A partitioned solution approach for the simulation of dynamic behaviour and acoustic signature of flexible cavitating marine propellers

Ocean Engineering 197 (2020) 106854

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A partitioned solution approach for the simulation of dynamic behaviour and acoustic signature of flexible cavitating marine propellers Tobias Lampe a, *, Lars Radtke b, Moustafa Abdel-Maksoud a, Alexander Düster b a b

Institute for Fluid Dynamics and Ship Theory, Hamburg University of Technology, Am Schwarzenberg-Campus 4, 21073, Hamburg, Germany Institute for Ship Structural Design and Analysis, Hamburg University of Technology, Am Schwarzenberg-Campus 4, 21073, Hamburg, Germany

A R T I C L E I N F O

A B S T R A C T

Keywords: Propeller Fluid-structure interaction Ffowcs Williams-Hawkings equation

Upcoming legislative developments and modern propeller designs call for numerical methods which are able to assess the complex interaction of the propeller’s hydrodynamic and structural dynamic behaviour as well as the resulting influence on the acoustic signature. In this work, the problem is engaged making use of a partitioned, strongly coupled solution approach, thus enabling the usage of specialized solvers for each domain. On the fluid side, a boundary-element-method is applied while on the structural side a finite element method is employed. Information transfer between each subproblem is handled by a separate coupling tool. The acoustic evaluation is performed by means of a Ffowcs Williams-Hawkings equation based technique. In the present paper, the main focus is to establish the method’s capability to simulate the interaction of all fields and its’ physical consistency. The results show that an external excitation of the blade can be captured in the resulting acoustic spectrum.

1. Introduction Recently, the increase of noise pollution in the oceans due to humanrelated activities has attracted a lot of attention from corresponding authorities. Unfortunately, ships and their activities are among the main causes of noise generation in the oceans. It has been claimed that this industry alone has caused at least 20 dB increase in the ambient noise in the oceans. Consequently, specific regulations concerning the noise emission of commercial ships are to be expected in the next couple of years. For instance, one can point out the recent deliberations of the International Maritime Organization (IMO) with regard to marine noise pollution (IMO, 2014) or those by the Bureau Veritas (BV), which recently introduced the optional class notation “NR 614” to address concerns related to underwater radiated noise generated by ships (BV, 2014). These deliberations are also highlighted by findings of the Eu­ ropean Commission (EU, 2010) regarding the preservation of the marine water environmental status. It is therefore expected that, in the near future, one of the major concerns of the marine industry will be to evaluate ship-induced far-field noise at the design stage. Naturally, marine propellers are subjected to inhomogeneous inflow conditions due to their operation in the hull wake, ship maneuvers as well as other influences such as wave motion. It is therefore required to consider blade loads as well as the resulting blade deformations as

transient in time. In conventional propeller design, however, the blades are usually considered as rigid or simplified methods utilizing mean blade loads and cantilever beam theory (Carlton, 2018) are employed to assess the influence of blade elasticity. As propeller geometries as well as materials used in the past were not likely to allow large deformations or vibrations, and the acoustic signature of the propeller was of secondary interest, this approach was mostly sufficient. For modern propeller design procedures, this is not the case. Highly skewed geometries and composite materials may lead to severe deformations and vibrations, which affect propeller performance as well as acoustic signature and must be treated in a wholesome manner. It is therefore mandatory to develop simulation methods which are able to predict the dynamic behaviour and acoustic far-field of flexible marine propellers in inhomogeneous wake conditions under consider­ ation of fluid-structure interaction (FSI). These techniques need to provide a balance between accuracy and efficiency suitable for the design stage. Considering the modelling of the fluid domain, the boundary element method (BEM) offers a balance between accuracy and efficiency most suitable for design purposes (Katz and Plotkin, 2001). In this work, the potential theory based first-order panel method panMARE is employed, which is an in-house development. Considering the coupled simulation tool presented in this work is intended to be used in the early stages of propeller design, it will mainly be used to investigate

* Corresponding author. E-mail address: [email protected] (T. Lampe). https://doi.org/10.1016/j.oceaneng.2019.106854 Received 6 June 2019; Received in revised form 4 November 2019; Accepted 8 December 2019 Available online 23 December 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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propellers operating close to the design point. For these configurations, pressure based effects are dominant as, for example, leading edge sep­ aration or other viscosity based effects should not be present. For off-design conditions, the applicability of the method has to be carefully evaluated. With regard to the acoustic signature of marine propellers, cavitation effects are considered as one of the most distinct influences on the acoustic spectrum. In the scope of a BEM, a partially non-linear model without mesh modification (Vaz and Bosschers, 2006) can be employed to efficiently simulate sheet cavitation phenomena, which are considered as the dominating factors for the acoustic spectrum in the frequency range below 200 Hz (Collier, 1997). In the described frequency range, the highest sound pressure amplitudes are usually associated with the blade frequency range and multiples thereof, which in turn are mostly dependant on pressure fluctuations due to blade motion and loads as well as the evolution of sheet cavitation on the blade. The growth and collapse of cavitation bubbles, which is largely responsible for the broadband sound in higher frequency ranges, should not hold a large influence for the described application cases. On the structural side, effort can be made to utilize high-order finite element methods (p-FEM) (Szab� o and Babu�ska, 1991; Düster et al., 2017) in order to decrease the computational effort. Here, the high-order finite-element code AdhoC (Düster and Kollmannsberger, 2010) is employed, which is also developed at the TUHH. The various solution approaches for FSI problems may roughly be divided into two groups. In monolithic approaches, the coupled problem is discretized and solved as a whole, i.e. the individual subproblems are solved simultaneously, usually using a common numerical method. In partitioned approaches, the subproblems are solved separately. The interaction is accounted for by exchanging coupling quantities between the subproblems, which, for FSI problems, are the structural displace­ ments and the fluid loads. During the solution of one subproblem, the coupling quantities depending on the other subproblem are held fixed which demands for an iterative solution and exchange of data if an implicit coupling is desired. For an in depth explanation of monolithic and partitioned solution approaches for FSI, we refer to Bungartz and €fer (2006), Bungartz et al. (2010) or Bazilevs et al. (2013). Here, we Scha like to emphasize that partitioned solution approaches allow to utilize existing software and specialized numerical methods for the individual subproblems. In addition, different methods can be utilized to stabilize and accelerate the solution process. To this end, the most promising method was found to be the quasi-Newton least squares (QNLS) method proposed in Degroote et al. (2009). An acoustic evaluation of the coupled problem is performed in a post processing fashion. Considering the fluid problem an approach based on incompressible potential theory is used to simulate the hydrodynamic domain, which assumes an infinite magnitude for the speed of sound. Accurate far-field acoustic properties can only be obtained by the use of an additional technique. Here, the Ffowcs Williams-Hawkings equation (FWHE) in Formulation 1A (Brenter and Farassat, 2003) is used since the required input consists only of body movement and load related terms. This formulation facilitates an efficient and easily applicable usage in the context of a BEM and allows the consideration of fluid compressibility. The paper’s main intention is to present a simulation framework capable of handling the complex interaction of the structural, fluid and acoustic domains. With regard to these problems, no validation data is currently available and the chosen simulation setups are therefore intended to highlight the general capabilities of the simulation tech­ nique as well as its physical consistency. The remainder of this paper is structured as follows. In the next section, the governing equations for fluid, structural and acoustical mechanics are summarized and the initial boundary value problem underlying the FSI simulation is introduced. In Section 3, the dis­ cretization techniques utilized for each domain are described. In order to establish confidence in the modelling of the individual domains as well

