Underwater radiated noise reduction technology using sawtooth duct for pumpjet propulsor

Ocean Engineering 188 (2019) 106228

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Underwater radiated noise reduction technology using sawtooth duct for pumpjet propulsor Denghui Qin a, b, Guang Pan a, b, *, Seongkyu Lee c, Qiaogao Huang a, b, **, Yao Shi a, b a

School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, 710072, China Key Laboratory for Unmanned Underwater Vehicle, Northwestern Polytechnical University, Xi’an, 710072, China c University of California, Davis, One Shields Avenue, Davis, CA, 95616, USA b

A R T I C L E I N F O

A B S T R A C T

Keywords: Pumpjet propulsor The underwater radiated noise The sawtooth duct Ffowes-Williams and Hawkings model

Marine Environmental Protection Committee (MEPC) and other organizations have made calls to develop reg­ ulations for limiting noise levels at sea. The propeller noise plays an important role in underwater radiated noise. Pumpjet propulsor (PJP), as a new type of propulsion system, is widely used in underwater vehicles. A propeller noise reduction technology using sawtooth duct is proposed and applied for the first time to PJP. In this paper, a normal PJP model and a PJP model with the sawtooth duct are presented to investigate effects of the sawtooth duct on hydrodynamic and noise performances. Flow of the models is simulated using a commercial CFD soft­ ware, and noise is predicted based on DES simulation and FW-H equation. Results show that the sawtooth duct results in 2% of maximum water efficiency loss. The sawtooth duct directly changes the morphology of wake vortices, and reduce the TKE value at most of the wake region of PJP. Furthermore, the sawtooth duct can significantly reduce the noise generated by PJP from 10Hz to 5000Hz, especially in the low-frequency below 1000 Hz. The overall SPL of PJP is reduced about 2.5–5 dB by sawtooth duct with the maximum reduction 4.88 dB at the design point.

1. Introduction Over the past few decades, due to the rapid increase in the number of vessels, the sea ambient noise contributed by the commercial shipping fleet has increased by about 12 dB (Hildebrand, 2009). It is known that fish utilize sound as the main means when they navigate, mate, and communicate with others (Bass and Mckibben, 2003). Also marine mammals mainly use sound for underwater communication and sensing (Wartzok and Ketten, 1999). The underwater radiated noise (URN) from commercial shipping may overlap their communication frequency band, which will severely affect the fundamental living activities of marine life (Hildebrand). In order to ensure sustainable shipping, more and more international organizations, such as International Maritime Organiza­ tion (IMO) and Marine Environmental Protection Committee (MEPC) have made calls to develop regulations about the URN from commercial shipping to limit the noise level at sea. (MEPC 60th session Agenda item 18). The underwater noise of ships can be divided into three main classes:

1. Hydrodynamic noise; 2. Machinery noise; 3. Propeller noise (defined by The Specialist Committee on Hydrodynamic Noise). Amongst the three classes, propeller noise usually dominates the URN of ships (Ross, 2013). With the development of technology, the hydrodynamic noise and machinery noise of ships becoming less and less by the use of various methods. The propeller noise is becoming one of the main traceable signals of noise radiated by ships. For naval ships and submarines, the smaller radiated noise, the less likely that the possibility of being detected by sonar. Therefore, the propeller noise determines the € detectability and survivability of the ship (Ozden et al., 2016). Conse­ quently, noise prediction and noise reduction technology for propeller has always been an important topic for naval ships. Pumpjet propulsor (PJP), as a new underwater propulsion system, is typically composed of the rotor, the stator, and the duct. The PJP usually has a great balance of the rotating moment due to the existence of the stator blades (Pan et al., 2013) (Suryanarayana et al., 2010a). Therefore, PJP is widely used in submarines such as the American ”Sea Wolf” class submarine, British ”Vanguard” class submarine and other Unmanned

* Corresponding author. School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, 710072, China. ** Corresponding author. School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, 710072, China. E-mail addresses: [email protected] (D. Qin), [email protected] (G. Pan), [email protected] (S. Lee), [email protected] (Q. Huang). https://doi.org/10.1016/j.oceaneng.2019.106228 Received 19 December 2018; Received in revised form 1 May 2019; Accepted 16 July 2019 Available online 31 July 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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Ocean Engineering 188 (2019) 106228

