Prediction of submarine scattered noise by the acoustic analogy

Journal of Sound and Vibration 426 (2018) 186e218

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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Prediction of submarine scattered noise by the acoustic analogy C. Testa*, L. Greco CNR-INSEAN Marine Technology Research Institute, Via di Vallerano 139, 00128 Rome, Italy

a r t i c l e i n f o

a b s t r a c t

Article history: Received 6 November 2017 Received in revised form 5 April 2018 Accepted 9 April 2018 Available online 26 April 2018 Handling Editor: Y. Auregan

The prediction of the noise scattered by a submarine subject to the propeller tonal noise is here addressed through a non-standard frequency-domain formulation that extends the use of the acoustic analogy to scattering problems. A boundary element method yields the scattered pressure upon the hull surface by the solution of a boundary integral equation, whereas the noise radiated in the fluid domain is evaluated by the corresponding boundary integral representation. Propeller-induced incident pressure field on the scatterer is detected by combining an unsteady three-dimensional panel method with the Bernoulli equation. For each frequency of interest, numerical results concern with sound pressure levels upon the hull and in the flowfield. The validity of the results is established by a comparison with a time-marching hydrodynamic panel method that solves propeller and hull jointly. Within the framework of potential-flow hydrodynamics, it is found out that the scattering formulation herein proposed is appropriate to successfully capture noise magnitude and directivity both on the hull surface and in the flowfield, yielding a computationally efficient solution procedure that may be useful in preliminary design/ multidisciplinary optimization applications. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Sound scattering Acoustic analogy Underwater noise

1. Introduction The prediction of hull-pressure fluctuations and flowfield noise induced by screw-propellers in behind-hull conditions are of primary interest for civil and military applications, in that, sound scattering due to the interaction between hull surface and acoustic waves emitted by the propeller(s) may be relevant. Theoretically, sound scattering occurs when an obstacle (scatterer) is present in the path of an acoustic wave and produces secondary sound spread in a variety of directions. Under the assumption that the wavelength of the impinging sound is comparable with a characteristic size of the scatterer, the acoustostructural interaction causes a re-distribution of the energy content of the impinging wave into reflected and diffracted secondary waves that may remarkably alter magnitude, waveform and directivity of the overall noise field, on both hull surface and in the flowfield, with respect to the unbounded space propagation. Sound scattering analysis is a complex matter, involving propeller/hull hydrodynamics and hydroacoustics, hull structural dynamics and acousto-structural interactional phenomena. The issue is relevant for civil applications for what concerns noise regulations and the impact on marine mammals; it plays a crucial role for submarines as their detection and classification is mostly due to the radiation of the sound energy towards the enemy units. In the low-frequency range (less than 200 Hz) dominating the sound spectrum at large

* Corresponding author. E-mail address: [email protected] (C. Testa). https://doi.org/10.1016/j.jsv.2018.04.011 0022-460X/© 2018 Elsevier Ltd. All rights reserved.