as the coupling software, preliminary studies conducted in our previous work are considered in Section 4 before the setup investigated in this work is presented in Section 5. At last, in Section 6 the P1356 propeller (Richter and Heinke, 2006) is simulated in configurations of varying complexity and the results are discussed. 2. Governing equations of fluid, structural and acoustic subproblem As the partitioned solution approach enables the use of separate solvers for each domain, the governing equations for the fluid, structural and acoustic subproblem are introduced here separately, too. The coupled solution of the structural and fluid field is realized using coupling conditions, which in each subproblem can be represented by standard boundary conditions. Since the acoustic evaluation is obtained by means of a post-processing technique, a cohesive mathematical description is utilized but no further coupling conditions need to be introduced. A schematic description of the coupling process is given in Fig. 1, which also illustrates the respective domains and the associated terminology. 2.1. Fluid subproblem The hydrodynamic domain is simulated utilizing a potential theory based simulation method. Thus, an incompressible, irrotational and inviscid fluid is assumed. The potential flow can therefore be computed by solving the Laplace equation r2 Φ ¼

∂2 Φ ∂2 Φ ∂2 Φ þ þ 2 ¼ 0 inΩft ∂x2 ∂y2 ∂z

(1)

for the velocity potential Φ. The potential Φ is a linear combination of the potential due to the movement of the body relative to a fixed frame coordinate system, the influence of the perturbation potential induced by the body and an arbitrary background flow. The unsteady Bernoulli equation is used to obtain the total pressure p ¼ p∞ þ

ρ ��� 2

rΦ∞ j2

� � �rΦj2

ρ

∂φ þ ρgðz∞ ∂t

zÞ:

(2)

Therein φ and Φ∞ denote the disturbed and the free stream potential, respectively, such that Φ ¼ Φ∞ þ φ. Here, p∞ and z∞ denote to the (atmospheric) reference pressure and the reference height (where p∞ is prescribed, here it refers to the position of the free water surface), respectively. The constants ρ and g describe fluid density and the grav­ itational constant. By utilizing a distribution of sources σ and doublets μ on the boundary Γf ¼ Γf;b [ Γf;w , a general solution to the described continuity equation is given, see e.g. (Katz and Plotkin, 2001), viz � � � � Z Z 1 1 1 1 ΦðxÞ ¼ dΓf dΓf;b þ Φ∞ : μ n ⋅ grad σ (3) 4π Γf r 4π Γf;b r In this case, the surface of immersed bodies and associated wake surfaces, correspond to the boundaries Γf;b and Γf;w , see Fig. 1. The vector n refers to the unit outward normal of the respective surface while r is the distance between x and a point on the surface. The introduced doublets μ and sources σ are defined as

μ¼Φ

Φi ​ ​ and ​ ​

σ¼

∂Φ ∂n

∂Φi ; ∂n

(4)

respectively. Therein Φi denotes the internal potential of the body. In order to yield a solvable system of equations from a discretization of Equation (3), boundary conditions need to be prescribed. As the disturbance due to the body should not be present in the far field, lim rφ ¼ 0 onΓf;∞ t

r→∞

2

(5)

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Ocean Engineering 197 (2020) 106854

Fig. 1. Illustration of the initial boundary value problems for structural dynamics (left) and fluid dynamics (right) as well as acoustic evaluation by post-processing.

expression for the potential on the cavitating part of the body is ob­ tained, viz # Z "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ∂Φ 2 kvk2 þ ðp∞ pv Þ þ 2gðz∞ zÞ 2 vi;η ve;ξ dξonΓc : Φ ¼ Φ0 þ ρ ∂t ξ

must hold, which is already in place due to the general nature of the formula given in Equation (3). Naturally, another condition is to allow no flow normal to the body surface, which can be prescribed by: gradðΦ∞ þ φÞ⋅n ¼ 0 onΓft :

(6)

(10)

For coherent bodies, this requires that the potential inside the boundary will not change, such that Φi ¼ const

inΩf;b t :

Here, the integration is performed along coordinate ξ of a local

orthogonal coordinate system with coordinates ξ ¼ ðξ; η; ζÞT . ξ is ori­ ented in direction of the local streamline and ζ in outward pointing normal direction. An illustration of the coordinate systems used for the sheet cavitation model is given in Section 3.1.1 in Fig. 2. The integration starts at the cavity inception or detachment point where Φ0 is present, which in this work is always the leading edge. Pressure and suction side are handled separately. The variable v denotes the effective velocity vector due to body movement (vmove ) and inflow conditions (vin ),

(7)

By choosing Φi ¼ Φ∞ , a Dirichlet-like boundary equation is obtained, which reads � � � � Z Z 1 1 1 1 μ n ⋅ grad σ (8) dΓf dΓf;b ¼ 0: 4π Γf r 4π Γf;b r In this work, only propeller blades which produce lift and hence also shed a wake from a sharp trailing edge (TE) are investigated. Thus, the Kutta condition (Katz and Plotkin, 2001) is employed, which enables us to specify the doublet strength in the wake. In this context, the Kutta condition states that there should be no velocity component normal to the camber line at the trailing edge of the respective section. Assuming a continuous velocity, a stagnation point must therefore be present at the trailing edge and the pressure difference between suction and pressure side must be zero. This infers a stagnation point at the trailing edge, where the pressure difference between suction and pressure side is zero: ΔpTE ¼ 0:

v ¼ vmove þ vin :

(11)