Underwater Vehicles (UUV). At present, researches of pumpjet pro­ pulsor mainly concentrate on the experiment and numerical simulations of the hydrodynamic performance. Ch. Suryanarayana et al. (2010b) conducted an experiment on hydrodynamic performance of the under­ water vehicle equipped with a pumpjet propulsor. Stefan Ivanell (Iva­ nell) studied the hydrodynamic performance of the torpedo with a pump jet using computational fluid dynamics (CFD) method. Pan Guang et al. (Pan et al., 2013) carried out the numerical calculation of one vehicle equipped with a pump jet propulsor. Lin Lu et al. (2016) investigated the hydrodynamic performance of a pumpjet propulsor using CFD method. Qin et al. (2018) presented the influence of tip clearances on PJP hy­ drodynamic performance based on CFD method. For underwater radiated noise predictions, the acoustic analogy based on the Ffowcs Williams and Hawkings (FWH) Equation (Williams and Hawkings, 1969) is one of the most widely used approaches. The noise source information is usually obtained from CFD results, such as Detached-Eddy-Simulation (DES) or Large-Eddy-Simulation (LES). Due to the increased computational power, this method is becoming avail­ able for marine propellers. Varney et al. (Varney and Martino) proposed that adding a perforated wear liner inside the rotor housing can alter the spectrum of noise produced by the pump jet. In addition, Bagheri et al. (2015) studied the noise trend of marine propellers. FWH equation is used to predict the noise of a propeller under non-cavitating and cavi­ tating conditions. Sai et al. (Sai et al., 2017) proposed a semi-empirical modulation model based on a generalized acoustic analogy theory to € predict counter-rotation propeller non-cavitation noise. Ozden et al. (2016) investigated the noise of the INSEAN E1619 propeller at different operational conditions. Also the noise results of propeller in open water and behind the DARPA suboff submarine are compared. In general, most papers focus on the underwater noise of the single propeller or counter-rotation propeller, and there are few publications about the noise of pumpjet propulsor. Therefore, more attention can be paid to the study of pumpjet propulsor noise. The flight of most of the owl species is not audible to their prey. The silent fly of owl has always been a source of inspiration for scientist to find solutions for quieter machinery. The theoretical and experimental researches on mechanisms of the nearly silent flight owls were carried out by lots of researchers in past decades (Sarradj et al., 2011). The noise reduction mechanisms for the special morphological features of owl wing had been studied and are used for airfoils, resulting in sawtooth (Howe, 1998) or serrated trailing edges (Howe, 1991) airfoils. Subse­ quently, the sawtooth trailing edge technology was applied to the aero-engine exhaust nozzle, called chevron exhaust nozzles, which was proved to be an effective way to reduce jet noise and was widely used in the aviation industry. Since 1996, GE (General Electric) company and NASA had been exploring and investigating it for noise reduction and have achieved good results. This drew more attention to the sawtooth trailing edge. In order to reduce jet noise of airplane exhaust systems, GE (Martens) developed the chevron nozzle. It was found that the sawtooth trailing edge technology can achieve the 2–3 EPN (Effective Perceived Noise) dB noise reduction. Callender (Callender et al., 2005) carried out the far-field noise test of Chevron nozzles in the University of Cincinnati Nozzle Acoustic Test Facility (UCNATF). The results show that the overall sound pressure level (OASPL) is reduced from 3 to 6 dB. The sawtooth trailing edge nozzle which developed jointly between GE Aircraft Engines and NASA has proven to be an effective and efficient means to reduce jet noise with minimal impact on engine performance, operability, weight, and cost, for some aircraft systems (Martens). Hence it is widely used in the aviation field, but to authors’ best knowledge, it has not been used for reducing noise of the underwater propeller. However, given the similar geometry and flow between the pumpjet propeller and the aircraft engine jet nozzle, it is worth exploring whether the sawtooth duct can have the same noise reduction on the PJP. In this paper, effects of the sawtooth duct on hydrodynamic and noise performances of PJP is studied. The goal of this paper is: 1) Assessing the applicability of DES with FW-H equation in predicting the

underwater radiated noise of propeller. 2) Investigating the effect of the sawtooth duct on the flow of PJP. 3) Comparing the difference for wake vortices structures of normal PJP and the sawtooth duct PJP. 4) Quan­ titatively evaluating the noise reduction of the sawtooth duct on PJP at different working conditions. The structure of this article is as follow. The background is introduced in Section 1. Section 2 gives the numerical methodology. In Section 3, the flow and noise simulation methods are validated. Section 4 shows the geometry, the numerical set-up and the grid independence test. The effects of sawtooth duct on the hydrody­ namic coefficients of PJP are discussed in Section 5. Also, the structures and evolution of PJP wake vortices are compared. Moreover, Section 6 quantitatively evaluates the noise reduction by using sawtooth duct. Finally, the conclusions are summarized in Section 7. 2. Methodology The governing equations of incompressible and single-phase fluid flow are Reynolds Averaged Navier-Stokes (RANS) equations as follows:

∂ρui ¼0 ∂xi

(1) �

� �

∂ðρui Þ ∂ ρui uj ∂p ∂ ∂ui ∂uj ¼ þ μ þ þ ∂t ∂xj ∂xi ∂xj ∂xj ∂xi

�� þ

∂ ∂xj

�

ρu’i u’j þ Sj

(2)

where ρ is the fluid density, t is the time. xi and xj (i ¼ 1, 2, 3, j ¼ 1, 2, 3) denote the cartesian coordinate components. ui and uj are the velocity

components and p is the pressure. ρu’i u’j is the Reynolds stresses, and μ denotes the dynamic viscosity. Sj denotes the generalised source term for the momentum equation. Firstly, the steady flow of PJP is simulated based on the RANS with Shear-Stress-Transport (SST) k ω model to study the hydrodynamic performance (Menter, 1994). The superior performance of this model has been discussed in a large number of papers (Huang et al., 1104) (Ji et al., 2010). Hence, the SST k ω turbulence model is applied in this paper for the numerical simulation of propeller. No further elaboration of the model will be made here. Secondly, based on the steady flow, the DES simulation of PJP is carried out and then time dependent flow data is used to predict the underwater radiated noise of PJP based on the FWH equation. The DES model combines both RANS and LES. The DES method based on the Spalart & Allmaras model (Spalart and Allmaras, 1992) is shown as below. The eddy viscosity νt is defined as follow:

νt ¼ ~νfν1 fν1 ¼

χ¼

χ3 χ 3 þ Cν 31

~ν

ν

(3) (4) (5)

where the subscript ν stands for viscous. Cν 31 equals to a constant value. The transport equation has become � o � ∂~ν ∂~ν 1 n ~ν cw1 fw r⋅½ðν þ ~νÞr~ν� þ Cb2 �r~νj2 þ Cb1 ½1 ft2 �S~ þ uj ¼ ∂t ∂xj σ �� �2 ~ν Cb1 ft2 þ ft1 ΔU 2 (6) 2 κ d A more detail presentation of the DES method can be seen in spalart et al. (Spalart and Allmaras, 1992). Finally, the noise is predicted based on FWH equation, which is shown as follow:

2

Ocean Engineering 188 (2019) 106228

D. Qin et al.

1 ∂2 p’ c20 ∂t2

r2 p’ ¼

∂ f½ρ v þ ρðun ∂t 0 n

vn Þ�δðf Þg

� � � � ∂2 vn Þ δðf Þ þ Tij Hðf Þ ∂xi ∂xj

is used, and the pressure and momentum terms are discretized by second order scheme. The time term is discretized using the bounded 2nd order implicit time discretization and the time iteration step is set as 1 x 10 4 s. The hydrodynamic results are the mean value of the DES results for the last 20 rotation period. The numerical simulation results of hydrodynamic coefficient are compared with the experimental data (Baltazar et al., 2018) (see Table 1). As shown, all the relative errors of the thrust and torque co­ efficient are always less than 2% with maximum relative error ΔKT ¼ 1:58% at J ¼ 0:5. So the numerical simulation results have a good agreement with the experimental data, and the numerical simulation method with the DES turbulence model is applicable and reliable for the propeller flow simulation.

∂ � P n þ ρui ðun ∂xi ij j (7)

where c0 is the sound velocity in the far field. pij denotes the fluctuating stress tensors acting on fluid units. ui and uj is the flow velocity in the i and j axis direction. δðfÞ is the Dirac delta function and HðfÞ is the Heaviside function. p’ denotes the acoustic pressure. nj is the component of outward-directed unit normal vector of a surface in the j axis direc­ tion. Vn is the velocity component normal to a surface. ρ0 denotes the density of free-stream, and ρ is the density of fluid. Tij is the Lighthill stress tensor defined as follows: c20 ðρ

Tij ¼ ρui uj þ Pij

ρ0 Þδij

(8)