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distances from the hull, a significant part of submarine radiated noise is generated by propellers; they emit sound about 30e50 dB greater than the other sources of noise, with a spectrum dominated by distinctive tonal contributions due to high unsteady blade-loads induced by the flow distortions of the upstream wake and periodic impacts of vortical structures against the blades (i.e., bilge vortices, rudder horseshoe vortices and corner vortex originating in the hull-sail intersection region) [1]. Even though the need of minimizing propeller-induced noise has led towards complex sickle-shaped blade configurations to reduce any coupling with the incoming flow, the design of an acoustically stealth submerged vehicle requires the analysis of the whole configuration to capture the hull-scattered sound when it is impinged by the noise hydrodynamically generated by the propeller(s). The wide literature concerning sound scattering problems (see, for instance, [2e6]) shows that the scattered pressure field is typically predicted through linear approaches based on boundary integral formulations solving the Helmholtz equation for the velocity potential or the acoustic pressure. Beside them, alternative formulations based on the linear version of the Ffowcs Williams and Hawkings Equation (FWHE) have been proposed [7e10]. These extend the use of the acoustic analogy to scattering problems either through time-domain approaches [8] or frequency-domain methodologies [7,9,10]. However, the number of literature papers on underwater noise facing the problem of hull sound scattering in hullpropeller configurations is limited. Hydroacoustic investigations on whole configurations typically focus on the propeller system, hydrodynamically affected by the presence of the hull (see, for instance [11]). Among the available works, Ianniello et al. [12] propose a physically-consistent approach, not very common for marine problems, to tackle the hydroacoustic analysis of a fully-appended hull in steady motion, propelled by rotating subcomponents; noise computation is decoupled from flow simulation: all hydrodynamic sources of sound are first detected through a CFD (Computational Fluid Dynamics) analysis of the whole configuration whereas noise propagation in the flowfield is then described by the acoustic analogy for permeable surfaces [13]. In this kind of simulation, the hydrodynamic solver provides an overall description of the flowfield around the whole configuration and inherently captures any mutual interaction among pressure disturbances generated by different ship components (propeller, hull, rudder, appendages, etc); hence, hull scattering effect prediction becomes a hydrodynamic issue instead of a hydroacoustic one. Although well-posed, such a hybrid approach is CPU-time demanding and thus, not compatible with pre-design stages and optimization studies that require fast and reliable simulations. However, for those configurations and operating conditions where a primary source of noise may be (clearly) identified, and under the assumption that the impinging pressure field is independent of the presence of the scattering surface(s), the overall noise field may be decomposed into incident and scattered components and an acoustic scattering modelling may be used, thus avoiding time-consuming computations. Within the limits of this approach, the acoustic scattering analysis may be conceived as a two-step problem, where the primary source of noise is seen as frozen, thus generating an incident pressure field which may be computed by a prior (hydrodynamic and hydroacoustic) analysis as if it were isolated, whilst scatterer bodies are involved in the second stage of the problem, concerning with the scattered noise prediction. Aeronautical configurations are well suited for this kind of approach [14,15]; differently, submarines, ships or vessels, where the primary source of noise is undoubtedly the propeller, inherently suffer of the mutual hydrodynamic interactions between propeller(s) and turbulent/ vorticity-fields released by the fully-appended hull. Nevertheless, in absence of relevant blade-vortex interactions, hullepropeller hydrodynamics may be limited to consider the hull wake incoming to the isolated propeller in unbounded fluid domain, thus allowing the application of a scattering formulation as previously described. In this context, Kehr and Kao [6] propose a time-domain iteration method to predict ship hull and free surface acoustic scattering effects on propeller noise due to unsteady sheet cavitation and fluctuating forces arising when the propulsor operates in the nonuniform ship wake. The same scattering formulation is applied in Wei et al. [16] to capture the effect of a submarine hull on propeller non-cavitating noise; a suitable CFD analysis of the isolated propeller in the hull wake, combined with the FWHE (limited to thickness and loading noise terms) predicts the incident radiated sound field. In van Wijngaarden [17] a chain of computational methods is proposed for the prediction of the ship hull pressure fluctuations induced by a cavitating propeller: a Boundary Element Method (BEM) for the acoustic potential satisfying the Kirchhoff-Helmholtz equation is used to capture the scattering effects from hull and free surface once the wake field and propeller noise source strengths are known. In this framework, the need to gain a deeper insight into submarine noise generation mechanisms has inspired the present paper that proposes the non-standard pressure-based formulation presented in Gennaretti and Testa [9] for the prediction of non-cavitating propeller tonal noise scattered by a submarine in cruise motion. Specifically, by renouncing to model those tonal noise components associated to periodic impacts of vortical structures against propeller blades, the overall pressure field is computed by a frequency-domain boundary integral solution of the linear version of the FWHE, solved by a BEM scheme, where: i) the nonuniform propeller onset flow is assumed to be the hull nominal wake, known from a devoted CFD analysis; ii) the incident pressure field, on the scatterer and in the flowfield around it, is evaluated by a prior hydrodynamic/ hydroacoustic analysis of the isolated propeller working in the previously defined nonhomogeneous onset flow; iii) scattered pressure distributions, on the hull and in the flowfield, are obtained by the definition of suitable acoustic transfer functions matrices between the incident and the scattered sound. The submarine is assumed to operate in deep water, hence no scattering contributions from bottom and free surfaces are considered. This formulation has been applied to helicopters [18,19] and propeller-driven aircraft configurations [20,21], whereas an early attempt of application to a hull-propeller system has been addressed in Ref. [22]. Akin to [16], in Refs. [18e22] the impinging aero/hydro-borne sound comes from the 1A Farassat formulation [23] once the unsteady blades pressure field is known from an aero/hydro-dynamic analysis of the rotor/ propeller system. Differently, in order to capture those potential wake-induced pressure contributions, only indirectly modelled by the loading-noise term of the FWHE, in the present work the incident pressure fluctuations are predicted by a fully-validated potential-based panel method, combined with the Bernoulli equation for incompressible flows [24]. According

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to the above simplifications of the underlying fluid dynamic problem, such a potential flow-based analysis is suited to yield a fast and reliable description of the propeller impinging field, especially on the tail-cone that is expected to scatter the incident sound relevantly. In this context, the well-posedness of the incompressibility assumption is discussed in the theoretical modelling section. The validity of the FWH-based scattering results is established by comparison with outcomes from a direct hydrodynamic approach that solves propeller and scattering body jointly, in the framework of potential flows [24]. Note that recent studies on acoustic scattering predictions of moving bodies by pressure-based formulations, [25] and [26], demonstrate how the inclusion of the nonlinear sources of sound may be highly recommended, depending on the velocity and shape of the scattering body, as well as the wave number of the impinging perturbation signal. However, the very low Mach number of the translating hull (less than 0.001) suggests that these effects, strictly related to the presence of relevant gradients of the velocity potential associated to the hydrodynamic steady-state condition of the moving hull not affected by impinging waves [27], may be here reasonably negligible. To the authors' knowledge, the application of a scattering modelling based on the acoustic analogy combined with a unified hydrodynamic/hydroacoustic BEM analysis of the isolated propeller in a wake field is a novelty for underwater acoustic applications. Moreover, the trade-off between accuracy of predictions and computational efforts makes such approach appealing for multidisciplinary optimization applications aimed at stealth acoustic pre-design purposes. The paper is organized as follows. Section 2 outlines the main aspects of the acoustic scattering formulation, herein revised. Section 3 summarizes some details of the potential-based hydrodynamic approach used for comparison purposes, whereas Section 4 shows a case-study on the hull-plate propeller configuration faced in Ref. [22], and the results on the noise scattered by a submarine subjected to the tonal noise of its rotary-wing propulsive system. The comparison between computational costs required by the direct hydrodynamic approach and the proposed scattering formulation to achieve converged solutions is also provided, whilst the main conclusions are summarized in Section 5.