It should not be confused with the effective velocity term often used in reference to the nature of wake fields. ve;ξ refers to the component of the effective velocity acting in direction of ξ. Lastly, vi;η describes the induced velocity component acting in direction of η. The KBC states that the normal component of the velocity on the cavity sheet must be zero, such that D Fðs1 ; s2 ; s3 ; tÞ ¼ 0 onΓct ; Dt

(9)

(12)

with F ¼ s3 t cav ðs1 ; s2 ; tÞ representing the cavity surface. Here, t cav re­ fers to the cavity thickness. Contrary to the approach for the DBC, the KBC is formulated making use of a local non-orthogonal coordinate sys­

Physical considerations yield that the wake is force-free, which in turn infers that Equation (9) holds on Γf;w . With regard to BEM, special care has to be given to the deformation of the wake. In this case, a first order accurate explicit Euler scheme is used which enforces the wake geom­ etry to be locally tangent to the direction of the flow. The general methodology is given in Wang et al. (2016).

tem with coordinates s ¼ ðs1 ; s2 ; s3 ÞT and basis vectors t1 ; t2 ; t3 . While directions s1 and s3 coincide with ξ and ζ, direction s2 depends on the local discretization, see Section 3.1.1. The KBC can now be rewritten which yields � � ∂tcav ∂tcav ∂tcav ðvs1 ðt1 ⋅ t2 Þvs2 Þ þ ðvs2 ðt1 ⋅ t2 Þvs1 Þ ¼ kt1 � t2 k2 vs3 : ∂s1 ∂s2 ∂t (13)

2.1.1. Sheet cavitation As one of the main contributors to the overall acoustic signature of marine propellers, the occurrence and development of sheet cavitation on the blades is considered. The boundary conditions utilized in the model are prescribed on the body surface instead of the cavity and the discretization is not altered during the simulation to optimize the grid with regard to the cavitation model. Thus, the model is termed partially non-linear without mesh modification (Vaz, 2005). A thorough description of the cavitation model and its implementation would be out of the scope of this work. For further information, the reader is referred to Gaschler (2017). At this time, only a schematic view of the underlying theoretical considerations is given in order to illustrate the integration of the sheet cavitation model in the larger scope of the complete method. Two additional boundary conditions are utilized in the model, namely the dynamic (DBC) and kinematic (KBC) boundary conditions. For the DBC, it is assumed that the pressure in the cavity region, denoted by Γct , is equal to the vapour pressure pv , such that p ¼ pv on Γct . This condition can be transformed using the unsteady Bernoulli equation such that an

Therein, vs1 ;s2 ;s3 denote the total velocity components, which harbour contributions of the effective velocity due to body movement and inflow conditions as well as those of the induced velocity, in direction of the coordinate system’s respective basis vectors. These velocities are

Fig. 2. Local orthogonal (left) and local non-orthogonal (right) coordinate sys­ tem utilized in the sheet cavitation model. 3

Ocean Engineering 197 (2020) 106854

T. Lampe et al.

described by vsk ¼

∂Φ þ v⋅tk ∂sk

2.4. Acoustic subproblem for k ¼ 1; 2; 3:

An acoustic evaluation of the simulations is performed in a postprocessing fashion making use of the FWHE in Formulation 1A (Brenter and Farassat, 2003). The required input consists only of load and movement related data, to be collected at the surface of the evalu­ ated body, thus enabling an easy usage in the scope of BEM in general and the coupled problems considered in this work. In the present algo­ rithm, all data is collected from the fluid domain. Due to the enforce­ ment of the coupling conditions described in Equations (19) and (20), structural deformations and vibrations are accounted for. With regard to the acoustic solution, a distinction between contributions of fluid displacement p’T and body loading p’L is made to obtain the overall acoustic pressure p’; � Z � ρðv_a ⋅na þ va ⋅n_ a Þ 4π p’T ðxo ; tÞ ¼ dΓf;b 2 kekð1 m⋅eÞ ret Γf;b # (22) Z " _ þ cm⋅e ckmk2 Þ ρva ⋅na ðkekm⋅e f;b � þ dΓ 3 kek2 �1 m⋅ej Γf;b

(14)

2.2. Structural subproblem The underlying differential equation of the structural subproblem, is the balance of linear momentum. As described, for example, in Wriggers (2008), the balance of linear momentum can be stated in the unde­ formed reference configuration as

ρ0 d€ ¼ DivðF SÞ þ ρ0 b inΩs0 ;

(15)

with d€ denoting the acceleration. ρ0 is the density in the undeformed configuration and b denotes a volume load. d is the unknown displace­ ment field. F ¼ ∂∂Xx denotes the deformation gradient and S is the second Piola-Kirchhoff stress tensor. We employ the St. Venant-Kirchhoff model to relate S with the Green-Lagrange strain E ¼ 12 ðFT F 1Þ by S ¼ λ trðEÞ I þ 2 μ E ¼ C E:

(16)

ret

Therein, C corresponds to the elasticity tensor known from linear the­ ory, where it describes the relation between the engineering strain and the Cauchy stress. In addition to Equation (15), Dirichlet (displacement) and Neumann (traction) boundary conditions d ¼ d onΓs;d 0 ;

(17)

F S N ¼ t onΓs;t 0

(18)

(23)

p’ðxo ; tÞ ¼ p’T ðxo ; tÞ þ p’L ðxo ; tÞ:

(24)

Γf;b

kekð1

Equations (22) and (23) make use of the FWHE velocity vector va , the distance vector e, which points from sound source point to the observer point xo , the velocity vector normalized by the speed of sound m ¼ vca , the speed of sound c and the loading vector l ¼ pna . Here, na refers to the surface unit normal vector. In order to account for FSI effects as well as cavitation, the FWHE velocity vector va contains contributions attrib­ uted to rigid body movement and structural deformations as well as those associated with the development of cavitation on the blade, viz

Having introduced the governing equations for the structural and fluid domain, the interaction of both domains can be achieved by the introduction of coupling constraints. Obviously, the enforcement of the coupling constraints must be carried out on the shared boundary of both

domains, denoted as ΓFSI . The motion of the fluid boundary Γf;b must t adhere to that of the structure, which is easily achieved by specifying the position of nodes of the fluid mesh according to the displacements yielded by the structural simulation. If b dðX; tÞ is used to describe the

va ¼ vmove þ na

∂tcav : ∂t

(25)

cav

The term ∂t∂t is obtained according to the considerations in Section 2.1.1. All values are taken at a retarded time

displacement of Γf;b t from its initial position at t ¼ 0,

tret ¼ t

(19)

kek c

(26)

which refers to the time of emission. Naturally, this time does usually not coincide with the chosen time steps, and a linear interpolation is utilized to obtain the required information.