3.2. Noise prediction methodology validation

3. Validation

After conducting DES simulations, the FW-H equation is used for acoustic prediction of propeller. As for choosing of the noise contribu­ € tion sources for propeller noise prediction, Ozden et al. (Ozden et al., 2016) carried out a study for a navy propeller. Firstly, the surface of blades and hub were set as the sound sources. As a contrast, a rotating domain was created around the propeller, where the outer surface (porous surface) of this domain was employed as the noise source sur­ face. Compared with the noise results using the blades and hub surface as source surface, the results using the porous surface made it possible to include nonlinear terms and showed better noise prediction values. Also, Sandro Ianniello et al. (Ianniello and De Bernardis, 2015) carried out the research about the potential of the Acoustic Analogy in the analysis of the underwater noise generated by a marine propeller. Results show that a reliable hydroacoustic analysis of a marine propeller seems to require the computation of the FWH equation’s nonlinear quadrupole sources. And the noise contribution due to the nonlinear sources can be assessed through the porous formulation, by using the outer surface of a rotating domain. So for the noise prediction of PJP, the outer surface of domain around PJP is defined as the noise source surface. Due to that there is no experimental noise data of the propeller Ka470 with the 19A duct, the noise prediction methodology is validated using another benchmark case, and the noise results are compared with the experimental data. Jacob et al. (2005) carried out one noise test of � rod-airfoil at Ecole Centrale de Lyon (ECL). This experiment is now considered as a canonical benchmark for air noise prediction. A NACA0012 airfoil of chord C ¼ 0.1 m is placed downstream of a cylin­ drical rod with a diameter of d ¼ 0.01 m. The spanwise length of the airfoil is 0.3m. The incoming velocity U0 ¼ 72 m=s. The noise is tested through two bruel & kjaer 4191 microphones and bruel & kjaer 2669 preamplifier. The noise receiver is located in the middle plane of the airfoil, and the radius is 1.85M. The numerical verification is carried out based on the experiment of Jacob, and the models are in the same with the experiment. Taking into account the relatively large amount of calculation, the computational spanwise length of rod-airfoil is Lsim ¼ 3:5d ¼ 0:035 m. The computational domain and boundary conditions are shown in Fig. 2. The computational domain extends from -6C to 7C in the streamwise direction and from -3C to 3C in the crosswise direc­ tion. The total computational domain is divided into two domains: near field domain and far field domain. The mesh of near field domain is obviously encrypted to better simulate the flow and turbulent eddy structures around rob and airfoil. The domain size is shown in Fig. 3

3.1. Flow simulation methodology validation In this paper, numerical validation is carried out with a widely known duct propeller, the propeller Ka4-70 with pitch-diameter ratio P/ D ¼ 1.0 inside the duct 19A (Baltazar et al., 2018). The diameter of propeller is 200 mm, and the tip clearance between blades and duct is uniform and equal to 0.8 mm. The computational domain of duct pro­ peller (see Fig. 1) is a cylinder with 5D diameter, extending from -3D to 7D in the streamwise direction. Structural mesh is used here. Specif­ ically, the first boundary layer height around the walls is 0.001 mm which ensures the yþ value of walls is less than 1, and the growth rate is 1.05. Details of the blades mesh are illustrated in zoom-in view in Fig. 1. The total number of cells of this mesh is about 8 million. The advance ratio J is J ¼ U∞ =ðnDh Þ, where U∞ is the free stream velocity. n is the rotor blade rotating speed (r/s) and Dh is the diameter of the propeller. ρf is the density of fluid. The propeller thrust coefficient is KT ¼ Thrust=ðρf n2 D4h Þ, and the torque coefficient is KQ ¼ Torque= ðρf n2 D5h Þ. The relative errors ΔKT and ΔKQ are defined as below.

ΔKT ð%Þ ¼

ΔKQ ð%Þ ¼

KT

KT KT EXP

� 100

(9)

KQ CFD KQ CFD � 100 KQ CFD

(10)

CFD

EXP

In the DES simulation of duct propeller, the boundary of inlet is velocity inlet with U∞ ¼ 2:57m=s and the boundary of outlet is pressure outlet with an average relative static pressure 1 MPa. The rotating speed is changed in different work conditions. In addition, SIMPLEC algorithm

Table 1 The hydrodynamic coefficients at different work conditions.

