PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Analysis Methods and Design Measures for the Reduction of Noiseand Vibration Induced by Marine Propellers
Julian KIMMERL1; Paul MERTES1; Vladimir KRASILNIKOV2; Kourosh KOUSHAN2; Luca
SAVIO2; Mario FELLI3; Moustafa ABDEL-MAKSOUD4; Ulf GÖTTSCHE4; Nils REICHSTEIN5
12SCHOTTEL GmbH, Mainzer Straße 99, 56322 Spay, Germany

SINTEF Ocean, Otto Nielsens veg 10, 7052 Trondheim, Norway
3

4

CNR INM, Via di Vallerano 139, 00128 Rome, Italy

Technical University Hamburg-Harburg, Am Schwarzenberg-Campus 4, 21073 Hamburg, Germany
5

Fr. Lürssen Werft GmbH & Co. KG, Zum Alten Speicher 11, 28759 Bremen-Vegesack, Germany

ABSTRACTThe paper presents the ongoing work in the research project ProNoVi. The objective of the project is toimprove the numerical and experimental methods for the prediction of noise and vibrations induced by apropeller operating behind ship hull in full-scale conditions. Based on the improved methods practical
recommendations for the reduction of noise and vibration levels for diơerent classes of vessels are a further
objective. The propeller cavitation noise is identiﬁed as the dominating noise source, which coincides with
important frequencies of perception of marine fauna and thus may have negative impact on marine life. As aside eơect, reducing the noise under water will increase comfort and thus safety for crew and passengers on
board. The project aims at delivering a better understanding of fundamental physical mechanisms related toturbulence, induced vorticity and cavitation dynamics, which play a decisive role in generation of tonal andbroadband propeller noise. The project partners will give a joint presentation, focusing on state-of-the-art inthe ﬁeld, today’s challenges in both the experimental and numerical approaches, and how the ProNoVi
project is addressing those. ProNoVi is funded by the European Union. Further information is available atwww.martera.eu/projects/pronovi.
Keywords: Underwater Acoustics, Propeller cavitation, Turbulence, CFD, Ffowcs-Williams Hawkings

1. INTRODUCTION
Ships feature multiple sources of noise, with the machinery typically emitting in the frequency
ranges between 1 and 50Hz and the propeller creating peaks in the orders of multiples of its propeller
blade frequency between 10 and 200Hz. While the sources of each of these peaks can be identified
easily by their frequency, there is an additional broadband noise spectrum typically attributed topropeller induced cavitation. Both of the aforementioned noise sources factor into human health andcomfort onboard ships, the main contributor to the underwater noise spectrum is the propeller
cavitation. It appears in many applications with high area load and low cavitation numbers, withincreasing extent in off-design operation. There are cavitation types which are avoided during
propeller design, when they are known to lead to significant noise or erosion on the blade surface, forexample pressure side cavitation. Contemporary propeller designs al low for stable suction side sheet
and tip vortex cavitation. The underwater noise generation by these cavitation types and secondary
effects in combination with the turbulence, structure and rudder are the main focus in this research
1jkimmerl@schottel.devladimir.krasilnikov@sintef.no3
mario.felli@cnr.it
4m.abdel-maksoud@tu-harburg.de
5nils.reichstein@luerssen.de27397

project.
The radiated underwater noise due to shipping activities raises the natural underwater backgroundnoise level in the frequency range from 10 to 300Hz by an estimated 20 to 30 dB. With an increase ofabout 3dB per decade, this development is extremely fast in comparison to evolutionary timescales forsome of the affected sea fauna to adapt. Low frequency noise covering the 63 Hz to 125 Hz 1/3 octave
bands is dominated by propeller cavitation. This frequency band of emission coincides with important
frequencies of perception of whales, marine mammals and fish, which can be observed in masking,behavioral responses and physiological stress in the affected specimen (1). These concerns are
addressed in EU Commission decisions (2,3) and IMO Guidelines (4), which in turn leads tointernational regulations pushed by classification societies (5). Besides the noise emissions interferewith acoustic sensors of naval, research and oceanographic vessels and underwater monitoring
systems.
The project “Analysis Methods and Design Measures for the Reduction of Noise and Vibration
Induced by Marine Propellers” (ProNoVi) aims to deliver reliable tools for predicting different typesof propeller cavitation and the dynamics in operation behind a ship hull, to advance hydroacoustic andnoise propagation models in this context and to generate practical recommendations for noise
mitigation measures regarding both the radiated and on-board noise levels.