must be satisfied. Equation (19) also facilitates the imposition of the second constraint, which enforces equal velocities of fluid and structural boundary, such that for the body in the fluid subproblem the velocity (20)

3. Numerical methods

holds. The boundary conditions described by Equation (19) and Equa­ tion (20) need to be prescribed in the fluid domain. On the structural side, an equilibrium of the tractions t must be enforced such that t ¼ p n þ τ w onΓFSI t :

_ l⋅e

�

ret

2.3. Coupling conditions

vmove ¼ d_ onΓFSI t

�

dΓf;b 2 m⋅eÞ ret � Z � l⋅e l⋅m þ dΓf;b 2 m⋅eÞ2 ret Γf;b kek ð1 # Z " _ þ cm⋅e ckmk2 Þ 1 l⋅eðkekm⋅e � þ dΓf;b 3 c Γf;b kek2 �1 m⋅ej 4πp’L ðxo ; tÞ ¼

and homogeneous initial conditions are prescribed. Here, N represents the unit outward normal vector on the boundary of Ωs0 while t and d are given surface tractions and displacements, respectively.

b d ¼ d on ΓFSI t

1 c

Z

In this work, a partitioned solution approach is pursued, as this al­ lows the use of specialized and well developed simulation software for each involved domain. Information transfer between the subproblems is €nig et al., managed by the in-house coupling software comana, see (Ko 2016). In the following, the discretization methods for all involved subproblems are summarized and a coherent description of the coupling process is given.

(21)

Therein, τ w denotes the wall shear stress. With regard to potential the­ ory, which makes use of the assumption of an inviscid fluid, τ w is approximated on a panel-wise basis within the BEM using an empirical friction coefficient.

3.1. Boundary element method The hydrodynamic domain is handled by means of a boundary element method. For detailed information regarding the method’s 4

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Ocean Engineering 197 (2020) 106854

fundamental principles and its’ implementation, the reader is referred to Katz and Plotkin (2001). For the purpose of this work, it is important to note that the surface of body and wake are discretized by first-order panels on which a constant variable strength is assumed. A discretized formulation of Equation (3), which is suitable for numerical solution, is given by

! Φðl; mÞ ¼ Φ ld ; m

edges’ midpoints while direction ζ is normal to the panel surface. The coordinate systems utilized in the sheet cavitation model are illustrated in Fig. 2. It is assumed that the orientation of the radial panel strips coincides with the direction of the local flow and thus the integral in the DBC is evaluated along each strip. Pressure and suction side are evaluated separately as the DBC and KBC are prescribed and evaluated for each side. A discretized formulation of the DBC is then given by

" sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ∂Φ v2i;η kvk2 þ ðp∞ pv Þ þ 2gðz∞ zÞ 2 ρ ∂t w¼ld þ1 l X

þ

(31)

ve;ξ Þξd �jðw;mÞ :

0¼

� � � � Z Z nb nw X X 1 1 1 1 dΓf;b dΓf;w μ n⋅grad þ μ n⋅grad i i f;b f;w 4 r 4 r π π Γ Γ i¼1 i¼1 i i � � Z nb X 1 1 dΓf;b σ i : 4π Γf;b r i¼1 i

In this context, m and l denote the indices of panel strip and panel in that strip, respectively. While m covers the range of all radial strips, here l ¼ 0; …; lr holds with lr referring to the index of the reattachment panel. The potential at the detachment point is denoted by Φðld ; mÞ and ξd refers to the distance between each panel’s chordwise edge midpoints. Regarding the KBC, with jðl;mÞ indicating an approximation of the spatial and tem­

(27)

poral derivatives by a central difference scheme and a backwards dif­ ference scheme, respectively, a discrete formulation is obtained, viz � � ∂tcav ∂tcav kt1 � t2 k2 vs1 ðv ðt1 ⋅t2 Þvs2 Þjðl;mÞ jðl;mÞ ¼ ∂t ∂s1 s1 (32) ∂tcav þ ðvs2 ðt1 ⋅t2 Þvs1 Þjðl;mÞ : ∂s2

Here, nb and nw denote the number of panels used to discretize the body and wake surface, respectively. Considering Equation (27), only the doublet strength on the body panels is unkown, since the doublet strength in the wake can be prescribed employing the Kutta condition (Katz and Plotkin, 2001) and the source strength on the body is given by (28)

σ ¼ n⋅v onΓf;b t :

In this context, rigid body motions, influences due to structural defor­ mation as well as the inflow field of the propeller are accounted for in the effective velocity v, which governs the body panel source strength. If a non-uniform inflow field is specified, the point set describing the inflow is projected onto the propeller blades and the respective veloc­ ities are prescribed at the thus identified panels. Evaluating Equation (27) at each collocation point on Γf;b yields an equation system and its’ solution delivers the unknown doublet strength for all body panels. The now available doublet and source strengths can be inserted in a discrete formulation of Equation (3), yielding the solution for the potential: � � � � Z Z nb nw X X 1 1 1 1 ΦðxÞ ¼ dΓf;b dΓf;w μ n⋅grad þ μ n⋅grad i i 4π Γf;b r 4π Γf;w r i¼1 i¼1 i i (29) � � Z nb X 1 1 f;b dΓi þ Φ∞ σ 4π Γf;b r i¼1 i

In the coordinate system the KBC is formulated in, the normal and chordwise direction coincide with the orthogonal coordinate system. Direction s2 differs and is oriented in direction of the vector pointing from each panel’s center towards the midpoint of the corresponding southern edge, see Fig. 2. Following the considerations in Gaschler (2017) and Vaz (2005), prescribing the discrete KBC at all cavitating panels yields a system of equations for the cavity thickness, such that Mk tcav k ¼ f k:

Here, k denotes the discrete time step. A schematic description of the solution algorithm for the hydrodynamic domain is then given by: 1. Compute time steps using the fully wetted solution process depicted in Sections 2.1 and 3.1 until a periodical solution is acquired. 2. Activate cavitation model. 3. Estimate initial cavity shape based on p � pv . 4. Prescribe dipole strength on cavitating panels according to Equation (31). 5. Utilizing the computed dipole strength, set up and solve the new system of equations for an updated pressure and velocity field. 6. Solve Equation (33) for the cavity thickness at all panels. 7. If the cavity thickness at the reattachment panels exceed a certain threshold, move cavity boundaries up/downstream at the respective panel strip. The conditions for the modification of the cavity boundaries are described in Gaschler (2017) and Vaz (2005). 8. Repeat Steps 4. - 8. until convergence is achieved. 9. Proceed to next time step and repeat Steps 4. - 9.