Fig. 1. The computational domain and boundary conditions of duct propeller Ka4-70 with duct 19A 3

J

KTCFD

KTEXP

ΔKT

10KQCFD

10KQEXP

ΔKQ

0.1

0.4723

0.4680

0.92%

0.4415

0.4387

0.64%

0.3

0.4184

0.4140

1.06%

0.4325

0.4279

1.08%

0.5

0.2529

0.2490

1.58%

0.3552

0.3506

1.30%

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Ocean Engineering 188 (2019) 106228

Fig. 4. Sound pressure level result at receiver comparing with the experi­ ment data.

Fig. 2. The computational domain and boundary conditions.

Fig. 5 (a). The propeller has 11 rotor blades and 9 stator blades. The rotor is in front of the stator, and the rotors rotate clockwise (seen from the front of the model). The diameter of pumpjet propulsor is D ¼ 0:26 m and the length of original duct is L0 ¼ 0:17 m. The design point of PJP is in the case the rotational speed of rotor blade N ¼ 3200 RPM and inflow velocity 25.72 m/s (J ¼ 1.90). An O XYZ cartesian coordinate system is built. The origin O is located at the center of the propeller. and the positive direction of the z axis coincides with the inflow direction. The direction of x and y axes are defined by the right-hand rule. Inspired by the sawtooth nozzle, the sawtooth duct is used for pumpjet propulsor (Fig. 5 (b)). As shown Fig. 6 (a), the sawtooth is attached to the trailing edge of original duct. Fig. 6 (b) shows the sawtooth size. The length of sawtooth is Ls ¼ 0:025 m and the angel of sawtooth is β ¼ 70∘ . The number of teeth of the sawtooth duct is N ¼ 18. To simulate real flow of propeller, one half ellipsoid type flow-guide cap has been added in the front of the propulsor model. The total length of PJP with sawtooth duct after adding flow-guide cap is L ¼ 0:4 m. Two abbreviations, NPM (the normal pumpjet propulsor model) and SDPM (the sawtooth duct pumpjet propulsor model), are used for clarifying. The geometries of rotor and stator blades are all the same for both models.

Fig. 3. The mesh of the center plane and zoomed in view around airfoil.

which is determined by making reference to the research of Jean-Christophe Giret (Giret et al., 2012). The outer surface of near field domain is selected as the sound source (see the red line in Fig. 3). This sound source choosing is applied in order to be consistent with the choosing of the noise source surface of the pumpjet propeller noise prediction. Fig. 3 shows the grid of the center plane. There are 181 nodes along the cylinder and 281 nodes on the airfoil. The initial wall spacing is 0.001 mm which ensures the yþ value on the walls is less than 1, and the growth rate is 1.05. The zoom-in view shows the boundary layer grid of the cylinder, the airfoil and the local encrypted mesh nearby. The total number of grids is about 2.85 million. The DES simulation is carried out. The boundary conditions of calculation are set as same as the experiment. The time step is Δt ¼ 2:5 � e 6 , and the 10000 time steps CFD results are sampled. The receiver is located same as the microphone in the experiment. Based on the SPL results in Fig. 4, the noise results agree well with the experimental value of Jacob (Jacob et al., 2005). The tonal peak can be obviously seen in f ¼ 1375:6 Hz (St � 0:19) which is caused by the impingement of large vortices on the leading edge of airfoil. Conse­ quently, it is concluded that the numerical approach using DES and FW-H is reliable for the application considered in this paper.

4.2. Computational domain and mesh Fig. 7 shows the computational domain of PJP and boundary con­ ditions setting. The computational domain is a cylinder with 5D diam­ eter, extending from -3L to 7L in the streamwise direction. The total computational domain is divided into four parts: the rotor domain, the stator domain, the noise domain and the external domain. The rotor domain is rotating domain, and the other three domains are stationary. The contacting surfaces among those domains are set as the interface to exchange flow information. In order to get accurate acoustic results, a mesh refine domain around the propeller called “noise domain” is created. This domain is a cylinder with 1.5D diameter and 4D length, whose rotating axis aligns with the symmetry axis of propulsor. The outer wall of this domain is selected as the noise source. The structured grid is favorable for the boundary layer simulation. Therefore, the structured grids are used in this research. Fig. 8 shows the mesh of NPM in longitudinal section. Specifically, the grids of rotating domain, stator domain and noise domain are denser than that in the far field. Also, the meshes of blades and duct for NPM and SDPM are shown in Fig. 9. For both models, the meshes of rotor and stator blades are identical and only the mesh of duct and hub of SDPM is different from NPM. For SDPM, the grid of duct consists of 18 rotational copies of a sawtooth channel mesh. Additionally, the rotor and stator blades are surrounded by O-grids (see the zoom-in view) which ensures a good

4. Numerical set-up 4.1. Geometry model The pumpjet propulsor geometry 3D model in this study is shown in 4

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Fig. 5. The pumpjet propulsor geometry 3D models.