2. Acoustic formulation 2.1. Background Let us consider a body, bounded by a surface SB defined by f(x, t) ¼ 0, moving with velocity v throughout a compressible fluid undergoing transformations with negligible entropy changes. If the boundary surface is assumed to be impermeable the following form of the FWH equation [28] can be written cx2R3

,2 p 0 ¼


 v ½r v,Vf dðf Þ   V,½P Vf dðf Þ  þ V, V,½T Hðf Þ  vt 0

(1)

where p0 ¼ c2 b r is the acoustic disturbance, with br ¼ ðr  r0 Þ representing the density perturbation, and c, r0 denoting, respectively, the speed of sound and the density of the undisturbed medium. The overbars denote generalized differential 2 2 b I þ VÞ stands for the operators and ,2 ¼ ð1=c2 Þðv =vt 2 Þ  V is the generalized D'Alembertian operator. In addition, P ¼ ð p compressive stress tensor, V represents the viscous stress tensor (neglected under inviscid flow assumption), b p ¼ p  p0   indicates the gauge pressure, T ¼ r u5u þ ð b p  c2 b r Þ I þ V is the Lighthill stress tensor whereas u, H and d are the fluid velocity, the Heaviside and Dirac delta functions, respectively. Following the Green function approach presented in Refs. [29] and [30], assuming the nonlinear perturbation field terms negligible and the surface SB undeformable, for f such that jVfj ¼ 1, in a space rigidly connected with SB the boundary integral solution of Eq. (1) is given by

p0 ðx; tÞ

Z y 

h

^ þ ½v,n ð1  v,VqÞ_ G ^ r0 v,n v,VG

SB



Z h

  i ^  P_ n ,Vq G ^ ðP nÞ,VG

i ret

dSðyÞ (2)

ret

dSðyÞ

SB

where the two integrals represent the thickness and loading noise contributions, respectively. Here, n is the outward unit h i 1 ^ normal on the surface SB and Gðx; y; tÞ ¼ 41p r ð1M the retarded Green function. Moreover, (x(t), t) and (y(t), t) represent rÞ ret

the observer position at the observer time and the source position at the emission time, r ¼ jrj with r ¼ x(t) y(t), whilst ð1  Mr Þ is the Doppler factor, with Mr ¼ v , (r∕r)∕c denoting the surface Mach number in the direction of radiation. The symbol ð _ Þ indicates time derivatives performed in the space rigidly moving with SB whereas the symbol [ ]ret denotes that the integrand must be evaluated at the retarded emission time t ¼ tq, where q is the time taken by an acoustic disturbance released from y to reach the observer location, x, at current time, t.

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2.2. Scattering modelling Let us consider the boundary SB of an arbitrarily moving (rigid) body impinged by an incident pressure disturbance p0I coming from a source of sound that is here considered to be unaffected by the presence of the scattering body itself (frozen noise source). Under these assumptions, noting that the acoustic disturbance and pressure perturbation coincide in case of small perturbation fields, the total acoustic disturbance field may be decomposed into an incident component p0I (known) and a body-induced contribution p0B (unknown) due to the presence of the scatterer

p0 ðx; tÞ ¼ p0B ðx; tÞ þ p0I ðx; tÞ

(3)

Assuming inviscid flows, following [9], the application of Eq. (2) for x2SB combined with the above pressure decomposition yields the following Boundary Integral Equation (BIE) for the acoustic disturbance

½1  lðx; tÞp0B ðx; tÞ

i  ¼ lðx; tÞ r0 v2n ðx; tÞ þ p0I ðx; tÞ Z h i ^ þ ½v,n ð1  v,VqÞ_ G ^  r0 v,n v,VG

ret

SZB

 SZB



h h

^  p_ 0 n,Vq G ^ p0B n,VG B

i ret

i ^  p_ 0 n,Vq G ^ p0I n,VG I

ret

dSðyÞ (4)

dSðyÞ

dSðyÞ

SB

where vn ¼ v , n, l ¼ 0:5=ð1  M 2n Þ and Mn ¼ vn∕c. At the right-hand-side, the additional terms evaluated at the observer ^ arising for x2S . From the knowledge of the position come from the contribution of the singularities of the kernel function VG B

incident pressure field on SB and scatterer motion, the solution of Eq. (4) provides p0B on SB ; further, the noise emitted by the body for x outside SB is computed by the following Boundary Integral Representation (BIR)

p0B ðx; tÞ

Z ¼ SZB

 SZB



h

^ þ ½v,n ð1  v,VqÞ: G ^ r0 v,n v,VG h h

^  p_ 0 n,Vq G ^ p0B n,VG B ^  p_ 0 n,Vq G ^ p0I n,VG I

i ret

i ret

i ret

dSðyÞ

dSðyÞ

(5)

dSðyÞ

SB

whereas Eq. (3) yields the total sound. Equations (4) and (5) state that the pressure field upon an arbitrarily moving body, along with the sound it radiates, is given by three noise terms related to: i) the rigid-body motion of SB ; ii) the scattererhydroborne noise interaction process; iii) the incident pressure field. By recalling the linearity of the integral operator and decomposing p0B into a rigid-body motion term p0R and a scattering component p0S (namely, p0B ¼ p0R þ p0S ), the following BIEs for p0R and p0S hold

½1  lðx; tÞ p0R 

2 ðx; Z tÞh¼ lðx; tÞ r0 vn ðx; tÞ i ^ þ ½v,n ð1  v,VqÞ_ G ^ r0 v,n v,VG

ret

SZB



h

p0R

^  p_ 0 n,VG R

^ n,Vq G

i ret

dSðyÞ (6)

dSðyÞ

SB

0 ½1  lðx; tÞp0S ðx; tÞ ¼ lðx; Z htÞ pI ðx; tÞ i 0 ^  p_ 0 n,Vq G ^ pS n,VG  S SZB



h i ^  p_ 0 n,Vq G ^ p0I n,VG I

ret

ret

dSðyÞ (7)

dSðyÞ

SB

However, the above FWH-scattered formulation is not intended for the prediction of pressure perturbation generated by rigid-body motion since an accurate evaluation of the corresponding surface pressure would require the inclusion of the contribution from the Lighthill stress tensor in Eq. (1). Within the framework of linear FWH formulations, this is a key-point distinguishing an acoustic scattering problem from a standard hydroacoustic analysis where the pressure distribution upon the emitting surface comes directly from a hydrodynamic solver and the inaccuracy mentioned above tends to vanish when