The pressure at each collocation point on the body can now be computed making use of the Bernoulli equation (2). In order to obtain the infor­ mation required for the coupling process, a panel force is derived and the respective traction is calculated by means of t¼

ft ; aΓf;b

(30)

i

with f t as the total force vector acting on the panel and aΓf;b denoting the panel area.

(33)

i

3.1.1. Partially non-linear cavitation model without mesh modification Due to the technique chosen to model sheet cavitation on the pro­ peller blades, the discretization of the fluid mesh is not altered. The local orthogonal coordinate system’s base vectors are defined using the respective panels’ geometry. Direction ξ is given by the chordwise

Considering the integration of the cavitation model in the overall FSI algorithm, high fidelity is pursued. The dynamic interaction of blade movement and deformation with the cavities developing on the pro­ peller blade is considered in each time step as well as each of the FSI 5

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Ocean Engineering 197 (2020) 106854

Fig. 3. Data flow within one time step of the coupling algorithm.

iterations, see Fig. 3.

corresponding operator refers to. An operator formulation for the fluid subproblem is introduced and reads

3.2. Finite element method Since the discretization process with respect to FEM is well known, in this work only a brief description is given in order to ensure a coherent description of the coupling algorithm, see Section 3.3. The structural domain Ωs0 is described by a set of elements Ωse . Within each of these elements, the unknown displacement field can be obtained as a weighted sum of shape functions n X

d�

Ni ðξ; η; ζÞ di inΩse ;

Considering the structural subproblem described by Equation (35), the displacement field can directly be evaluated at the required points. For the fluid problem, which is managed by a BEM, this is not the case, and an interpolation of the tractions from the collocation points of the fluid mesh to the quadrature points of the finite elements must be performed. The interpolation method utilized here is a mesh based inverse distance €nig et al., 2016). Denoting the interpolation by the oper­ weighting (Ko ator I yields

(34)

i¼1

(37)

tk ¼ I ∘tk :

where ξ, η and ζ denote the local element coordinates. Here, di corre­ sponds to the degree of freedom i. As commonly described in literature, integrated Legendre polynomials are used to obtain a hierarchical basis. A weak form is acquired from the discretized problem using the formulas for the geometrical description as well as the unkown displacement field, see e.g. Wriggers (2008). The generalized-α method (see e.g. Kuhl and Crisfield (1999)) is utilized as the time integration method while a mass-proportional damping is employed. The resulting nonlinear system is solved using the Newton-Raphson method.

Consequently, it may be worthwhile to note that for the displacements dk ¼ dk holds while for the tractions tk 6¼ tk . The coupling algorithm used here can now be described by the introduced operators, such that in terms of a fixed-point interpolation (38)

i i diþ1 k ¼ A k ∘S k ∘I ∘F k ∘dk

holds. Fig. 3 illustrates the coupling algorithm. The operator A ik refers to a convergence acceleration method, which in this case is the quasi-Newton least squares method. For a thorough description of the method, the reader is referred to Degroote et al. (2009) or Radtke et al. (2016). In the context of the coupling algorithm, the convergence acceleration infers a manipulation of the deformation field obtained by the structural solver before it is transferred to the fluid solver. It is possible to further simplify the notation by combining the existing operators into P k , yielding

3.3. Coupling algorithms Regarding the structural domain, the solution process of acquiring a discretized displacement field due to specified loads can be described in each time step k as dk ¼ S ∘tk :

(36)

tk ¼ F ∘dk :

(35)

Here, dk refers to the displacement vector of the fluid nodes. Considering the basic nature of the coupling process, as depicted in Fig. 1, it is apparent that the fluid solver needs to be supplied with displacements while the input to the structural solver consists of the occurring trac­ tions. While the displacements need to be prescribed at the nodes of the fluid mesh, the tractions are required at the quadrature points of the structural mesh. Naturally, these point sets do not coincide, requiring the use of dedicated techniques to obtain the information at the required positions. The displacements at the required points are obtained by projecting the fluid vertices onto the structural mesh and using the shape functions of the FEM to acquire the displacement at the respective po­ sitions. Having obtained the required displacement field, it is applied to the vertices of the fluid mesh on a pointwise basis. The current position of a fluid mesh point is thus prescribed as the addition of the point’s original position and it’s current displacement. The operator S de­ scribes the computation of the displacement field as well as its’ evalu­ ation at the projected points. The loads are described by the vector tk which collects the tractions at the quadrature points of the structural mesh. In this work, vectors marked with an overbar indicate that the therein collected quantities are prescribed within the subproblem the

iþ1

i i ~ i diþ1 k ¼ A k ∘P k ∘dk ¼ A k ∘dk :

(39)

iþ1

~ Here, d denotes the (unmodified) solution obtained from the solver. k On the basis of this, a residual can be formulated concerning the occurring displacements vectors such that ~iþ1 rik ¼ d k

dik :

(40)

By defining a threshold, the residual is used to assess the convergence in each iteration of the calculated time steps. 3.4. Ffowcs Williams-Hawkings equation The discretization used to evaluate the acoustic part of the simula­ tions coincides with that of the fluid mesh. The contributions of body displacement and loading, as well as the resulting overall pressure at the observer point, can therefore be obtained by

6

T. Lampe et al.

Ocean Engineering 197 (2020) 106854

4πp’T ðxo ; tÞ ¼

m⋅eÞ

þ

� kek2 �1

Γf;b i

4πp’L ðxo ; tÞ ¼

i¼1 nb Z X þ i¼1

Γf;b i

�

c

m⋅ej3

�

Z nb X 1 Γf;b i

l⋅e

kek2 ð1

ret

# ckmk2 Þ

_ þ cm⋅e ρva ⋅na ðkekm⋅e

5. Setup In this work, the SVA-P1356 propeller (Richter and Heinke, 2006) of the KCS container ship is investigated. The fluid as well as structural mesh used in the following simulations is shown in Fig. 4. In all simu­ lations the time step for both, structural and fluid domain, is chosen such that the propeller rotates by 1∘ in each step. Table 1 gives an overview of all investigated configurations’ com­ mon parameters. Naturally, material properties are only applied in simulations when elasticity is taken into account. For simulations involving inhomogeneous inflow conditions, the hull wake field illustrated in Fig. 5 is prescribed. The hull wake field is considered as stationary with respect to time. For the acoustic evaluation of all configurations, only one of the propeller blades is considered. This choice was made in order to facili­ tate the assessment of the influence of elasticity on the acoustic char­ acteristics, which is hard to distinguish for cases in which the interaction of inhomogeneous inflow conditions, cavitation effects, the effect of the propeller blades on each other and the blade flexibility are taken into account simultaneously. The observation point underlying all acoustic evaluations is located at a distance of 30 m in direction of the y-axis, see Fig. 4, to the propeller center. As the evaluation point is still well within the near-field, a solution can, for non-cavitating cases, also be obtained by a purely potential theory based approach utilizing the unsteady Bernoulli Equation. Throughout the simulations, the relative position of propeller and observation point remains fixed.