Fig. 6. The size definition of sawtooth.

Fig. 9. The mesh details of the blades and duct for NPM and SDPM.

Fig. 7. The computational domain and boundary conditions.

regions are determined and re-changed based on the vortex distribu­ tion area of preliminary flow field. The yþ value of the walls is less than 1. The total number of cells for NPM is approximate 12; 000; 000, while the number of cells for SDPM corresponds to 14; 000; 000. 4.3. Boundary condition For boundary conditions setting, the inlet is set as velocity inlet with normal speed, V ¼ 25:72 m/s, while the rotational speed is adjusted with range of 2400 � 4200 revolutions per minute (RPM) to alter the physics conditions. The turbulence intensity is 5% as the default. The averaged static pressure of outlet is set as 10 MPa. The flow of PJP is solved by ANSYS FLUENT (Version 15, ANSYS Inc.) based on finite volume method solver. In the DES simulation, the SIMPLEC algorithm is used and the pressure and momentum terms of equations are discretized by the second order scheme. Specifically, the time term is discretized using the bounded 2nd order implicit time discretization. The CourantFriedrichsLewy (CFL) number is defined as CFL ¼ ΔtU=Δmin , where U is

Fig. 8. Longitudinal section view of mesh for NPM.

boundary layer near the walls. Furthermore, to accurately capture the formation and evolution of small vortices using as few grids as possible, the meshes in the region of rotor tip, downstream of duct and hub wake are refined (see the rectangular box in Fig. 8). Those mesh-refined 5

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Ocean Engineering 188 (2019) 106228

the local velocity and Δmin ¼ minðΔx ; Δy ; Δz Þ. The time step setting should satisfy that CFL < 1. In case N ¼ 3200 RPM, the time iteration step is set as 5 � 10 5 , the time required for a 1∘ of rotor blade rotation.

Table 3 The Grid convergence analysis for NPM.

4.4. Grid independence validation In order to facilitate the discussion, the non-dimensional physical quantities are defined as shown in Table 2. In the table, v is the far field flow velocity; n denotes the rotor speed (r/s); D is the diameter of the propulsor; ρ is the fluid density; Tt denotes the thrust of rotor; Ts is the thrust of stator and duct; Mt denotes the torque of rotor and Ms is the torque of stator and duct. In this section, mesh independence validation is conducted. Based on the two-grid assessment procedure proposed by Roache (1997), two different meshes of NPM, coarse and fine meshes, are generated. For both meshes, the geometric dimensions and topological structures are maintained as identical and the only difference is the mesh size setting. For the coarse mesh, the yþ value of the walls is about 1, while the yþ value of the walls for fine mesh is about 0.5. In addition, the nodes number of blades and duct surfaces are refined with a factor 1.2 for fine mesh. Also, the mesh in the vortex refinement region (see Fig. 8) is refined to more accurately capture the development of propeller tur­ bulence. For the coarse mesh, the maximum size of the vortex refine­ ment regions, Δmax ¼ maxðΔx ; Δy ; Δz Þ, is 1:5%D, while Δmax ¼ 1%D for fine mesh. The total cells number of coarse grid Ncoarse corresponds to about 12; 000; 000 while the number of fine mesh Nfine is about 24; 000; 000. The grid-convergence index (GCI) based on the theory of Richardson Extrapolation (Roache, 1997) is used to report grid-convergence tests. A coarse-grid Richardson error is defined as Ecoarse ¼

