IS- Computational Environmental Mechanics for the Blue GrowthAn Overview on the use of the Ffowcs Williams-Hawkings Equation for the Hydroacoustic Analysis of Marine PropellersS. Ianniello and C. Testa

VIII International Conference on Computational Methods in Marine Engineering MARINE 2019

R. Bensow and J. Ringsberg (Eds)

AN OVERVIEW ON THE USE OF THE FFOWCS WILLIAMS-HAWKINGS EQUATION FOR THE

HYDROACOUSTIC ANALYSIS OF MARINE PROPELLERS

IANNIELLO S. AND TESTA C.

Institute of Marine Engineering (INM) CNR - National Research Council

Via di Vallerano, 00128 Rome, Italy e-mail: sandro.ianniello@cnr.it, claudio.testa@cnr.it

Key words: Hydroacoustics, Ffowcs Williams-Hawkings, Acoustic Analogy

Summary. Since the last 70s, the integral formulations solving the Ffowcs Williams-Hawkings equation are the standard approach for the prediction of noise generated by a body moving in a fluid flow and, in particular, propulsion and/or lifting devices based on rotating blades. This methodology represents the base of research and commercial software used by aeronautical industry and is more and more being applied to naval sector too, in the attempt of providing the shipbilding industry with effective predictive tools, which to fulfill the stringent regulations on underwater noise emission with. The paper offers a brief overview on the use of the Acoustic Analogy for marine propeller hydroacoustics. At first, we propose a comprehensive numerical analysis which emphasizes the intrisinc, nonlinear nature of the problem. Then, some possible computational strategies to evaluate the noise induced by a sheet cavitation phenomenon are proposed and compared. Some numerical results are presented, by avoiding as much as possible any mathematical detail on the adopted, integral formulations and focusing the attention of the significant capabilities and the effectiveness of the methodology.

1 INTRODUCTION The Ffowcs Williams-Hawkings (FWH) equation published in 1969 [1] represents an extension of the original work of Lighthill on the aerodynamically generated sound [2] and governs the noise generated by any body moving in a fuid flow. It may be easily derived from the fundamental conservation laws of mass and momentum, expressed in terms of generalized functions, by representing the presence of the body as a “discontinuity” in the fluid field. Under the assumption of negligible effects of viscosity on sound generation (which reduces the compressive stress tensor to the scalar pressure field) and isentropic transformations for the fluid (which allows to approximate the pressure-density relationship with the linear term of its series expansion, so that the acoustic pressure is expressed by 𝑝𝑝′ = 𝑐𝑐0

2�̃�𝜌, being �̃�𝜌 the density perturbation and 𝑐𝑐0 the constant speed of sound), this differential, non-homogeneous wave equation reads

𝔻𝔻2𝑝𝑝′ = 𝜕𝜕𝜕𝜕𝜕𝜕 {[𝜌𝜌0𝑣𝑣𝑛𝑛 + 𝜌𝜌∆𝑛𝑛

𝑢𝑢𝑢𝑢]𝛿𝛿(𝑓𝑓)} − 𝜕𝜕𝜕𝜕𝑥𝑥𝑖𝑖

{[𝑝𝑝�̂�𝑛𝑗𝑗 + 𝜌𝜌𝑢𝑢𝑖𝑖∆𝑛𝑛𝑢𝑢𝑢𝑢]𝛿𝛿(𝑓𝑓)} + 𝜕𝜕2

𝜕𝜕𝑥𝑥𝑖𝑖𝜕𝜕𝑥𝑥𝑗𝑗[𝑇𝑇𝑖𝑖𝑗𝑗𝐻𝐻(𝑓𝑓)] (1)

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where the D’Alembert operator is expressed by

𝔻𝔻2 = 1𝑐𝑐0

2𝜕𝜕2

𝜕𝜕𝜕𝜕2 − ∇2, and the function 𝑓𝑓 = 0 (being 𝑓𝑓 is the support of the Dirac and Heaviside funcions, 𝛿𝛿 and 𝐻𝐻), represents a radiating surface 𝐷𝐷 in the three-dimensional space. In equation (1), 𝜌𝜌0 is the fluid density of the undisturbed medium; ∆𝑛𝑛

