic fieldgoverned by Eq. (7) is given by

p(x, t) = S

0

[vn vG

+[vn (1 v)] G]

dS

S

[(Pn) G (P n) G

]dS

S

[u n u+ G

+[u n (1 u+ )] G]

dS (7)

where each integral is expressed in a frame of referencethat is fixed with S. In the equation above, u = (u v), u+ = (u+ v), whereas

G =

[14pi r

1 + r vc0 r1

]

where r = y x and r = r, with the vectors x and ydenoting the observer and source position, respectively.In addition, the symbol ( ) denotes time derivation, whereasthe subscript indicates quantities that are evaluated at theemission time, t, which represents, given observer time,t, and location, x, the instant when the contribution to thecurrent acoustic disturbance was released from y.The above boundary integral representation for permeablesurface requires the knowledge of the kinematics of the sur-face S as well as the pressure and the flowfield velocity onthe surface itself to evaluate the pressure disturbance every-where in the field. The application of Eq. (7) to cavitatingpropellers subject to sheet cavitation is straightforward byobserving that the cavity thickness hc is very thin comparedto blade chord and assuming the surface S to be coincidentwith the blade surface S

Bwith porosity contributions from

blade regions SCB

where transient sheet cavitation occurs.Indeed, the fluctuating cavity volume produces a differencebetween the normal components of the rigidbody veloc-ity, v, and of the fluid velocity, u, that, in the body frameof reference, corresponds to

(u v) n = dhcdt

(8)

Such term, defined as cavitating transpiration velocity , isthe term through which, in Eq. (7), the effect of the dynam-ics of the bubble is included without arbitrarily introducingeffects related to (not compatible, in the integral formula-tion for rigid surfaces applied) surface deformations due tothe growth and collapse of the cavity. Hence, decomposingthe fluid density as

= 0 + (9)

where indicates the (small) density perturbation withrespect to the undisturbed medium density, and assuming

are taken into consideration, as shown in Figs. 2 where pro-peller and trailing wake are shown (flow incoming from theleft). Hydrophone coordinates with respect to the propellerare also given.

Hydrophone x/DP

y/DP

p1,p2,p3 -1.21 -1.06, 0.01, 1.06p4,p5,p6 -0.22 -0.48, 0.01, 0.48p7,p8,p9 0.43 -0.48, 0.01, 0.48

Figure 2: Propeller-hull plate configuration and location ofhydrophones. Top: hull-plate configuration (left) and un-bounded configuration (right). Middle: view from the top.Bottom: coordinates of hydrophones. (x-axis parallel topropeller axis and pointing downstream, x = 0 at intersec-tion with propeller reference plane).

Propeller operating conditions reflect a test case describedin Salvatore et al (2006). The INSEAN E779A modelpropeller operates in a non-homogeneous wakefield real-ized through a wake generator. Freestream velocity isV0 = 6.22 m/s, and propeller rotational speed is n = 30.5rps. The advance coefficient referred to free stream isJ = 0.897. See Salvatore et al. (2006) for details onthe model propeller geometry and wakefield measurementsthrough Laser-Doppler Velocimetry.Non cavitating conditions and cavitating conditions corre-sponding to cavitation number values n = 2.835, 3.645and 4.445 are considered. Figure 3 shows the cavitationpattern at n = 3.645 when the blade is in the top rightpositions (angle = 0). The extension of the cavitating

portion of the blade as a function of blade angular positionfor the three different n values is also depicted.

Figure 3: Propeller in cavitating non-homogeneous flow:J = 0.897. Left: cavity pattern at = 0, n = 3.645.Right: variation of cavity area with blade angular position.