2. PROJECT STRATEGYIn previous projects, propeller induced cavitation and the following collapse of cavities are
identified as the primary sources of underwater radiated noise. These projects usually focus on either
single phenomena as source of underwater noise or are limited to specific parts of ships, particularly
the propeller. Due to the complexity of the effects causing underwater noise, these approaches givevaluable insight on the underlying physics. In ProNoVi the entire system of ship and propeller andtheir interaction is considered, in an effort to make the noise prediction technology ready for industry
use in a design tool. The output consists of recommendations and guidelines for industry, research andacademia, and regulating societies. In order to generate industry relevant expertise, the investigationconsiders three existing ship and propeller geometries, a twin screw mega yacht with fixed pitch
propellers, a 155m single screw container ship with controllable pitch propeller and a 24m twin screw
catamaran for servicing wind farms. The first is designated the reference target case, which is
examined in two experimental facilities and full-scale, and provides validation for numericalinvestigations regarding cavitation and noise radiation.
ProNoVi has four interconnected sub-projects that are aiming to advance methods in theirrespective area beyond the current state of the art. In the “experimental investigations” sub-project,methods to accurately measure noise generation are developed, validation studies for the simulations
are conducted, fundamental mechanisms of propeller cavitation and noise generation are analyzed, andthe noise reduction measures found in the other work packages are validated in two model test
facilities, SINTEF Ocean (SO) and National Research Council - Institute of Marine Engineering(INM). In the “flow simulation models” sub-project SO, SCHOTTEL, and Lürssen numericallyanalyze the target cases and identify noise mitigation measures. To reach this objective, the propeller
is simulated in open water and behind ship condition for various loadings and cavitation numbers inmodel and full-scale, mirroring the model tests. The quantification of full-scale effects on propeller
generated noise with numerical methods is an additional area of interest within the project. Tools forthe numerical investigations are StarCCM+ and OpenFOAM. Examined aspects are turbulence andcavitation models, grid and time step refinement, interface influence by sliding mesh and overset grid,
adaptive mesh refinement and modification of conservation equations to reduce numerical diffusion.
The sub-project “design oriented tools” deal with the improvement of the acoustic prediction with aboundary element method (BEM) called panMARE developed by Technical University
Hamburg-Harburg. Using the validation data from experiments and simulations, the cavitation modelis extended. “Noise reduction measures” are extracted during the project runtime from all findings,
and in parallel developed and examined with the flow simulation models and design oriented tools.

3. EXPERIMENTAL METHODS
3.1 Model-scale Data
There are few test cases with 3D data of propeller cavitation, thus validation of numerical methods7398

is often sought in visually comparing cavitation tunnel photographs. The experimental methods
employed in ProNoVi are aimed at providing high quality model scale experimental data to go in -depth
into the fundamental underlying mechanisms of noise generation and propagation, and at validating
the computational tools for cavitation and noise prediction. To this end, standard methodologies forpropeller wake flow, cavitation and underwater radiated noise assessment (i.e. op tical techniques, suchas PIV and LDV, time resolved visualizations, noise and pressure pulse measurements) are integrated
with unconventional approaches including stereometry and advanced velocity-pressure/visualization–pressure conditional techniques (6). The experimental program is conducted in the large free surface
cavitation channel of CNR-INM and in the cavitation tunnel of SINTEF OCEAN on a reference
propeller model operating in open water at different shaft inclinations and in behind -hull condition,with and without the rudder, under a comprehensive range of operative conditions in terms of propellerloading and cavitation number.
3.2 Full-scale Data
For verification purposes the project uses full-scale measurement data available for the target case.
The full-scale data is collected with the vessel passing a hydrophone on a track , as shown in Fig. 1.