𝑢𝑢𝑢𝑢= 𝑢𝑢𝑛𝑛 − 𝑣𝑣𝑛𝑛, where 𝐮𝐮 and 𝐯𝐯 are the velocity of the fluid and the surface 𝐷𝐷, respectively, and the subscript 𝑛𝑛 indicates the projection along the outward normal direction to 𝐷𝐷; the same direction is identified by the 𝑗𝑗-component of the unit normal vector �̂�𝐧, while 𝑝𝑝 = 𝑝𝑝 − 𝑝𝑝0 is the pressure disturbance and 𝑇𝑇𝑖𝑖𝑖𝑖 = 𝑢𝑢𝑖𝑖𝑢𝑢𝑖𝑖 + (𝑝𝑝 − 𝑐𝑐0

2�̃�𝜌)𝛿𝛿𝑖𝑖𝑖𝑖 the Lighthill tensor, with 𝛿𝛿𝑖𝑖𝑖𝑖 the Kronecker symbol. The role played by the domain 𝐷𝐷 is essential and strongly characterizes the solving approach. When 𝐷𝐷 coincides with the surface 𝑆𝑆 of the (rigid) moving body, the impermeability condition (∆𝑛𝑛

𝑢𝑢𝑢𝑢= 0) simplifies equation (1) and the three source terms on the right-hand side identify the well known thickness, loading and quadrupole noise components [3, 4]. Otherwise, the integration domain is immersed in the flow field and the solution is achieved by adding the contribution from the first two surface integrals, determined on the porous, radiating domain 𝐷𝐷: 𝑓𝑓 = 0, and the third, volume term, computed in the whole flow region 𝑉𝑉: 𝑓𝑓 > 0, affected by the body motion [5]. Theoretically speaking, the choice of a domain 𝐷𝐷 embedding all possible noise sources, makes the volume term’s contribution null and enables the evaluation of 𝑝𝑝′ by surface integrals only. In any case, equation (1) is rewritten in an integral form through the Green method and the use of the free-space function

𝐺𝐺(𝐱𝐱, 𝜕𝜕; 𝐲𝐲, 𝜏𝜏) = 𝛿𝛿(𝑔𝑔)4𝜋𝜋𝜋𝜋 ; 𝑔𝑔 = 𝜏𝜏 − 𝜕𝜕 + 𝜋𝜋

𝑐𝑐0, (2)

where 𝐱𝐱 and 𝒚𝒚 represent the observer and the source locations, 𝜕𝜕 and 𝜏𝜏 are the observer and emission times and 𝜋𝜋 = |𝐱𝐱(𝜕𝜕) − 𝐲𝐲(𝜏𝜏)| the source-observer distance. As said, the FWH-based integral formulations represent the standard approach in Aeroacoustics and are widely used by aeronautical industry. In the last years, this methodology is catching on in the naval sector, where the stringent regulations on underwater noise emission push the shipbuilding industry to acquire and use hydroacoustic predictive tools, possibly at a design stage. Of course, equation (1) governs the sound generated by a body moving both in air and underwater; nevertheless, the deep differences of the physical characteristics of these fluids and, consequently, of the conditions at which analogous devices can operate, should lead to a careful and proper interpretation of the numerical solutions and to take nothing for granted. In this context, it has been shown that, unlike analogous aeronautical devices and regardless of the low rotational speed, the hydroacoustic field of a marine propeller is dominated by the non linear (quadrupole) term of equation (1). This behavior is related to the intrinsic features of propagation mechanisms of a rotating source and the significant destructive interference occurring in multibladed devices. A pseudo-analytic demonstration of these assertions may be found in [6]. Here, the same conclusions are carried out through a numerical simulation.

2 NON CAVITATING PROPELLER IN OPEN WATER A FWH-based hydroacoustic analysis of a conventional marine propeller in open water (uniform, nocavitating flow) is here taken into account. The tested device is the INSEAN

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E779A propeller and the analysis refers to an advance ratio 𝐽𝐽 = 0.71, for which some complete sets of data are avalaible, coming from both a DES (Detached Eddy Simulation) and a RANS (Reynolds Averaged Navier Stokes) hydrodynamic simulation. The rotational speed is fixed to n = 25 rps, corresponding to 𝑀𝑀𝑅𝑅 = 0.0118. Due to the lack of underwater noise measurements and despite the incompressibility assumption characterizing both the RANS and DES data, the FWH noise predictions is somehow validated through a direct comparison with the pressure signatures provided by the hydrodynamic solvers (see [7]).

Figure 1- The E779A propeller, the porous domain 𝑆𝑆𝑃𝑃 and hydrophones.