3.1 Radiated pressure: noncavitating conditionsIn this Section a numerical comparison between the hy-droacoustics predictions performed through the Bernoulliequation model, Section 2.3, and the FWHE model, Sec-tion 2.4, is shown. The noise field generated by an iso-lated non-cavitating propeller in a non-homogeneous wakeis predicted in three representative locations, whose coordi-nates (with respect to the propeller) are specified in Fig. 2.For the sake of clarity, time domain pressure signals in-duced by one blade of the four-bladed screw are consid-ered. The analysis of Fig. 4 highlights a very good agree-ment for hydrophone 2 and 5 located, respectively, up-stream and close to the propeller disk. The agreement isworse for hydrophone 8 located downstream the propellerdisk because of the more intense acoustic effect of the trail-ing wake. Discrepancies in the acoustic signature, relatedto the spatial position of the hydrophones, have been ana-lyzed in the past by the authors (Testa et al. (2008), Testa(2008)).The guidelines derived from those studies are that, in theframe-work of potential flows and linear acoustics, theFfowcs Williams and Hawkings equation yields noise sig-nature predictions not directly affected by the presence ofthe potential wake. Specifically, only the loading noiseterm somehow accounts for the precence of the wakethrough the pressure distribution upon the blades. Theacoustics effects of the wake may be completely modeledthrough the quadrupole contribution, that is, by includingnon linear terms associated to the flow velocity. On thecontrary, the Bernoulli-based approach is directly able tocapture the acoustic influence of the shed wake. Follow-ing Testa (2008), from a theoretical standpoint the use of awake locally aligned with the fluid velocity should improvethe agreement between the two formulations.To analyse the effect of hydrophone positions on pressurefluctuations, the processing of time signals in frequencydomain is very helpful. To this purpose, the actual four

Figure 4: Non-cavitating propeller: time histories of single-blade pressure signals at hydrophones no. 2, 5, 8.

Figure 5: Non-cavitating propeller: spectrum of the pressure signals at hydrophones no. 2, 5, 8.

bladed propulsor is considered and Fourier analysis is per-formed for each corresponding time signal. Amplitudes ofFourier harmonics referred to propeller revolution periodT as the reference frequency (PF = 1/T ) are shown inFig. 5. Due to typical cancellation of signals from multiplesources (blades) whose contributions differ only in phase,Fourier harmonics are non zero only at multiples of bladepassing frequency BPF = 4 PF .The comparison among the two hydroacoustic models isgood over the whole range of frequency addressed. Com-paring different hydrophones, first harmonic intensities aremuch affected by hydrophone location upstream, at ordownstream propeller disk. Moreover, the quantitative im-portance of higher order harmonics is also dependent onhydrophone locations (radiated pressure directivity).Both Fig. 4 and Fig. 5 show pressure levels in Pascal. Adifferent way to present results is to convert intensity intoDecibel. Given a pressure signal p(t), conversion to Deci-bel is obtained as

dB = 20 log10 (1/20 p/pref ) (10)

where pref = 1 106 Pa is typically used for acoustics inwater. For hydrophone no. 5, the result is shown in Fig. 6.Comparing Fig. 6 and the second picture of Fig. 5 it maybe noted that, data presented in dB are useful to appreciatehigher harmonics having small intensity in Pascal.

Figure 6: Spectrum (in dB) of the pressure signal at hy-drophone 5

3.2 Radiated pressure: cavitating conditionsThe analysis of pressure induced by a non cavitating pro-peller is now extended to the case of cavitating flow. Op-erating conditions are same as above except for free streampressure which is lowered from atmospheric pressure to avalue corresponding to n = 3.645. Under such condi-tions, transient cavitation is present on propeller blades.Cavitating and non cavitating flow results obtained by theBernoullibased approach are first compared to highlight

Figure 7: Single-blade pressure signal in non-cavitatingand cavitating conditions at hydrophones no. 2, 5, 8.

the impact of transient cavitation on radiated pressure.Specifically, Figure 7 shows time histories of pressure in-duced by a single blade on hydrophones no. 2-5-8 over apropeller revolution period T . Principal features are:

transient cavitation on blades yields a large increase ofpressure peak intensity with respect to non cavitatingconditions;

pressure pulses due to transient cavitation present aquasi-impulsive nature, with one main (negative) peakand one-two secondary ones;

time signals reveal a very different frequency contentof radiated pressure in case of cavitating and non-cavitating flow, with high frequency contributions dueto transient cavitation.