Figure 1 – Full-scale measurement principle (7) and schematic of measurement track (8)It is represented in a way that the ship is assumed to be a point source. To obtain the sound source
level at a normalized distance from the received level, the propagation losses are adjustedܵ‫ܮ‬ሺ݂ሻ ൌ ܴ‫ܮ‬ሺ݂ሻ ൅ ܶ‫ܮ‬ሺ݂ǡ ܶௌ ǡ ܶோ ǡ ‫ݎ‬ǡ ‫ܦ‬ǡ ߙሻ,(1)respecting the properties of the source, depth ܶௌ and frequency, and the measurement setup, depth of
the hydrophone ܶோ , horizontal distance ‫ݎ‬, water depth ‫ ܦ‬and reflection coefficient ߙ. The water
depth is measured with the target vessels echo sounder, the hydrophone depth is calculated from thesubmerged cable length and the angle at the water surface, the distance be tween ship and hydrophone
is calculated from GPS-data, the sea floor properties are taken from a geological map. The results ofthe measurement are corrected regarding the environment influences and given as monopole source
levels according Urban (9), which ensure comparability of measurements under different conditions orwith other ships.

4. NUMERICAL METHODS
4.1 Ffowks-Williams Hawkings Equations
There are two types of numerical tools utilized in the project, which are the finite volume methodwith a certain degree of turbulence modelling (CFD), and a boundary element method as a design toolfor iterative propeller und hull design (BEM). While the near-field pressure fluctuations can beevaluated directly by CFD methods on the surfaces adjoining the sound sources, due to the assumptionof incompressibility in the applied CFD methods, the solution accuracy decreases in the far-field andconsidering reflection at the sea floor or free surface is inaccurate. Thus another method for evaluating
the far-field noise independently of the flow solution is required.
In aeroacoustics, models that do not require any flow assumptions, which are reducing the soundsources to point emitters, are frequently in use. This generalized theory is called acoustic analogy andcan be directly derived from the mass and momentum equations leading to the inhomogeneous wave
equation. While the Lighthill acoustic analogy considers sound sources in a free stream, the Curle
analogy extends this approach to consider static surfaces with reflection and diffraction. Finally the
Ffowks-Williams Hawkings analogy has no such constraints, and can be applied to moving surfaces.
The original Ffowks-Williams Hawkings (FWH) formulation (10) distinguishes the acoustic sources7399

by types, with a monopole associated with thickness ‫݌‬Ԣ ் , a dipole for the lifting forces ‫݌‬Ԣ௅ and aquadrupole for non-linear contributions ‫݌‬Ԣொ caused for example by cavitation-turbulence interaction‫݌‬ᇱ ሺ‫ݔ‬Ԧǡ ‫ כ ݐ‬ሻ ൌ ‫்݌‬ᇱ ሺ‫ݔ‬Ԧǡ ‫ כ ݐ‬ሻ ൅ ‫݌‬௅ᇱ ሺ‫ݔ‬Ԧǡ ‫ כ ݐ‬ሻ ൅ ‫݌‬ொᇱ ሺ‫ݔ‬Ԧǡ ‫ כ ݐ‬ሻ.
(2)
ᇱThe first two acoustic pressure disturbances ‫ ݌‬are evaluated on the surfaces of the noise source,
the latter is computed as a volume integral. Evaluating the spectrum for high frequencies in thisformulation requires excessive storage, as the whole solution field has to be saved for very small time
steps, which is conflicting with the goal of investigating the broadband noise appearing in highfrequency ranges, caused by propeller induced cavitation. In addition , this method demands highcomputational effort to solve the volume integral in each time step. Hence, the porous formulation of
the FWH equation (11), where the sound sources are evaluated by a surface integral on a control
volume around the propeller and its wake, is superior. In turn this necessitates a high mesh resolution
within the control surface to eliminate dispersion and dissipation errors. In this formulation the terms
lose their original physical meaning and ‫்݌‬ᇱ is called pseudo-thickness and ‫݌‬௅ᇱ pseudo-loading (12).
Assuming the control surface surrounds all non-linear sources, which is valid if the length
encompasses the full turbulent wake, their contribution ‫݌‬ொᇱ ሺ‫ݔ‬Ԧǡ ‫ݐ‬ሻ ՜ Ͳ and is thus included in thesurface integral. As the ship’s hull itself is another source, the location of the control surface is a
critical aspect of ProNoVi. For both BEM, as in house implementation, and CFD, for OpenFOAM
from the libAcoustics library (13), the Farassat 1A formulation (14), with the time derivatives as an
input into the surface integrals, is utilizedଵ ఘ ൫௩ሶ ା௩ ൯బ೙೙ሶ‫்݌‬ᇱ ሺ‫ݔ‬Ԧǡ ‫ כ ݐ‬ሻ ൌ ସగ ‫׬‬ௌ ൤ ௥ሺଵିெ
൨
ሻమ 
ೝ