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the acoustic disturbance is evaluated at points located far from the emitting surface (see the quadrupole expression in Refs. [31] and [32], for instance). In addition, for the purposes of this work, the rigid-body motion is a uniform translation yielding a constant pressure field upon the body surface that, in turn, does not produce any noise disturbance at points located in a frame of reference fixed with it. For these reasons, Eq. (6) (and the corresponding integral representation for p0R in the field) is

Fig. 1. Overall pressure - 1st BPF. BEM-hydrodynamics (a). Scattering modelling (b).

Fig. 2. 3D view of the propeller/submarine configuration.

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not considered in the following. Thus, by transforming the problem in the frequency-domain, the scattering analysis is e0 ðx; uÞ ei u t , Eq. (7) recasts performed for each harmonic of the incident pressure wave; for p0 ðx; tÞ ¼ p 0

½1  lðxÞe pS ðx; kÞ

0

¼ lðxÞ pI ðx; kÞ Z e h i 0 ^  i k n,Vs G ^ e n,VG  p ðy; kÞ ei k s dSðyÞ S

SZB



h

i ^  i k n,Vs G ^ p e0I ðy; kÞ ei k s dSðyÞ n,VG

SB

whereas the corresponding BIR is given by

Fig. 3. Sketch of the E1486 model propeller (a). Nominal wake-field (b).

(8)

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0 e pS ðx; kÞ

Z h

i 0 ^  i k n,Vs G ^ e n,VG pS ðy; kÞ ei k s dSðyÞ

SZB

i ^  i k n,Vs G ^ p e0I ðy; kÞ ei k s dSðyÞ n,VG

¼ 

h

(9)

SB

where ðeÞ means Fourier transformation, u denotes the angular frequency, k ¼ u∕c is the wave number and s ¼ c q. 2.2.1. Discretization procedure The numerical solution of Eqs. (8) and (9) is achieved by a zero-th order BEM. It consists in dividing SB into quadrilateral 0 ~S and p ~I 0 piecewise constant. The integral equation is solved by requiring that it is satisfied at the center panels and assuming p of each body element (collocation method, see also [30]). Specifically, discretizing SB into M panels Sm B , at the center of j-th element Eq. (8) yields, for a given value of k,

Fig. 4. Open water performance. BEM grid (a). Computations vs experiments (b).

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Fig. 5. 1st BPF, starboard overall SPL. BEM hydrodynamics (a). FWH formulation (b).



0 eS ðkÞ 1  lj p j

M  P

¼

m¼1

s ~ m ðkÞ Bjm þ i k Cjm p

þ

M  X



lj djm þ Bjm þ i k Cjm p~ Im ðkÞ

(10)

m¼1

~ sm ¼ p ~S 0 ðxm ; kÞ , p ~ Im ¼ p ~I 0 ðxm ; kÞ, and lj ¼ l(xj). In addition, djm is the where, for xm denoting the center of the m-th panel, p Kronecker delta function, while the coefficients are defined as

Z

Bjm ðkÞ ¼ ei k sjm Cjm ðkÞ

¼ ei k sjm

Z

^ dS n,VG

Sm B

Sm B

^ dS n,Vs G

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Fig. 6. 1st BPF, starboard. Incident SPL (a). Scattered SPL (b).

with sjm denoting the time delay of the propagation of signals between the source point at xm and the observer point at xj. By e I , respectively, and coefficients in e S and p collecting scattered and incident pressures at the M panels in the column vectors p the matrices B and C, the solution of Eq. (10) may be written in the following matrix form

e S ¼ EI ðkÞ p eI p

(11)

where, for L denoting the diagonal matrix collecting the lj's,

EI ðkÞ ¼ ½I  L  BðkÞ  i k CðkÞ1 ½L þ BðkÞ þ i k CðkÞ

(12)

is the matrix of the transfer functions (FRF) between incident and scattered pressures at panel centers. Similarly, for q observer points in the flowfield, the discretization procedure applied to Eq. (9), combined with Eq. (11), yields

~ SF ¼ EðkÞ p ~I p

(13)

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Fig. 7. 1st BPF, rear. Impinging (a), scattered (b) and overall (c) isobars.

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Fig. 8. 1st BPF, starboard aft. Incident SPL (a). Scattered SPL (b).

where EðkÞ ¼ HðkÞ½I þ EI ðkÞ  is the FRF between the incident pressure field and the q scattered pressure values collected in e SF , being H(k) the FRF between the scattered sound in the flowfield and the incident pressure upon the scatterer. p 3. Hydrodynamic approach A hydrodynamic standpoint is here assumed to predict the pressure field generated by a propeller in the presence of a hull in front of it. This is addressed by a potential hydrodynamic formulation for incompressible, inviscid and irrotational flows, that solves propeller and hull jointly (in time-domain) in terms of velocity potential perturbation. At this stage it is worth noting that the pressure signal at (x, t) evaluated by an incompressible solver represents a pseudo sound not physically compatible with any acoustic analysis. In fact, it is characterized by signals emitted by source points overlapping at x simultaneously because of the lack of compressibility delays that account for the time shifts among the different contributions. What affects the resulting signal (in amplitude and waveform) is not so much the magnitude of the time shifts but rather the relative differences among them (depending on the mutual distances and motion of the source points with respect to the observer). As shown in Ianniello et al. [33], the operating conditions of a marine propeller (i.e., very low rotational speeds respect to the sound speed) and the typical size of propeller blades (model or full scale) yield negligible values of time shifts between the signatures induced by the single sources. In addition, for the configurations analyzed in Section 4, where