Correlation of pressure pulses and time evolution of tran-sient cavity volume yields that the main pressure peak oc-curs when the cavity collapses (t/T = 0.5 circa, in Fig. 8).In the present case study, the wakefield incoming to the pro-peller presents an intense velocity defect with sharp bound-aries (see Salvatore et al (2006b)). This traduces into arapid collapse of blade cavity with a resulting strong pres-sure peak.Akin to non-cavitating propellers analysis, the FWHE isnow applied to evaluate the noise field generated by sheetcavitation on propeller blades, and results are comparedto those obtained through the Bernoulli equation model.Figure 8 shows that both models predict the same trendof noise signatures although some significant differences

in terms of pressure peak amplitudes appear. Such dis-crepancies are largely explained because of different com-putational schemes used by the two hydroacosutic ap-proaches to compute noise emissions due to transient cav-ities. Specifically, the Bernoulli equation captures cavita-tion noise through spatial and time first order derivatives ofthe velocity potential evaluated through boundary integralequations as Eq. (1). Local sharp variations in time andspace of the velocity potential and its normal derivative onSB

due to the transient cavity dynamics are smoothed whenusing Eq. (1) to evaluate the effects at the observer location.Differently, the integral solution by the FWH/TVM modelis characterized by the presence of time derivatives up tothe second order in the kernel of the integrals, see Eq. (10).Due to the quasi-impulsive nature of cavitation dynamicsexpecially in the collapsing phase, the overall pressure sig-nature might be affected in case of non sufficiently time-accurate hydrodynamics predictions of the cavitating flow.The investigations of modelling as well as numericalsources of noise predictions by the two approaches are un-derway and further studies are necessary. However, presentresults allow to claim that once a BEM-based hydrody-namics model provides the required input to both Bernoulliand FWH/TVM approaches: (i) both the Bernoulli equa-tion and TVM enable to describe noise radiation from anacoustic source, (ii) quasiimplusive pressure disturbancedue to cavity formation, growth and collapse phases canbe described by the two approaches although quantitativepredictions are affected by different numerical uncertainty.As a matter of fact, this is the the main difference betweenthe hydroacoustics solvers herein compared, making theTVM model more sensitive to the occurrence of cavita-tion. In the frequency domain, Fig. 9 shows how the en-ergy associated with the cyclic collapse of the cavity is re-distributed over a wide range of frequencies. Coherently tothe analysis in the time domain, the spectrum of cavitationnoise signals obtained through Bernoulli and FW-H modelsexhibits differences in terms of harmonics magnitude overthe examined frequency range (20BPF). In terms of Deci-bel noise mesurements, Fig. 10 shows how the same levelof noise is predicted throughout the overall spectra of fre-quencies. The different waveshape of pressure time his-tories determines a poor agreement at 4BPF and 8BPF fre-quencies. The agreement improves from 12BPF to higherfrequencies. Previous conclusions are confirmed by resultspresented in Fig. 11 where different cavitation conditionsare investigared.

3.3 Pressure fluctuations on hull plateAn important issue in the framework of propeller hydroa-coustics is the evaluation of pressure fluctuations inducedto the hull plate. This is a challenging task involving thehydrodynamic interactions between the hull wakefield andthe propeller, and the effects of the hull plate scattering the

Figure 8: Cavitating propeller: noise induced by a single blade at hydrophones no. 2, 5, 8.

Figure 9: Cavitating propeller: spectrum in Pascal of induced noise at hydrophones no. 2, 5, 8.

Figure 10: Cavitating propeller: spectrum (in dB) of induced noise at hydrophone no. 5.

propeller-induced noise. Here, a preliminary study is pre-sented in which the two methodologies described above areapplied to analyse the notional propeller-plate configura-tions in Fig. 2 under non-cavitating flow conditions.In the present case where a flat horizontal plate is con-sidered, the Solid Boundary Factor Sbf concept by Huse(1996) can be applied to estimate the intensity of pressurefluctuations on a solid wall through a simplified model in

which the solid wall is not explicitely taken into account inhydrodynamics calculations. Then, pressure signals deter-mined on the plate surface are very close to those evaluatedin the same locations by removing the solid plate:

ppropeller+plate = Sbf psingle propeller (11)according to Huse (1996), a factor Sbf = 2 can describewith reasonable accuracy the case of a flat solid plate. Once

Figure 11: Time signals (top) and spectra (bottom) of cavi-tation noise at hydrophone no. 5 for two different cavitationnumbers.