� 2

m⋅eÞ

kekð1

m⋅eÞ2

(41) dΓf;b i ;

ret

_ l⋅e

l⋅m

comana’s basic methodology with regard to information transfer and €nig et al. (2016). details of its implementation can be accessed in Ko

dΓf;b i

2

kekð1

Γf;b i

i¼1

�

ρðv_a ⋅na þ va ⋅n_ a Þ

"

nb Z X i¼1

�

nb Z X

dΓf;b i ret

� dΓf;b i ret

(42)

# Z " nb X _ þ cm⋅e ckmk2 Þ 1 l⋅eðkekm⋅e � þ dΓf;b i ; 3 2� c Γf;b 1 m⋅ej kek i¼1 i ret p’ðxo ; tÞ ¼ p’T ðxo ; tÞ þ p’L ðxo ; tÞ:

(43)

4. Preliminary work Considering the presentation of a partitioned solution algorithm for multi-field problems in general, it would be best to investigate the behaviour of each field’s solver on its own before assessing the perfor­ mance of the coupled technique. As this would be out of the scope of this paper, the reader is referred to several previously published papers in order to establish confidence in the modelling of the structural, fluid and acoustic domain as well as the coupling process. panMARE has been used in the context of a wide range of application cases, covering con­ figurations involving propellers as well as propeller acoustics but also wind turbines and others. The capabilities of the code with regard to simulations involving propellers in unsteady inflow conditions has been demonstrated in Scharf et al. (2015) by means of a comparison of pressure fluctuations obtained from panMARE and a RANS calculation. €ttsche et al. (2017), the acoustic signature of a pulsating sphere is In Go evaluated and compared to a reference solution based on the Bernoulli equation. Furthermore, results of panMARE’s acoustic evaluation tool €ttsche et al. (2019) and where compared against measurements in Go Kleinsorge et al. (2017). In Gaschler (2017), the setup which is inves­ tigated here is simulated for a rigid propeller and thus findings regarding the discretization of the fluid domain could be transferred. comana, which handles the coupling process, has likewise been used in various application cases, such as the simulation of flexible wind turbines (Wiegard et al., 2019) or the berthing maneuver of offshore service ships (Ferreira Gonz� alez et al., 2015). An investigation of the coupling process as well as the convergence behaviour of propeller related structural meshes used in AdhoC has been performed in Radtke et al. (2018).

Table 1 P1356 propeller and environment. Propeller Nr. of blades

nB

Skew

s

½deg�

31.83

Diameter

D

½m�

7.9

Rotation rate

n

½s

Material properties Material density

ρM

½kgm

Poisson ratio Environment Fluid density Speed of sound

5

1

1.25

� 3

�

ρF c

8150.0 0.3

ν ½kgm ½ms

1

3

�

�

1025.0 1500.0

Fig. 4. P1356 fluid mesh and coordinate system (left, green), structural mesh (right, blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 7

T. Lampe et al.

Ocean Engineering 197 (2020) 106854

Table 2 Thrust and torque coefficients for varying stiffness. Advance coefficient J ¼ 0:75. Young’s modulus [Pa]

kT

kQ

η0

Rigid 2⋅1011

0.1632 0.1632

0.0266 0.0266

0.732 0.732

2⋅1010

0.1626

0.0265

0.732

2⋅109

0.142

0.023

0.735

In this case, the blade deformation causes a reduction of both blade thrust and torque. The effect becomes increasingly pronounced for high blade flexibility. While the efficiency remains relatively unaffected for moderate elasticity parameters, the method predicts a slight increase in efficiency for the case with a Young’s modulus of E ¼ 2⋅109 Pa. Since stiffness values of this amount would cause unacceptable vibrations of the blades, a transfer of these findings to actually operating propellers is infeasible. The reduction of thrust and torque is also reflected in the occurring deformation field. Fig. 7 presents the deformations of the propeller blades’ median plane at different relative radii. The blades experience different amounts of deformation at leading and trailing edge, thus changing the pitch angle, which is considered as the dominating influence responsible for the altered performance characteristics of the propeller. The change in pitch angle, estimated employing the deformation values at leading and trailing edge and the assumption of a constant chord length, is also given in Fig. 7 via the annotations associated with each section plot. All sections experience a reduction in pitch angle, which in turn causes the reduction of propeller thrust and torque. The progression of pressures calculated based on the FWHE at the observation point is given in Fig. 8. Although the hydrodynamic results were obtained for the whole propeller, only one blade is taken into ac­ count in the acoustic evaluation for the reasons given in Section 5. For better comparability, both graphs are shifted to their mean values. Although thrust and torque coefficient were severely reduced for the flexible propeller, the progression of the FWH pressures is almost un­ affected. As homogeneous inflow conditions are prescribed, no pertur­ bation of the pressure plots due to structural vibrations is present. Consequently, the resulting acoustic spectrum at the observation point, which is presented in Fig. 9 for the rigid case, is almost identical.

Fig. 5. P1356 in stationary inflow.

6. Results and discussion In the following, configurations of varying complexity are simulated in order to demonstrate the capabilities and physical consistency of the presented approach. 6.1. Homogeneous inflow, full propeller For an initial assessment of the presented method, the full propeller is simulated in homogeneous inflow conditions. After a correct model­ ling of the hydrodynamic domain has been ensured, blade elasticity and its effect on the acoustic signature of the propeller is considered. Fig. 6 presents the thrust and torque coefficient at multiple advance co­ efficients, calculated by panMARE for a rigid propeller, and compares them to results obtained in experiments by Richter and Heinke (2006). Both thrust and torque coefficient show a slight deviation from the ex­ periments at the entire range of advance coefficients. For the torque coefficient, an increasing deviation from the experimental results is observed for decreasing advance coefficients. Since the propeller thrust is expected to have a dominating influence on the occurring de­ formations, the integrity of the results should not be affected, though. The propeller is then simulated at an advance coefficient of J ¼ 0:75 under consideration of varying magnitudes of elasticity. Table 2 presents the utilized material properties as well as the resulting coefficients for thrust, torque and efficiency.