The prediction of noise is pursued in time domain through both the linear formulation 1A [4] and the porous approach [5] in order to identify the actual contribution of linear and nonlinear sources. The hydrodynamic mesh is constituted by several 3D blocks. In particular, the flow around the propeller is determined on subsequent, cylindrical layers which can act as porous domains 𝑆𝑆𝑃𝑃; furthermore, also the noise measurement points are selected among the computational nodes located outwardly 𝑆𝑆𝑃𝑃, as to avoid any data-fitting procedure. Figure 1 shows one of these surfaces embedding the propeller and the numerical hydrophones, aligned to the flow and located on two horizontal lines, named 𝐻𝐻 and 𝐾𝐾. These points have to fulfill two conflicting requirements: they must be positioned outside the porous domain and, then, far enough from both the body and all nonlinearities occurring in the flow; at the same time, they should be also located in a region far from the computational boundaries, where the pressure is hopefully determined with a good accuracy. The differences between the RANS and DES simulations are well known. A standard RANS approach averages the value of the velocity everywhere: the turbulent viscosity is convected in the flow by a transport equation and the modeling of the velocity turbulent component always gives rise to a significant diffusion of the numerical solution. In particular, the vorticity in the downstream region is soon smeared out. In a DES approach, on the contrary, the eddy viscosity depends on a link between the mesh spatial resolution and the distance from the rigid surfaces: the more the computational cell is small, the more the viscosity reduces moving far from the body. This means that by using a rather fine mesh, the blade detached eddies are determined in a sort of direct way and the undesirable numerical damping is notably reduced. It is important to point out that both the approaches provide the same pressure distribution upon the blade surface, as confirmed by the

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very good comparison with experimental data (not shown here for brevity). Then, if the interest is focused on the propeller performances (thrust, torque, efficiency), as well as on the computation of the FWH linear terms, their effectiveness and reliability are, in essence, equivalent. On the contrary, the capability to model the vorticity in the downstream region is significantly different, as it clearly appears in figure 2 (see [8]).

Figure 2- RANS (left) and DES (right) solutions, at 𝐽𝐽=0.71.

Figure 3- RANS, DES and FWH (linear) pressure signals in the proximity of the propeller.

It is worth noting that the mesh used for the two runs is exactly the same, so that the only difference stands in the aforementioned way the turbulent component of the velocity is taken into account. For additional details about the hydrodynamic solution, the reader can refer to [9, 10]. Figure 3 shows the pressure time histories determined by DES and RANS simulation and the corresponding noise prediction carried out by the FWH linear formulation 1A (using the

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DES hydrodynamic data) at different points. Very close to and upstream the propeller (points 𝐻𝐻1 and 𝐾𝐾1, top pictures), all the signatures are totally characterized by the BPF and the linear noise prediction matches very well the two (coincident) time histories provided by the hydrodinamic solvers. As we move from the body, however, the DES solution starts to exhibit a more irregular waveform and to differ from the other signatures. This difference immediately becomes visible at more distant points, like 𝐾𝐾1 or 𝐾𝐾3, (bottom pictures). There, the DES pressure is always characterized by more pronounced fluctuations and, apparently, by a higher frequency content, while the RANS pressure still agrees well with the linear noise prediction and exhibits a much more regular shape, with four well defined peak values corresponding to the propeller blades. Such a persisting, good agreement between the RANS and the FWH-based solutions is rather self-explained. Both of them, in fact, are not able to account for the same, fundamental phenomena occuring in the field, that is the vorticity spreading in the downstream region. The RANS solution is uncapable to capture this effect because of the numerical diffusion, while the linear FWH solution does not include it explicitly. Not surprisingly, the differences with the DES pressure become more and more pronounced at the furthest points from the body, where the pressure is more affected by the whole 3D vorticity field, compared to points where 𝑝𝑝 is dominated by the tonal component, ruled by the blades' passage. In fact, behind points 3, both the RANS and the FWH linear solutions lose any reliability.

Figure 4- RANS, DES and FWH (linear) numerical solutions at points 𝐻𝐻4 and 𝐾𝐾5.