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Fig. 9. 1st BPF, starboard aft, overall SPL. BEM hydrodynamics (a). FWH formulation (b).

any acoustic phenomenon affected by compressibility (i.e., the collapse of vapour bubbles in cavitating flows) is totally absent, a preliminary analysis addressed by the Farassat 1A formulation (not shown here for conciseness) proves that accounting for an instantaneous propagation of sound does not alter the resulting signals in a significant way, at least within a distance of about 10 propeller diameters from the hub. Thus, in spite of the theoretical acoustic inconsistency of the incompressibility assumption, the pressure signals predicted by the potential hydrodynamic solver hereafter described can be numerically compared with the corresponding FWH-based acoustic signals and, in turns, can be used to check the consistency of the numerical solutions coming from the scattering modelling. In view of the above discussion, from here on the term noise shall be used for the pressure signatures carried out by the scattering and hydrodynamic approaches, indistinctly. The same considerations justify the use of the incompressible Bernoulli equation, combined with a propeller BEM hydrodynamics, to detect the input data of the scattering modelling (that is the incident pressure field). Starting from the potential hydrodynamic formulation presented in Greco et al. [24], the velocity potential perturbation 4 associated to the propeller-hull multibody configuration derives from the following integral solution of the Laplace equation obtained by the unbounded-space Green function technique [34]

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Fig. 10. 1st BPF, rear, overall SPL. BEM hydrodynamics (a). FWH formulation (b).

I

EðxÞ 4ðxÞ ¼

SB

 Z v4 vG vG G4 dSðyÞ  D4 dSðyÞ vn vn vn

(14)

SW

Here, SB includes both propeller and hull surfaces, SW denotes the convected wake surface, D4 indicates the potential jump across SW , G and vG∕vn represent the unit source and dipole in the unbounded three-dimensional (3D) space whilst E(x) is a domain function equal to 1, 1∕2 or 0 for x inside the flowfield, on SB or inside it, respectively. Assuming the hull a non-lifting translating body and the propeller wake SW a prescribed helicoidal surface (rigid-wake model), the enforcement of Eq. (14) on SB provides a BIE for the velocity potential upon propeller and hull where the impermeability condition on SB and the absence of pressure discontinuity and fluid particles crossing on SW define the corresponding boundary conditions. Following [24], the application of a zero-th order BEM yields a linear set of algebraic equations for the velocity potential upon propeller and hull centroids, 4B and 4H, respectively, that may be manipulated as to recast directly

4B ¼ Q BB cB þ WBB D4W þ Q BH cH

(15)

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Fig. 11. 1st BPF, starboard sail. Impinging SPL (a). Overall SPL: BEM hydrodynamics (b), FWH formulation (c).

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Fig. 12. 1st BPF, portside. Impinging SPL (a). Overall SPL: BEM hydrodynamics (b), FWH formulation (c).

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Fig. 13. 1st BPF, portside sail. Impinging SPL (a). Overall SPL: BEM hydrodynamics (b), FWH formulation (c).

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Fig. 14. 2nd BPF, starboard. Impinging SPL (a). Overall SPL: BEM hydrodynamics (b), FWH formulation (c).

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Fig. 15. 2nd BPF, starboard aft. Incident SPL (a). Scattered SPL (b).

4H ¼ Q HB cB þ WHB D4W þ Q HH cH

(16)

where cB, cH and D4W collect the kinematic boundary conditions and potential-jump distribution at body and wake centroids. In Eqs. 15 and 16 the time-varying matrices QBH and QHB describe the mutual hydrodynamic influence between hull and propeller, the time-constant matrices QBB and QHH account for propeller and hull hydrodynamics forced by the kinematic boundary conditions whereas the time-constant matrix WBB and the time-varying matrix WHB describe the effect of blades wake on propeller and hull hydrodynamics, respectively (matrix WBB would be time-varying in case of free-wake modelling). The solution of Eq. (15) implicitly accounts for hull-induced effects in terms of potential flow; however, to capture the influence of the hull viscous wake on propeller hydrodynamics, a suitable viscous onset flow has to be introduced. Invoking the natural velocity decomposition introduced in Morino [35], such an incoming flow may be decomposed into a rotational (viscous) contribution and a potential one, whose effect on the propeller hydrodynamics is inherently modeled by Eq. (15). Hence, the inflow physically-consistent with the present formulation is the selected viscous flow, without the velocity potential field contribution due to the translating hull that is here computed by a BEM hydrodynamic analysis of the unpropelled moving hull. Akin to subsection 2.2, the hull nominal wake is considered as selected onset flow. Once 4B and 4H are known

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Fig. 16. 2nd BPF, starboard aft, overall SPL. BEM hydrodynamics (a). FWH formulation (b).

from Eqs. (15) and (16), the discretized Eq. (14) for E(x) ¼ 1 yields the potential in the flowfield; the pressure fluctuations come from the Bernoulli equation

4_  v,V4 þ

1 p p kV4k2 þ þ gz0 ¼ 0 2 r0 r0

(17)

where gz0 denotes the hydrostatic head with respect to a reference vertical position z0. Similarly, the application of the Bernoulli equation for propeller blades and hull, yields the corresponding pressure pulses distribution.