6.2. Homogeneous inflow, sinusoidal load, full propeller The investigations conducted in Section 6.1 yielded a static

Fig. 7. Section deformations. Advance coefficient J ¼ 0:75. Young’s modulus E ¼ 2⋅109 Pa.

Fig. 6. Thrust and torque coefficient over advance coefficient for the rigid propeller. 8

Ocean Engineering 197 (2020) 106854

T. Lampe et al.

observation point which is not entirely based on a single frequency. Another distinct peak in the spectrum is observed at the excitation fre­ quency. The results indicate that the method presented here is capable of capturing the influence of structural vibrations on the acoustic spectrum. 6.3. Inhomogeneous inflow, single blade Having established a physically consistent behaviour of the simula­ tion approach with respect to configurations involving homogeneous inflow conditions, a setup in which a wake field is prescribed is simu­ lated. Cavitation is not considered yet. Here, only a single blade is considered for both the hydrodynamic and acoustic investigations. The undisturbed inflow velocity is vS ¼ 7:4 ms 1 while the actual inflow field is specified according to the velocity distribution illustrated in Fig. 5. Table 3 presents the mean thrust, torque and efficiency coefficients for the rigid case as well as three cases with varying elasticity. Contrary to what has been observed for homogeneous inflow con­ ditions, the mean thrust and torque coefficient increase with increasing blade flexibility. The mean efficiency remains almost unaffected though. Fig. 10 shows the progression of the thrust coefficient over one rotation period for the rigid blade as well as the case with E ¼ 2⋅1010 Pa. While the behaviour of the thrust coefficient is almost identical for large parts of the blade’s movement through the wake field, the thrust coefficient is increased in the vicinity of the 12 o’ clock position. An indication for the underlying mechanisms responsible for the increase in thrust can again be found in the deformation field of the blade, which, along with the change in pitch angle associated with each section, is given in Fig. 11. Due to the inhomogeneous inflow, the blade experiences a defor­ mation field fundamentally different to that found in open water con­ ditions. While for sections close to the hub a decrease in pitch angle is still present, the sections towards the tip are subjected to an increase in pitch angle. Since the upper part of the blade delivers a higher contri­ bution of the blade thrust, the increased load in this region prevails concerning the effect on the overall loads. For comparison, Fig. 12 shows the deformation field in open water conditions assuming thrust identity regarding the average thrust in the behind-hull configuration. The position of the blade relative to the hull wake as well as the resulting pressure field in comparison to that in open water conditions is shown in Fig. 13. In the vicinity of the evaluated blade position, the spatial gradients of inflow velocity are at their maximum, thus creating highly different inflow conditions on the respective parts of the blade surface. The described conditions in turn lead to the difference between pressure and deformation fields in open water and behind-hull conditions. In the following, an acoustic evaluation of the cases is performed and the resulting acoustic spectrum is given in Fig. 14. As expected, the dominating frequency for the rigid and flexible case is found to be the rotation rate of the propeller blade. In accordance with the small changes in thrust coefficient, the flexible blade’s rotation rate tonal remains almost unchanged. Due to the consideration of the structural deformations the amplitudes in the frequency range above the rotation rate change and new peaks are introduced to the spectrum. At

Fig. 8. Shifted FWH pressures at observation point over one rotation period. Advance coefficient J ¼ 0:75.

Fig. 9. Acoustic spectrum for homogeneous inflow and sinusoidal addi­ tional load.

deformation field for each propeller blade since homogeneous inflow conditions were prescribed and no gravitation is considered in the simulations. In order to verify the capabilities of the method with respect to capturing the influence of structural vibrations on the resulting acoustic signature, another simulation setup is evaluated. The full pro­ peller is simulated in homogeneous inflow conditions and a sinusoidally varying traction is added to the loads occurring due to the actual operation of the propeller. In this case, the added traction field is spatially constant and only acts in direction of the x-axis, see Fig. 4. While the excitation frequency is chosen as fEx ¼ 36 Hz, the maximum magnitude of the added tractions is tadd ¼ 25⋅103 Nm 2 . Coherent with the investigations in the previous section, the setup is evaluated using varying stiffness parameters. As a sinusoidally varying load was pre­ scribed, the mean thrust and torque coefficients closely match the final values computed in the previous section, see Table 2. In the following, the FWH pressure time series at the observation point is analysed by means of a fast Fourier transform (FFT) and the resulting acoustic spectrum is given in Fig. 9. The annotations in the plot denote the rotation rate and the excitation frequency. The most dominating frequency is found to be the propeller rotation rate, with the associated peak in the spectrum developing over multiple adjacent frequencies. The reason for the increased energy content in the adjacent frequencies lies in the choice of setup which only considers one of the propeller blades, leading to a pressure time series at the

Table 3 Mean thrust and torque coefficients for non-cavitating blade in inhomogeneous inflow. Varying stiffness parameters.

9

Young’s modulus [Pa]

kT

kQ

Rigid 2⋅1011

0.0699 0.0699

0.01 0.01

2⋅1010

0.0709

0.1007

2⋅109

0.0727

0.0105

T. Lampe et al.

Ocean Engineering 197 (2020) 106854

present, it could not be verified if these can be associated with the blade’s structural modes or if another mechanism is responsible for this behaviour. 6.4. Inhomogeneous inflow, single blade, cavitation Cavitation is considered as one of the governing influences on the acoustic signature of propellers. The setup described in Section 6.3 is therefore augmented in order to incorporate cavitation effects. The cavitation number, based on the blade’s rotational speed, is chosen as σ n ¼ 3:0 such that a coherent cavity is present on the blade when it passes the 12 o’ clock position of the inflow field and the blade loads are at their maximum. An impression of the cavity developing on the blade is given in Fig. 15. Table 4 summarizes the results of the cases investigated in this section. Due to the consideration of cavitation on the propeller blade, both mean thrust and torque coefficient are slightly reduced. For the cases involving flexible blades, a similar behaviour to what has been observed in Section 6.3 is present, as the mean blade loads increase with decreasing blade stiffness. The acoustic spectrum for the cavitating rigid and flexible blade is presented in Fig. 16. In the frequency range above the tonal associated with the rotation rate frequency, energy density is increased and the sound pressure levels are raised in comparison to the non-cavitating case. The distinct peaks in the frequency range in between 15 and 35 Hz, which were present in the frequency spectrum of the non-cavitating blade, see Fig. 14, are not observable any more. It is assumed that due to the increased energy density in this frequency range, the respective tonals are not as easily distinguishable as before.