Figure 4 just reports the results at point 𝐻𝐻4 (left) and 𝐾𝐾5 (right). While the DES pressure goes on to exhibit a more and more irregular, fluctuating waveform the RANS and the FWH solutions rapidly approach zero. Due to the closeness of the measurement points to the propeller, this annihilation of pressure is clearly unrealistic. Nonetheless, the two unreliable solutions are characterized by a significant dissimilarity. The RANS pressure is actually wrong, being related to the numerical diffusion (see figure 2) and the inherent inability of the computational model. The acoustic solution, on the contrary, is no doubt correct: the linear terms, in fact, only depend on blades' shape and hydrodynamic (DES) loads, that is the same quantities providing the very good and reliable noise predictions in the proximity of the blades' tip. Then, the vanishing of the FWH noise signature appears as an incontrovertible fact, due to the unambiguous lack of the nonlinear effects and, above all, the behavior of the linear terms for any propeller rotating at 𝑀𝑀𝑅𝑅~0.01. Let us focus, now, our attention on the actual role played by the nonlinear noise sources, here computable through the porous formulation. Figure 5

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shows the comparison between the DES hydrodynamic pressure, still the FWH linear noise prediction and the porous solution achieved by using the integration domain depicted in figure 1. Upstream the propeller (point 𝐻𝐻1, top-left picture) the porous and linear formulations provide the same noise signature and agree very well with the DES data; this confirms the negligible contribution of the quadrupole term in the strict proximity of the propeller .

Figure 5- Comparison between DES pressure and the noise predictions achieved by the FWH linear (1A) and porous formulations, at points 𝐻𝐻1, 𝐻𝐻3, 𝐻𝐻5 and 𝐻𝐻6.

Nevertheless, as we move toward the downstream region, the solution is affected by the well known end-cap problem and exhibits an undesired behavior. At the downstream point 𝐻𝐻3 (top-right), for instance, the linear formulation shows a slight underestimation with respect to the DES pressure, reasonably attributable to the effects of the blade tip vortex. The accounting for the nonlinear sources, however, does not improve the solution and, actually, seems to provide a worse result. Such a worsening becomes dramatic at points 𝐻𝐻5 and 𝐻𝐻6 (bottom pictures), where the noise predictions appear completely unreliable. As known, the reason for such a poor quality result is the crossing of the porous domain by the tip vortex, which represents an indirect demonstration of the fundamental role played by the vorticity field. In fact, from a theoretical point of view, the porous formulation imposes two conditions on the integration domain 𝑆𝑆𝑃𝑃: i) to be a closed surface and ii) to embed all possible noise sources. The domain depicted in figure 1 satisfies the first condition, but, due to the unavoidable limits of the computational domain, it cannot guarantee the fullfilment of the second one. Even worst. The downstream cap of the cylincrical domain 𝑆𝑆𝑃𝑃 not only does not embed the whole vorticity field, but it is totally immersed in the blade tip vortex spreading downstream the body, which there, presumably,

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represents the dominant source of noise. Thus, the results in figure 5 simply represent some not converged solutions. Figure 6 shows the same noise signatures at points 𝐻𝐻3, 𝐻𝐻5 (figure 5), here determined by splitting the integration domain 𝑆𝑆𝑃𝑃 in three parts: the open cylinder, encircling the propeller and the whole blade tip vortex, and the two caps located upstream and downstream the body. As clearly shown in the left picture, the slight discrepancy occurring at point 𝐻𝐻3 between the DES pressure and the porous formulation comes from the oscillating waveform imposed by the contribution of the downstream cap of 𝑆𝑆𝑃𝑃, while the signature provided by only the open cylinder perfectly overlaps the DES data.

Figure 6- The FWH porous solutions at 𝐻𝐻4 and 𝐾𝐾5, as splitted from different parts of 𝑆𝑆𝑃𝑃.

Moving toward the same downstream cap, its detrimental effects obviously increase, and at point 𝐻𝐻5 (right picture) the overall signature is dominated by a fictituos oscillation of notable amplitude. At the same time, the contribution from the upstream cap (where the flow is very smooth) is always negligible and the noise signature provided by the open cylinder seems to match the DES pressure in a satisfactory way. The numerical effects related to the end-cap problem may be faced in different ways [11, 12, 13, 14]. Nevertheless, what we wish to point out here is that these effects do not concern the FWH linear solution at all. They point out how the tip vortex released in the field represents the main source of noise and confirm the intrinsic, nonlinear nature of the hydroacoustic field generated by a marine propeller, regardless of the low values of the rotational speed.

3 SHEET CAVITATING PROPELLERS Another important and not standard application of the FWH equation concerns the assessment of noise induced by sheet cavitation phenomena. Acoustically speaking, cavitation is highly undesirable, as it induces and impulsive sound and deeply modifies the baseline acoustic signature of the propeller. These effects are inherently related to the spectrum of the high-energy radiated noise, that exhibits a low frequency range, governed both by tones (multiple of the blade passage frequency) and broadband hump (due to the large scale cavity dynamics), and a higher frequency broadband range due to the collapse of vapour bubbles [15, 16]. Furthermore, the occurrence of cavitation makes the detection of the sources of sound a very complicated and partially unsolved problem. In fact, the modern CFD is able to provide a satisfactory estimation of cavitation patterns [17], but a reliable simulation of important underlying phenomena (especially those related to cavities collapsing stage) is still far from being achieved.