a Solid Boundary Factor Sbf = 2 is applied, differencesin predicted pressure signals are primarily due to the dif-ferent numerical scheme used to evaluate pressure by theBernoulli equation in the two cases.It should also be noted that the Solid Boundary Factor con-cept is strictly valid only if the plate is not affecting theflowfield around the propeller. In the present case studyaddressing the propeller-plate configuration in Fig. 2, sucha condition is largely satisfied as illustrated in Fig. 12.This figure shows calculated pressure signals at three rep-resentative hydrophone locations, p2,p5,p8, immersed inthe open flow by using the Bernoulli hydroacoustic model.Different results are obtained by using as input hydrody-namic solutions from: (i) a single propeller configuration(label isolated propeller), and (ii) a propeller-plate config-uration (label propeller and plate). Differences betweenthe two modelling approaches are negligible.The result is not general for flat horizontal plates andmostly depends on vertical distance of plate from the pro-peller. If the plate is very close to propeller blade tips, thenpropeller flow confinement effects are present and SolidBoundary Factor correction from Eq. (11) is not valid. Thepresent computational model provides a tool to determinethe range of applicability of a simplified hydroacousticmodel based on single propeller hydrodynamics and SolidBoundary Factor correction.Figure 13 shows the intensity and distribution of pressureon the plate at a representative time step, corresponding toa propeller blade in the twelve oclock position. One posi-tive and one negative pressure peak are located in the plateregion just above the propeller. Less intense pressure fluc-

Figure 12: Propeller-hull plate configuration: hydrody-namic effect of solid plate on propeller induced pressure.Hydrophones no. 2, 5, 8.

tuations (wavelets) are present over the left side of the plate,above the propeller wake (in the picture, flow is from rightto left). The effect of propeller blades rotation is that pres-

Figure 13: Propeller-hull plate: propeller-induced pressuredistribution on plate surface. Time step corresponding toreference propeller blade in twelve oclock position.

sure peaks vary in intensity and locations. This effect maybe observed from pressure maps shown in Fig. 14, whereflow is from top to bottom and blades rotate from left toright when approaching the hull plate. Six different bladeangular positions, between = 0 and = 90 with step18 degrees are represented.4 CONCLUDING REMARKSTwo hydrodynamic-hydroacoustic methodologies for theanalysis of radiated noise and pressure fluctuations inducedby non cavitating and cavitating propellers have been pre-sented.

Figure 14: Propeller-hull plate configuration: propeller-induced pressure distribution on plate surface. From top left tobottom right: time sequence corresponding to key propeller blade between = 0 and = 90 with step 18 degrees.

Propeller hydrodynamics is described by a BEM coupledwith a nonlinear unsteady sheet cavitation model, andnumerical applications address a single propeller and apropeller-plate asseembly in a strongly non-homogeneouswakefield. Hydroacoustic models are based on a standardBernoulli equation for incompressible flows and on theFfowcs-Williams and Hawkings equation with a transpira-tion Velocity Model to account for blade cavitation effects.

Numerical results are analysed in time domain and in fre-quency domain. A fair agreement between results from thetwo formulations is found for a non cavitating single pro-peller configuration. When transient cavitation occurs, it isdemonstrated that the two methodologies are able to predicttypical emitted noise features characterized by large peakintensities and frequency content covering a wide range offrequencies at multiples of blade passing frequency.

Quantitative differences between cavitating flow noise pre-dictions by Bernoulli and Ffowcs Williams and Hawkingsmodels are observed and numerical issues in the hydrody-namic solution to motivate these discrepancies are formu-lated. Numerical uncertainty in the evaluation of cavity pat-tern can have a strong impact on radiated pressure levels. Inparticular, spatial and time discretizations used for the nu-merical solution of the cavitating propeller flow representan issue and consistency of solutions should be carefullyaddressed.

Further investigations are deemed necessary to clearly as-sess the range of applicability of the two hydroacousticmethodologies and to provide guidelines for the applicationof those models to automated design in which propellernoise emission and radiation represent primary constraints.ACKNOWLEDGEMENTSPart of the work described in this paper has been per-formed in the framework of the EU-FP6 Research ProjectVIRTUE, The Virtual Tank Utility in Europe under grantTIP5-CT-2005-516201.REFERENCESBatchelor, G., editor (1967). An Introduction to

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