 ఘ ௩ ሺ௥ெሶ ା௖ெ ି௖ெమ ሻଵೝೝ݀ܵ ൅ ସగ ‫׬‬ௌ ቂ బ ೙ ௥ మ ȁଵିெ
ȁయ 

ቃ

݀ܵ,

ೝ
௥௘௧
௥௘௧
ሶ
ଵ

௟
ଵ

௟
ି௟
ଵ
 ௟ೝ൫ೝಾሶೝశ೎ಾೝష೎ಾమ ൯ೝೝ ಾᇱ ሺ‫ כ ݔ‬ሻ‫݌‬௅ Ԧǡ ‫ ݐ‬ൌ ସగ௖ ‫׬‬ௌ ቂ௥ሺଵିெ ሻమቃ ݀ܵ ൅ ସగ ‫׬‬ௌ ቂ௥ మ ሺଵିெ ሻమ ቃ ݀ܵ ൅ ସగ௖ ‫׬‬ௌ ൤ ௥ మ ȁଵିெ ȁయ ൨ ݀ܵ.
ೝ
ೝ
ೝ
௥௘௧
௥௘௧
௥௘௧

(3)
(4)

Here ߩ଴ is density of the fluid, ‫ݒ‬Ԧ the relative velocity, ‫ݎ‬Ԧ the distance from the sound source to theobserver point ‫ݔ‬Ԧ, ‫ ܯ‬the Mach number, ܵ the body surface and ݈Ԧ ൌ ‫݊݌‬ሬԦ the pressure in surface unit
normal direction ݊ሬԦ. The subscripts represent a dot product of the vector with the unit normal vector ݊ሬԦ
on the surface into the fluid, its time derivative ݊ሬԦሶ, the unit radiation vector ‫ݎ‬Ԧመ or the surface velocityሬሬԦ. Integrands with ‫ି ݎ‬ଵ can be considered far-field and with ‫ି ݎ‬ଶ near-field
Mach number vector ‫ܯ‬terms. With the sound sources on the control volume, the far-field sound pressure is known for
arbitrary points, not confined to the initial simulation domain, and at all times within the simulation
time range, by interpolation between adjacent time steps.
Due to the sound velocity, the values on the control surface are generated from the time of emission,
by linearly interpolating the locations of surfaces at the retarded time between two adjacent time steps‫ݐ‬ଵ ൑ ‫ݐ‬௥௘௧ ൑ ‫ݐ‬ଶ considering the distance from source to observer‫ݐ‬௥௘௧ ൌ ‫ כ ݐ‬െหȁ௫Ԧሺ௧ ‫ כ‬ሻି௬ሬԦሺ௧ሻȁห௖

,(5)with ‫ݕ‬Ԧ as the sound source point.
Except for the cavitation tunnel test simulations, reflection modelling is neglected within the CFD
methods, due to the size of the domain. It is a very important factor in determining the influence of theexperimental facility and compare with the full-scale measurements. A comparison regarding
reflection at surfaces is shown in the non-cavitating propeller induced pressure field calculated with
panMARE in Fig. 2. Although it is not included in the flow simulation, reflection is modelled in thesolution of the FWH equation by mirroring the sources at the reflection surfaces, taking into accountthe acoustic properties of both materials at the interface. This results in a correction with a reflection
factor ܴ, ranging between 0 for no reflection and 1 for full deflection, and pressure ‫୻݌‬ᇱ on the sum ofpressures (15)ܴൌ௠ ୡ୭ୱమ ஀ି௡ටଵି