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Such a modeling uncertainty seems to be less critical in case of a sheet cavitation, which frequently occurs on conventional propellers operating in the hull wake field. It consists of a relatively thin vapour region which typically forms at blade leading edge, fluctuates in size in a limited azimuth range and eventually collapses, always remaining essentially attached to the blade surface (see figure 7). Under the assumption that: i) cavitation pockets remain attached to the blades surface and ii) the collapse of the cavity, due to condensation, does not imply violent implosions so that vapor bubble evolves in a smooth way (by progressively reducing its size up to disappear), a potential-flow hydrodynamics yields a reliable description of the cavity dynamics in terms of inception, growth and collapse [17].

Figure 7 – Sheet cavity time evolution.

In this framework, two alternative FWH-based formulations may be applied to predict the radiated noise. The first approach, named Equivalent Blade Modeling (EBM), simply adopts the standard linear formulation 1A and uses a time-varying integration domain corresponding to blade plus cavity (𝑓𝑓 = 0: 𝑆𝑆𝐵𝐵 + 𝑆𝑆𝐶𝐶). In this way, the possible presence of the cavity is essentially taken into account by the FWH thickness component, through the shape’s variations of an alternative (virtual) source-body. Actually, the method violates the basic assumption of a rigid body which the FWH-based formulation is based on, but in virtue of the aforementioned approximations, the cavity time evolution may be considered as an ordered sequence of steady-states. The second approach, referred to as Transpiration Velocity Modeling (TVM), is more rigorous: it basically arises from the porous formulation and adopts a velocity/acceleration transpiration term (extracted from the hydrodynamic solution) to impose a porous boundary condition on the portion of the body surface affected by the cavity (𝑆𝑆𝐶𝐶𝐵𝐵). This condition is expressed by the relation

(𝐮𝐮 − 𝐯𝐯) ∙ 𝐧𝐧 = 𝑑𝑑ℎ𝐶𝐶𝑑𝑑𝑑𝑑 ,

where ℎ𝐶𝐶 represents the time-dependent cavity thickness. Then, a 2D integral on 𝑆𝑆𝐶𝐶𝐵𝐵 adds to the classic FWH thickness and loading components and accounts for the hydroacoustic effects of the vapour cavity. The TVM approach establishes an important correlation between radiated noise and sheet cavitation pattern, yielding a mathematically-consistent description of blade-attached and fluctuating vapour pockets into an integral formulation for undeformable bodies. A schematic representation of the differences between these alternative schemes is depicted in figure 8, while the mathematical details may be found in [18, 19]. Hereafter, some numerical results concerning the analysis of the E779A propeller model, operating within the nonuniform onset flow depicted in the left picture of figure 9, are presented. The free-stream flow speed is 𝑈𝑈∞ = 6.24 m/s and propeller rotational speed is 𝑛𝑛 = 30.5 rps, corresponding to an advance

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coefficient 𝐽𝐽 = 0.9 and a cavitation number 𝜎𝜎 = 4.455. The large velocity defect in the core of the wake is expected to determine strong variations of pressure distributions on the blades and, hence, to induce a transient cavitation.

Figure 8 – Comparison between EBM (right) and TVM (left) schemes.

A fully-validated BEM (Boundary Elements Method) code, solving the Laplace equation for 3D unsteady flows around lifting bodies, working in arbitrary onset flows under unsteady sheet cavitating conditions, is used to provide the input data to the acoustic codes. The noise predictions refer to three numerical hydrophones located in the proximity of the body: point 𝑃𝑃2 is located in the disk plane, while points 𝐻𝐻4 and 𝐻𝐻5 are representative for the downtream and usptream regions, respectively (right picture of the same figure 9). Once the time histories of pressure and cavity thickness distributions are determined by the BEM solver, the noise is determined through the TVM method. It is interesting to note that the hydrodynamic solution refers to the actual four-bladed propeller, while the noise is determined (and shown) just for a single blade, as to emphasize the role of cavitation on the resulting waveform.