Fig. 10. Thrust coefficient over one rotation period for rigid and flexible blade.

7. Conclusion An approach for the assessment of the far-field acoustic signature of marine propellers under consideration of inhomogeneous inflow, cavi­ tation and blade elasticity was presented. A partitioned algorithm is utilized, enabling the usage of well developed solvers and discretization techniques for each domain. The respective solvers are chosen pursuing a balance between accuracy and computational effort suitable of the design stage. The fluid domain is handled by a BEM while for the structural domain a FEM is utilized. In the context of the BEM, cavitation is considered using an additional cavitation model. The interaction be­ tween fluid and structural domain is managed using a separate frame­ work which utilizes the quasi-Newton least squares method in order to accelerate and stabilize the solution process. An acoustic evaluation of the problems is performed with the FWHE in Formulation 1A. In the paper, a coherent mathematical description of each involved domain as well as the coupling process is given. The governing equations in their continuous formulation as well as the discretization techniques for each domain are described. As no validation data regarding the full complexity of the problem is available, a series of configurations intended to verify a physically consistent behaviour of the coupled simulations is simulated. For homogeneous as well as inhomogeneous inflow conditions, the correlation between loads experienced by the propeller blade and the corresponding deformation field shows a plau­ sible and sound behaviour. In general, these findings also apply to the acoustic evaluation. If the blade is subjected to an excitation by an additional sinusoidally varying load, the excitation frequency is clearly visible in the acoustic spectrum. For the non-cavitating blade in inho­ mogeneous inflow, peaks which might be attributed to the blade’s structural modes are present in the acoustic signature. As the same configuration with consideration of sheet cavitation does not exhibit these tonals, it is assumed that due to the dynamic behaviour of the sheet cavitation energy is introduced to the spectrum at a frequency range which coincides with the previously observed peaks, thus obstructing their extraction by means of FFT analysis. The method presented here

Fig. 11. Section deformations at 12 o’ clock position. Young’s modulus E ¼ 2⋅ 1010 Pa.

Fig. 12. Section deformations in open water assuming thrust identity. Young’s modulus E ¼ 2⋅1010 Pa.

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Ocean Engineering 197 (2020) 106854

Fig. 13. Blade position in wake and comparison of pressure fields in hull wake (kT ¼ 0:0709) and open water conditions (kT � 0:0709). Table 4 Mean thrust and torque coefficients for cavitating blade in inhomogeneous inflow. Varying stiffness parameters. Young’s modulus [Pa]

kt

kq

Rigid Rigid, cavitating 2⋅1011

0.0699 0.066 0.0691

0.01 0.0095 0.0099

2⋅1010

0.0706

0.0102

2⋅109

0.0754

0.0107

Fig. 14. Acoustic spectrum for inhomogeneous inflow and rigid as well as flexible blade.

Fig. 16. Acoustic spectrum for inhomogeneous inflow and rigid as well as flexible cavitating blade.

Author contributions section Fig. 15. Inhomogeneous inflow. σn ¼ 3:0. Cavitation extent at 12 o’ clock position.

Düster: Conceptualization, Resources, Supervision, Project admin­ istration, Funding acquisition, Writing – Review & Editing Maksoud: Conceptualization, Resources, Supervision, Project administration, Funding acquisition, Writing – Review & Editing Radtke: Methodology, Software, Validation, Formal Analysis, Inve­ sitgation, Data Curation, Visualization, Writing – Review & Editing Lampe: Methodology, Software, Validation, Formal Analysis, Inve­ sitgation, Data Curation, Visualization, Writing – Original Draft, Writing – Review & Editing

shows a stable and consistent behaviour. Future research priorities should be directed at the integration of the method in an optimization process regarding the acoustic properties of marine propellers. Natu­ rally, a validation of the complete method as well as its’ components should also be pursued.

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Ocean Engineering 197 (2020) 106854

Declaration of competing interest

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement The authors gratefully acknowledge the support provided by the DFG (German Science Foundation – Deutsche Forschungsgesellschaft) under the grant numbers AB 112/12-1 and DU 405/13-1. References Bazilevs, Y., Takizawa, K., Tezduyar, T., 2013. Computational Fluid-Structure Interaction: Methods and Applications. Wiley Series in Computational Mechanics. John Wiley & Sons. Brenter, K.S., Farassat, F., 2003. Modeling aerodynamically generated sound of helicopter rotors. Prog. Aerosp. Sci. 39, 83–120. https://doi.org/10.1016/S03760421(02)00068-4. Bungartz, H., Mehl, M., Sch€ afer, M. (Eds.), 2010. Fluid-Structure Interaction II, Modelling, Simulation, Optimisation. Volume 73 of Lecture Notes in Computational Science and Engineering. Springer. Bungartz, H., Sch€ afer, M. (Eds.), 2006. Fluid-Structure Interaction, Modelling, Simulation and Optimisation. Volume 53 of Lecture Notes in Computational Science and Engineering. Springer. BV, 2014. Underwater Radiated Noise (URN). Bureau Veritas. Carlton, J., 2018. Marine Propellers and Propulsion. Butterworth-Heinemann. Collier, R., 1997. Encyclopedia of Acoustics. Wiley. https://doi.org/10.1002/ 9780470172513 (chapter 46). Degroote, J., Bathe, K.J., Vierendeels, J., 2009. Performance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction. Comput. Struct. 87, 793–801. https://doi.org/10.1016/j.compstruc.2008.11.013. Düster, A., Kollmannsberger, S., 2010. AdhoC4 – User’s Guide. Lehrstuhl für Computation in Engineering, TU München, Numerische Strukturanalyse mit Anwendungen in der Schiffstechnik. TU Hamburg-Harburg. Düster, A., Rank, E., Szab� o, B., 2017. The p-version of the finite element and finite cell methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (Eds.), Encyclopedia of Computational Mechanics, second ed., Part 1. John Wiley & Sons, pp. 137–171. https://doi.org/10.1002/9781119176817.ecm2003g. Solids and Structures. (chapter 4). EU, 2010. COMMISSION DECISION on Criteria and Methodological Standards on Good Environmental Status of Marine Waters. European Commission. Ferreira Gonz� alez, D., K€ onig, M., Abdel-Maksoud, M., Düster, A., 2015. Simulation of safety-relevant situations regarding the interaction of service ships with offshore

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