౩౟౤మ ౸
೙మ௠ ୡ୭ୱమ ஀ା௡ටଵି

౩౟౤మ ౸೙మ,ᇱሺ‫ݔ‬Ԧǡ ‫ כ ݐ‬ሻ ൌ ‫݌‬ᇱ ሺ‫ݔ‬Ԧǡ ‫ כ ݐ‬ሻ ൅ ܴ‫୻݌‬ᇱ ሺ‫ݔ‬Ԧǡ ‫ כ ݐ‬ሻ,‫݌‬௦௨௠
ఘభ

(6)
(7)

௖

with ݉ ൌ ఘ and ݊ ൌ ௖ and the incident angle ȣ. On the boundaries to materials more dense thanభwater the root term in Eq. 6 becomes negative, so reference values from the literature are utilized tomodel measurement conditions (16).

7400

Figure 2 - Comparison of normalized pressure amplitudes of the non-cavitating propeller and hull with (right)
and without (left) reflections at the free surface and the sea floor
4.2 Frequency Domain Evaluation
The time domain results at the specified measurement points from the model and full -scalemeasurements are converted to a frequency domain spectrum with a fast Fourier transform. To reduce
spectral leakage, a Hahn window is using integer multiples of the most significant period in the time
domain, which typically means the propeller blade frequency. The pressure amplitudes in thefrequency domain are then converted to sound pressure levels with the reference pressure identical to
the full-scale measurements of ‫݌‬௥௘௙ ൌ ͳߤܲܽ௣ᇲܵܲ‫ ܮ‬ൌ ͳͲ  ൬௣
4.3

ೝ೐೑

ଶ

൰ .

(8)

Computational Fluid Dynamics

4.3.1 Turbulence modelling
For the scale-resolved turbulent finite volume investigations, the implicit large eddy simulation(ILES) in OpenFOAM and improved delayed detached eddy simulation (IDDES) in StarCCM+ arechosen, as they promise the best results with the least computational effort, such that they are still
possible to be applied in the industry in certain cases. Deducting from previous works by other authors(17) and simulations with a simple twisted hydrofoil shown in Fig. 3, the quality of the resolution of
turbulence regarding sheet cavitation in an ILES appears adequate for ProNoVi.

Figure 3 - Comparison experiment (18) and ILES of sheet cavitation pattern over Delft Twist11 Foil
Both CFD approaches rely on the mass and momentum equations to resolve the flow field up to a
turbulence scale defined by a filter function ‫ ܩ‬ൌ ‫ܩ‬ሺ‫ݔ‬Ԧǡ ȟሻ
(9)߲௧ ߩ ൅ ‫ ڄ ׏‬൫ߩ‫ݒ‬Ԧҧ ൯ ൌ Ͳ,ഥ െ ࡮ሻ.
(10)߲ ൫ߩ‫ݒ‬Ԧҧ ൯ ൅ ‫ ڄ ׏‬൫ߩ‫ݒ‬Ԧҧ ٔ ‫ݒ‬Ԧҧ ൯ ൌ െ‫݌׏‬ҧ ൅ ‫ ڄ ׏‬ሺࡿ௧In this derivation (19), which neglects commutation errors, filtered variables are shown by the overbarଵand ࡿ ൌ ૛ߤࡰ is the viscous stress tensor with the strain rate tensor ‫ ܦ‬ൌ ଶ ሺ‫ݒ׏‬Ԧ ൅ ‫ݒ׏‬Ԧ ் ሻ and the
dynamic viscosity ߤ. In the unresolved transport term ‫ ࡮ ڄ ׏‬the subgrid stress tensor is introduced(20)തതതതതതതതത
෩ ቁ.࡮ ൌ ߩ ቀ‫ݒ‬Ԧҧ ٔ ‫ݒ‬Ԧҧ  െ ‫ݒ‬Ԧҧ ٔ ‫ݒ‬Ԧҧ ൅ ࡮(11)෩
The ILES approach requires no subgrid viscosity to model the remaining ۰ as the dissipative
qualities can be mirrored by the truncation error ɒ of the convective terms (21). In ProNoVi theimpact of different discretization approaches on the numerical diffusion are in vestigated. The