Figure 9 – The nonuniform inflow inducing a sheet cavitation on the E779A propeller model and the hydrophones used for the corresponding hydroacoustic analysis. Left pictures in figure 10 show the comparison between the noise predictions achieved in absence of cavitation (wet condition) and in presence of a cavity (TVM scheme), at the three points of figure 7 (𝑃𝑃2, 𝐻𝐻4 and 𝐻𝐻5 at top, center and bottom pictures, respectively). As expected, the occurrence of cavitation induces a highly impulsive waveform on pressure signatures, along with a relevant increase of the magnitude of the acoustic disturbance. This exactly happens at the azimuthal positions corresponding to cavity occurrence on the blade and it is possible to

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demonstrate that the significant pressure pulses induced by cavitation are governed by the second time derivative of the cavity volume [18].

Figure 10 – On the left, the noise predictions in presence (TVM scheme) and absence (wet condition) of sheet cavitation; on the right, the comparison between the alternative approaches (TVM and EBM). Points 𝑃𝑃2 (top pictures), 𝐻𝐻4 (center pictures) and 𝐻𝐻5 (bottom pictures). Note that, due to the symmetrical location of 𝐻𝐻4 and 𝐻𝐻5 with respect to the disk plane, the signatures exhibit the expected sign inversion related to pressure values on face and back sides of the thrusting blade. The right pictures in the same figure 10 show the comparison between

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the noise signals determined through the TVM and EBM approaches. The agreement is very good, both in terms of phase and magnitude: the waveform is practically the same, albeit subtle discrepancies, due to the different modelling of cavitation noise sources. In this context, note that the acoustic integrals of the TVM approach are affected by the presence of external time derivatives (and, then, more sensitive to fluctuations of numerical nature, induced by the accuracy of the hydrodynamic input data), while these inaccuracies do not affect the EBM method, where no derivative act on the integrals. The lack of cavitation noise measurements does not enable a definite validation of the presented results; nevertheless, the excellent agreement of the numerical solutions, here achieved by two alternative and theoretically different computing techniques, bodes well regarding their own reliability (although limited to some specific aspects of cavitation phenomena).

4 CONCLUSIONS A brief overview on the use of the Acoustic Analogy for the hydracoustic analysis of a marine propeller has been proposed. The FWH-based integral formulations represent very powerful and adaptive computational tools: they allow to achieve a reliable assessment of the underwater noise and, at the same time, a deep understanding of the generating noise mechanisms taking place in the flow. In the first part of the paper, through the use of hydrodynamic datasets coming from both RANS and DES simulations, we have numerically demonstrated that (unlike analogous aeronautical devices) the underwater noise generated by a marine propeller is dominated by nonlinear sources This important feature of the hydroacoustic field is due to the occurrence and persistance of significant flow nonlinearities in the downstream region (the tip vortex) and, above all, the impressive decay of the FWH linear components, ruled by the low rotational Mach number. Then, the assessment of noise induced by a sheet cavitation has been carried out, through two alternative and rather different approaches. As expected, the occurrence of a cavity on the blade surface induces an impulsive waveform on pressure signatures and significantly alters its amplitude and frequency content.

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[7] Ianniello, S., Muscari, R., Di Mascio, A. Ship underwater noise assessment by the acoustic analogy. Part I: nonlinear analysis of a marine propeller in a uniform flow, Journal of Marine Science and Technology, 18(4), 547-570, 2013

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[13] Ikeda, T., Enomoto, S., Yamamoto, K. Quadrupole effects in the Ffowcs Williams-Hawkings equation using permeable control surface, Proceedings of the 18th AIAA/CEAS Aeroacoustics Conference, Colorado Springs, CO, 2012

[14] Nitzkorski, Z., Mahesh, K. A dynamic end cap technique for sound computation using the Ffowcs Williams and Hawkings equation, Physics of Fluids, 26, 2014

[15] Brown, N.A. Cavitation noise problem and solutions, International Symposium on Ship-board Acoustics, Noordwijkerhout, The Netherlands, 1976

[16] Breslin, J.P., Andersen, P. Hydrodynamics of ship propellers, Cambridge University Press, Cambridge, 1994

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[18] Salvatore, F., Ianniello, S., Preliminary results on acoustic modelling of cavitating propellers, Computational Mechanics, 32(4), 291-300, 2003

[19] Testa, C., Ianniello, S., Salvatore, F. A Ffowcs Williams and Hawkings formulation for hydroacoustic analysis of propeller sheet cavitation, Journal of Sound & Vibration, 413, 421-441, 2018

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