4. Numerical results Before showing the hydroacoustic results concerning with a submarine in cruise motion impinged by the sound field emitted by its propulsive system, a case-study is presented to assess the numerical comparison between the scattering modelling and the direct BEM-hydrodynamic predictions.

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Fig. 17. 2nd BPF, portside. Impinging SPL (a). Overall SPL: BEM hydrodynamics (b), FWH formulation (c).

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Fig. 18. 3rd BPF, starboard. Impinging SPL (a). Overall SPL: BEM hydrodynamics (b), FWH formulation (c).

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Fig. 19. 3rd BPF, portside. Impinging SPL (a). Overall SPL: BEM hydrodynamics (b), FWH formulation (c).

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Fig. 20. 3rd BPF, starboard aft. Incident SPL (a). Scattered SPL (b).

4.1. Case study The configuration examined consists of a single-bladed CNR-INSEAN E779A propeller model working below a rigid square plate located at a vertical distance of 0.66D from the propeller axis, centered respect to the disk plane (D ¼ 0.22727 m denotes propeller diameter). The plate length is 5D whereas the thickness is 0.1D; propeller and plate rigidly translate along the x axis (see Fig. 1) in open water at an advance ratio J ¼ V∕nD equal to 0.9, being n ¼ 30 rps the blade angular speed and V the advancing velocity. This configuration has been studied in Ref. [22] by comparing the overall pressure fluctuations at the 1st BPF (Blade Passage Frequency) carried out from the aforementioned formulations. However in that paper, some discrepancies due to the lack of acoustic effects induced by the blade wake (in the scattering modelling) and numerical inaccuracies at the plate edges (in the BEM-hydrodynamic approach) affected the results. Here, both issues have been overcome as follows: i) the impinging pressure upon the plate is computed by the Bernoulli equation that directly predicts the acoustic contribution from the shed wake; ii) the V4 term, needed to compute plate pressure fluctuations, comes from the gradient of Eq. (14) written for E(x) ¼ 1, avoiding grid sensitive finite differences schemes. The effects of such enhancements are shown in Fig. 1 where the top view of the plate highlights a very good agreement for the pressure amplitude and directivity. A devoted hydrodynamic investigation points out that the slightly smaller and weaker lobe ahead the propeller disk plane, just behind it and downstream, predicted by the scattering model, is ascribed to the non-modeled plate-propeller hydrodynamic interaction that is

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Fig. 21. 3rd BPF, starboard aft, overall SPL. BEM hydrodynamics (a). FWH formulation (b).

not relevant in terms of blade loads, but affects somehow the blade pressure distribution respect to the isolated-blade computation.

4.2. Submarine hydroacoustics The acoustic signature of a submarine impinged by the hydroborne tonal noise emitted by its propulsive system is investigated. The submarine shown in Fig. 2 is the model-scale CNR-INSEAN 2475 with forward planes, sail and stern appendages. It has been studied in the framework of the Research Project SUBMOTION II (contracted throughout the European Defence Agency) in Ref. [36], for maneuvering issues, and in Ref. [37] to characterize the steady-state hydrodynamic behaviour. The propulsive system, depicted in Fig. 3a, is the modified seven-bladed CNR-INSEAN E1486 model investigated experimentally within the activities of the WEAO EUCLID RTP 10.17 Research Project [38]. The operating conditions are defined by J ¼ 0.87, n ¼ 5.96 rps (D ¼ 0.2906 m) and an incoming flow shown in Fig. 3b. This nominal wake field is obtained by the RANS (Reynolds Averaged Navier-Stokes) solver applied in Ref. [37] and, as depicted, is characterized by an axial velocity component Vx (nondimensional respect to the advancing speed) highly affected by the presence of X rudders. As a preliminary step, the performance of the isolated propeller in open water are computed through the panel method presented in Ref. [24]; the computational grid used for this purpose, assuring BEM converged results in terms of nondimensional thrust and torque

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Fig. 22. 1st BPF, shielding factor. Starboard (a). Portside (b).

coefficients, KT ¼ T=r0 n2 D4 , KQ ¼ Q =r0 n2 D5 , is plotted in Fig. 4a, for the hub, blades and potential wake. The comparison between numerical and experimental results, depicted in Fig. 4b, is very good, in particular at high values of J. This demonstrates the capability of the BEM solver in capturing the main features of highly skewed blades hydrodynamics, making it well suited for detecting the sources of sound localized on the blades and to analyze, hydrodynamically, the propeller-hull configuration as a whole. 4.2.1. Hull pressures Here, the submarine in cruise motion is acoustically investigated in terms of hull pressure pulses. The hydroacoustic analysis is addressed for the first three BPFs of the impinging pressure field, corresponding to 41.7 Hz, 83.4 Hz and 125.2 Hz, respectively. The operating conditions are such that aft rudders and fore horizontal canard planes work at zero angle of attack and, the nominal wake incoming the propeller disk is not affected by the presence of the forward canards. In addition, being far enough from the propeller plane (approximately 8 diameters from the hub), a negligible contribution to the scattered noise is expected from canard surfaces respect to the sail and the rear portion of the hull. For these reasons, the canards are not included in the further acoustic investigation. At the 1st BPF, the wavelength of the impinging pressure wave is about 5 m (comparable with the hull length), so scattering effects are expected. Figure 5 depicts the starboard Sound Pressure Level (SPL); as shown, the scattering modelling (b)

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Fig. 23. 2nd BPF, shielding factor. Starboard (a). Portside (b).