7401

linear-upwind stabilized transport scheme (LUST), which combines the error of the central
differencing (CDS) and linear upwind differencing scheme (LUDS) ɒ୐୙ୗ୘ ൌ ͲǤ͹ͷɒେୈୗ ൅ ͲǤʹͷɒ୐୙ୈୗ,promising good results in accordance with the correct high wavenumber damping yielding the -5/3
slope of the logarithmic Kolmogorov turbulence energy spectrum in the inertial subrange .
For IDDES the values below the filtered scale are modelled using the RANS k-Z-SST turbulencemodel with All Y+ Treatment algorithm at solid boundaries. The DDES method introduces a delay
factor that enhances the ability of the model to distinguish between LES and RANS regions on meshes
where spatial refinement could lead to ambiguous behavior. Further, in the improved (IDDES)
formulation, the sub-grid length-scale includes a dependence on the wall distance. This approachallows RANS to be used in a much thinner near-wall region, thus providing some wall-modelled LES
capabilities. A hybrid 2nd-order upwind/bounded-CDS is used for modelling the convection terms.
Validation cases studied in ProNoVi, for example Fig. 4 (left), show that the present IDDES method
yields stable solution through the range of propeller loadings, in both non -cavitating and cavitatingcases, and the predicted propeller forces are close to the results obtained with the RANS method.
4.3.2 Vorticity confinement
To counter numerical diffusion in the tip vortex due to the convective term in OpenFOAM, theadaptive vorticity confinement method (VC) artificially convects vorticity back to the vortex center byadding a source term S on the right hand side of the incompressible momentum equation (22)
ሬሬሬԦȁ
‫׏‬ȁఠ
ܵԦ ൌ ߝ‫ݏ‬Ԧ ൌ ߝ
ൈ߱
ሬԦ.
(12)
ሬሬሬԦȁห
ห‫׏‬ȁன

In its original formulation the user has to define a proportionality factor ߝ to control the strength ofሬԦ magnitude contour lines, which
the source term, influencing a vectorial quantity along the vorticity ߱necessitates a calibration. To circumvent this, the adjusted model (23) estimates the factor byሬԦ between the CDS and LUST scheme
approximating the diffusive error ‫ܦ‬ଵሬԦ ‫ݏ ڄ‬Ԧ.
ߝ ൌ ȁ௦Ԧȁమ ‫ܦ‬
(13)
஼஽ௌ
௅௎ௌ்
ሬԦ ൎ ሾሺ‫ݑ‬The approximation is ‫ܦ‬ሬԦ ‫׏ ڄ‬ሻ‫ݑ‬ሬԦሿെ ሾሺ‫ݑ‬ሬԦ ‫׏ ڄ‬ሻ‫ݑ‬ሬԦሿ
and the absolute value of ߝ has to beentered into Eq. 12, as unphysical sign changes may appear near the vortex core. The method isimplemented in OpenFOAM (24) and comparison with RANS calculations, in Fig. 4 (right) for ahydrofoil test case (25), confirm the applicability in ProNoVi.

Figure 4 - PPTC propeller case (left), vorticity magnitude and resolved tip vortex: hydrofoil case (right)
In Star-CCM+ an extended rotating region (Sliding Mesh) around the propeller in combination with
Adaptive Mesh Refinement (AMR) is used instead of VC method, to counter the effect of numerical
diffusion. In the presence of a rudder, an Overset Mesh method is investigated, to extend the rotating
region over the rudder in an effort to reduce diffusion at the mesh interface. In preliminary propeller
open water tests with the AMR method it is found that the cell size in the tip vortex area needs to be as
fine as 0.125% of the propeller diameter in non-cavitating cases, and 0.06-0.08% in cavitating cases.
4.3.3 Phase-Change
To model cavitation the volume of fluid (VOF) method is selected, as it is still computationallyfeasible and is not oversimplifying. As the two phases are resolved on an Eulerian grid, the fluid
interfaces are reconstructed depending on the phase contents in boundary cells. With the exact value
unclear this is a topic of discussion, as it is expected that the resulting cavity size has significant
impact on noise generation. In tested cases the Full Cavitation Model by Singhal (26) as implemented
in OpenFOAM proves numerically unstable and in need of additional calibration. The Full
Rayleigh-Plesset cavitation model implemented in Star-CCM+ (27), which is based on the classical(non-simplified) Rayleigh–Plesset equation, is also tested with the PPTC propeller case. It is found7402