predicts total levels of sound in very good agreement with BEM hydrodynamics results (a), albeit subtle differences are present between the two starboard rudders, on the sail and upon the central part of the hull. Akin to Section 4.1, such differences reside in the hydrodynamic interactions between propeller and hull, not entirely modeled by the scattering formulation forced by the incident sound field of the isolated propeller rotating in the hull wake. However, within the limits of this approximation, the scattering modelling captures very well the main features of the acoustic phenomenon. At the same frequency, Fig. 6a shows the incident SPL distribution on the hull (assumed acoustically permeable) whereas Fig. 6b depicts the scattered one. The comparison between Figs. 5b and 6a highlights the scattering effect of the hull that is responsible of a local reduction of the overall SPL (respect to the impinging noise) fore, and an increase, aft, in particular on the rudders. An insight of the acoustic response of the tail-cone is given by Fig. 7 showing the qualitative structure of the incident (a), scattered (b) and overall (c) isobars. As depicted, the (constructive or destructive) interference between incident and scattered fields determines an overall SPL distribution mostly governed by the pattern of the incident field, with high gradients in particular on, and nearby, the first rudder. From a quantitative standpoint, the impinging and scattered SPL at the stern are plotted in Fig. 8 whilst the overall SPL from both hydrodynamic and FWH-based formulations are depicted in Figs. 9 and 10 for two different views. These figures highlight the shielding effect of the first rudder, the overpressure on the fourth rudder (just close to the tail-cone) as well as a slight change of the overall SPL spatial distribution elsewhere. The scattering modelling well describes such features of the acoustic phenomenon even though some discrepancies are observed in particular between the

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Fig. 24. 3rd BPF, shielding factor. Starboard (a). Portside (b).

first and second rudder. On the sail, the incident and the overall SPL carried out by the two approaches are shown in Fig. 11. The scattering effect is here less relevant respect to the stern and the FWH-based model confirms the very good agreement with the BEM hydrodynamic contour plot. Similarly, portside, Figs. 12 and 13 depict the incident and the overall SPL predictions upon hull and sail, respectively. As a general comment, the overall SPL directivity is very similar to the impinging one, almost everywhere; such a behaviour is captured by both formulations with a very good agreement, as well as the nonsymmetric response (starboard vs portside) that resides in the structure of the impinging pressure field (note that geometrically, the submarine is symmetric respect to the longitudinal plane). At the 2nd BPF the comparison of the starboard contour plots depicted in Figs. 6a and 14a highlights a lower energy content of the impinging sound than at the 1st BPF. The overall SPL maps computed by BEM hydrodynamic solver and FWHbased modelling, are presented in Fig. 14b and c, respectively. Akin to the 1st BPF, the FWH formulation captures magnitude and directivity of the overall sound with a goof level of accuracy. Figures 15 and 16 show that scattering effects are concentrated on the tail and rudders, confirming the effectiveness of the scattering formulation at higher frequencies. Portside, Fig. 17 plots the predicted incident SPL (a) and both the overall pressure levels coming from the two solvers ((b) and (c)), that confirm to be in very good agreement. At the 3rd BPF, the comparison between Figs. 18a and 14a demonstrates how the energy content of the impinging sound is greater than the 2nd BPF (even if less than the 1st BPF). The capability of the scattering modelling in providing predictions in

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Fig. 25. Shielding factor along the intersection between hull and XY coordinate plane.

Fig. 26. Hydrophones location on XY and XZ planes for noise field prediction.

very good agreement with the coupled hydrodynamic analysis is proven both starboard (see Fig. 18, (b) and (c)) and portside (see Fig. 19, (b) and (c)). The impinging and scattered SPL at the stern are plotted in Fig. 20 whilst Fig. 21 confirms the very good agreement between hydrodynamic and FWH-based results. At higher frequencies the magnitude of the impinging sound is considerably less than the first three BPFs; therefore, the acoustic analysis is not addressed. 4.2.2. Shielding factor analysis In order to get guidelines on the hull scattering response, the shielding factor gt defined as the ratio between the amplitudes of the total and impinging pressure field, for each frequency of interest, is introduced. A deviation from unity (if any) denotes scattering effects from any obstacle. Looking at the starboard shielding factor distribution predicted by the scattering modelling, in Fig. 22a, relevant scattering effects among the rudders and upon them, inducing, in turns, localized over/underpressures, are observed; note that all rudders scatter the impinging noise in a relevant way. Along the hull, the scattering determines a distributed underpressure with gt values increasing from the stern towards the nose. Portside, Fig. 22b confirms a lower acoustic scattering response than starboard, except close to the tail-cone, and highlights how the rudders strongly

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Fig. 27. SPL of the overall radiated pressure on XZ (a) and XY (b) planes.

scatter the propeller sound. At the 2nd BPF, Fig. 23a shows how, starboard, the hull scattering induces an underpressure with a maximum localized farther from the cone-tail. The comparison with Fig. 22a highlights a reduction of the scattering response along the hull, between the rudders and on the tail-cone, even though the directivity of the total pressure levels remains very similar. Portside, Fig. 23b confirms the same acoustic scattering response exhibited in Fig. 22b with some differences in terms of directivity and magnitude in particular for what concerns the overpressure among the third, fourth and first rudder as well as close to the tail-cone. Elsewhere, there are localized regions where the acoustic response of the hull is greater or lower than the 1st BPF. At the 3rd BPF, a further reduction of the overall pressure levels and differences in the directivity pattern are present starboard (see Fig. 24a) where the region of maximum underpressure moves farther than the 2nd BPF, and portside (see Fig. 24b). For the sake of clarity, Fig. 25 plots the shielding factor distribution along the intersection between the hull and XY coordinate plane (see Fig. 26) at the three BPFs hereby considered. 4.2.3. Radiated noise The pressure disturbance radiated by the submarine is computed on two circles of hydrophones centered at the middle of the hull, co-moving with it, having a radius of 5 m (see Fig. 26). On the XZ plane, Fig. 27a shows the comparison between the SPL predictions of the total radiated sound coming from the FWH-based scattering modelling and the BEM hydrodynamic approach. At the three BPFs, the agreement is excellent both in magnitude and directivity. Slight discrepancies arise at the 1st

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Fig. 28. SPL of the overall and incident radiated pressures on XZ (a) and XY (b) planes.