quite sensitive to the solution state at the moment when the model is enforced, and it requires a verysmall time step. For these reasons, the asymptotic Schnerr-Sauer cavitation model (28) is presently
selected for the studies in ProNoVi, which is independent of empirical values. The applicability tosound source modelling is validated by other authors (9) and in the simulations shown in Fig. 3. An
additional scalar transport equation is solved for the volume fractions of the phases with an equivalentmass transfer rate of݉ሶ ൌ ‫ܥ‬

ఘ೗ ఘೡ
ఘ

మ

ଷ
ሺͳ െ ߙሻߙ ඨయோሺ௣ି௣ೡ ሻ
ఘ೗

,

(14)

where R depends on the water quality properties, nuclei density and mean nuclei radius. In thederivation from the Rayleigh-Plesset equation, the effects of surface tension, viscosity andthermodynamics are neglected and higher order terms are ignored.
4.4 Boundary Element Method
The BEM approach is based on the assumption of irrotational, incompressible and inviscid flow
with the governing equations reducing to the Laplace equation for the velocity potential Ȱ and the
Bernoulli equation for the pressure ‫( ݌‬29)ȟȰ ൌ ‫׏‬ଶ Ȱ ൌ Ͳ,
(15)
ͳ
߲Ȱ
(16)
‫ ݌‬൅ ߩ݃‫ ݖ‬൅ ߩሺ‫׏‬Ȱሻଶ ൅ൌ ܿ‫ݐݏ݊݋‬Ǥʹ߲‫ݐ‬
Applying the Green identity with an impermeability condition on the body boundaries and thecavity surfaces and taking into account the Kutta boundary condition at the trailing edge to create theforce-free propeller wake, a unique solution to Eq. 15 is found (30). The cavity surfaces are calculated
iteratively with an additional dynamic boundary condition requiring the vapor pressure ‫݌‬௩ on thesurface. panMARE is solving this boundary value problem numerically with a low order panel-method
on quadrilateral panels with constant monopole and dipole strengths on the panels (16). From thisresults a velocity distribution and the corresponding pressure distribution from Eq. 16.

5. OUTLOOKPractically, the results of ProNoVi improve compliance of ships and propeller designs withregulations, reduce impacts on the environment, and improve safety and comfort on board ships. The
findings serve for elaboration of regulations and guidelines developed by classification societies. The
methods developed, facilitate the evaluation of design concepts and troubleshooting of existing
designs. For ship operators it results in long-term economic advantages, as compliance withinternational regulations can be achieved with efficient mechanisms and renovation costs can be
reduced. Scientifically the projects results have high value in ship hydrodynamics, numericalmodelling of full-scale ship flows, flow unsteadiness, turbulence, cavitation and hydroacoustics.
Subsequent research can be expected considering incompressibility effects, interface capturing of the
mixture and possibly energy transfer in cavitation processes.

ACKNOWLEDGEMENTS
The authors are very grateful for the support and funding by MarTERA, represented by
BMWi-project (03SX461C) “ProNoVi” for the German partners, the Research Council of Norway for
the Norwegian Partners (Project 284501) and MIUR-project “ProNoVi” for the CNR-INM.

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3. EU. Commission Decision 2017/848/EU. 2017;
4. IMO. Guidelines for the Reduction of Underwater Noise from Commercial Shipping to Address
Adverse Impacts on Marine Life. 2014.
5. DNV-GL. Silent Class Notation. 2010.
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