BPF, for an azimuthal angle ranging from 60 to 90 . Note that differently from the SPL distribution upon the hull, the spectrum of the radiated noise is characterized by a 1st BPF quieter than the third one. The same acoustic features are highlighted in Fig. 27b concerning with the XY plane. As shown, the polar plot at the second and third BPF is practically the same of the XZ plane whereas the directivity at the 1st BPF is rotated about 90 clockwise. The agreement between scattering and hydrodynamic predictions is excellent. In the mid-field herein investigated, Fig. 28 shows that, for both coordinate planes, at the second and third BPF, the total noise is practically unaffected by the hull noise whilst at the 1st BPF, the interference between the impinging and scattered fields modifies magnitude and directivity in the azimuthal ranges [270 , 120 ] and [240 , 0 ] for the XZ and XY planes, respectively. However, the analysis of the Overall Sound Pressure Level (OASPL) in both planes (see Fig. 29) shows that the total noise is mostly governed by the 3rd BPF harmonic component whose magnitude and directivity is practically equal to the incident pressure field at the same frequency. This result is not surprising because, even though for acoustic observers placed close to the hull the total noise may be highly affected by relevant scattering responses (if any) due to the surfaces nearby, such localized phenomena tend to vanish as the distance from the hull increases, also in virtue of the sound scattering effects by the rest of the configuration. Last but not least, a well-posed submarine acoustic design must minimize hull scattering effects on propeller induced noise; from this point of view, the investigated configuration seems to comply with this requirement.

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Fig. 29. OASPL of the total radiated noise on XZ (a) and XY (b) planes.

4.2.4. CPU-time comparison Once the pressure field radiated by the isolated propeller and impinging the scatterer is known, the demanding computational stage of the FWH-scattering modelling is the solution of the BIE, given by Eq. (7). However, it is found out that the CPU-time required to detect the scattered pressure distribution upon the hull surface is negligible (about 1∕100) respect to the prior hydrodynamic/hydroacoustic analysis of the isolated propeller. For the spatial and time discretization parameters listed in Table 1 and an Intel® Xeon® 2.93 GHz processor, the relative CPU-time needed to achieve the converged results shown in Section 4.2 is summarized in Table 2 (the scattering modelling run time includes the BEM-based impinging pressure calculation step). As a matter of fact, the efficiency of the scattering formulation is significant. Note that, if the incident pressure field were computed by a prior CFD/hydroacoustic analysis of the isolated propeller in imposed wake [11], the computational time-saving of the FWH-based formulation is expected to be incomparable respect to the CFD simulation of the whole propeller/hull configuration. These considerations make the scattering modelling appealing for preliminary/ optimal design processes, where a good trade-off between accuracy of predictions and computational costs is mandatory.

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Table 1 Discretization parameters for the submarine configuration. Hull panels Propeller panels Wake revolutions Wake panels Azimuthal steps

800 4480 3 280 2800

Table 2 Relative run time between the two solvers. Hydrodynamic solver Scattering Modelling

1.00 0.75

5. Conclusions A nonstandard frequency-domain formulation based on the acoustic analogy has been applied for submarine scattered tonal noise prediction. For each frequency of interest, the formulation identifies the transfer functions matrices between the impinging pressure on the hull and scattered sound on the scatterer and in the flowfield. The input data for such modelling is the incident pressure field impinging the hull, that has been here computed by the analysis of the isolated propeller working in the nominal hull wake through a 3D unsteady panel method, combined with the Bernoulli equation. For the first three BPFs of the incident field, the accuracy of the acoustic predictions on the hull surface has been investigated by comparing the SPL (referred to the total pressure) with those obtained by the hydrodynamic analysis of the whole propeller-hull configuration within the framework of potential flows. Numerical results highlight the effectiveness of the acoustic analogy scattering modelling in that, from a general standpoint, the quality of the comparison is very good. Slight discrepancies in magnitude and directivity are mainly due to the capability of the hydrodynamic approach in capturing all the potential hydrodynamic interactions between hull and propeller, partially modelled in the scattering solver. The good quality of the predicted SPL is obtained with a computational saving of about 25% respect to the joint hydrodynamic analysis; this makes the scattering modelling suitable for applications in preliminary/optimal design processes, where a good trade-off between accuracy of predictions and computational costs is mandatory. The comparison has proven to be very good also in the flowfield in terms of both SPL and OASPL magnitude and directivity. The main conclusions of the numerical investigation herein addressed are:  the scattering analysis is crucial for a reliable prediction of hull pressure fluctuations since the interference between incident and scattered sound gives rise to localized over/under-pressures, otherwise not modelled, that may induce severe propeller induced vibrations and, in turns, interior noise and structure-borne radiated sound.  the scattering response of the hull in the flowfield limits to modify the shape of the SPL directivity pattern (expecially at the 1st BPF) respect to propeller free-space noise propagation. Differently, no noticeable differences are shown in terms of OSPL.  the CPU-time required for the acoustic scattering analysis through the acoustic analogy-based formulation is appreciably less than the CPU-time to perform the coupled hydrodynamic and hydroacoustic analysis of the whole configuration.

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