ε ¼ f2 � r¼

1

rp ε rp

Nfine Ncoarse

Fine

Ecoarse

GCIcoarse

1.52

KTt

0.0035

1:05%

0.7052

0.7039

KTs

0.1163

0.1178

KMt

0.2993

0.2979

0.0038

1:14%

KTt

0.6788

0.6772

0.0043

1:30%

KTs

0.0465

0.0451

0.0038

1:14%

KMt

0.3026

0.3011

0.0041

1:22%

KTt

0.5546

0.5529

0.0046

1:38%

KMt

0.2814

KTs

0.1485

0.1469 0.2792

0.0041

0.0043 0.0059

1:22%

1:30% 1:78%

DES results for the last 20 rotation period, 7200 time steps). The results indicate that the numerical results of both grids are similar. The GCIs of coarse mesh results are always less than 2% when J ¼ 1.52, 1.79 and 2.54. And then, in order to validate the grid independence for noise pre­ diction, the pressure fluctuations of one probing point P1 (it will be described in Fig. 25) for coarse and fine meshes are compared in Fig. 10. As shown, the pressure fluctuations of coarse mesh agree well with that of fine mesh. In addition, the period of pressure fluctuation is about 0.0017 s, which is consistent with the blade passage period. It’s mean that the rotation periodicity of the rotor is well captured. Generally speaking, the hydrodynamic coefficients and flow pressure fluctuation of coarse mesh agree well with that of fine mesh and the gridconvergence test is great. In addition, for a typical case (J ¼ 1.79), with a coarse mesh about 12; 000; 000, the computational time consuming 1,468,800 s (approxi­ mately 17 days) for 12,000 time steps (using CPU Inter(R) Xeon(R) E52630 v4 @ 2.2 GHZ, 20 processors). For the fine grid, computing will take more time. so for the sake of reducing computing resources con­ sumption, the coarse mesh is more applicable in this research. Therefore, the coarse grid is used in the subsequent analyses.

(13)

5. Flow results and discussion

Here f1 and f2 denote the torque and thrust coefficient of stator-duct and rotor for coarse and fine meshes, respectively. p is the formal order of accuracy of the algorithm (p ¼ 2). r is the refinement factor between the coarse and fine grid. d is the dimensionality of the simulation. The GCI of coarse mesh is defined as

5.1. Results of the hydrodynamic performance of PJP The flow simulations of NPM and SDPM with inlet velocity V ¼ 25:72 m/s are carried out. And the rotational speed (N) varies from 2400 RPM to 4200 RPM. Fig. 11 shows the hydrodynamic coefficients and open water efficiency results for NPM. As shown, the thrust and torque coefficients of the rotor and stator-duct all decrease linearly with the

(14)

GCIcoarse ¼ Fs jEcoarse j

Coarse

2.54

(12) �1=d

Hydrodynamic coefficients

1.79

(11)

f1

J

Fs > 1 is a safety factor. Based on the recommendation of Roache (1997), Fs ¼ 3 is used here. The hydrodynamic coefficients of the fine and coarse meshes with different J are compared in Table 3 (the results are the mean value of the Table 2 Non-dimensional physical quantities. Physical quantities

Definition

Advance coefficient

J ¼

The thrust coefficient of rotor The torque coefficient of rotor The thrust coefficient of stator and duct The torque coefficient of stator and duct Total thrust coefficient Total torque coefficient The open water efficiency

v n�D Tt KTt ¼ 2 4 ρn D Mt KMt ¼ 2 5 ρn D TS KTs ¼ 2 4 ρn D Ms KMs ¼ 2 5 ρn D KT ¼ KTt þ KTs KM ¼ KMs

η ¼

J KT 2 π KM

Fig. 10. The grid convergence analysis of pressure fluctuation for NPM. 6

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Table 4 Hydrodynamic coefficients and open water efficiency results of SDPM contrast to NPM. J

2.54 2.34 2.18 2.03 1.90 1.79 1.69 1.60 1.52 1.45

increase of J. Also the torque coefficients of the rotor, KMt , is almost the same as the torque coefficients of stator-duct, KMs , which shows that this propulsor is well designed and has a good torque performance. Additionally, the thrust coefficient of rotor, KTt , is obviously greater than that of stator and duct system, KTs . In J ¼ 1:45, the stator-duct provide about 20% of the total thrust. But in work condition of maximum efficiency (J ¼ 1:79), the rotor blades supply about 90% of total thrust of propulsor, while the thrust of stator-duct accounts for 10% of total thrust. Furthermore, it should be noted that the thrust coefficient of the stator-duct is negative when J > 2:03. Hence the stator-duct provide about 10% � 20% thrust of propulsor at low J but changes from thrust to drag at high J conditions. Finally, the open water effi­ ciency firstly increases, then decreases with the increase of J, especially reaching peak value about 70:17% at J ¼ 1:79. In order to explore the effects of sawtooth duct on the hydrodynamic performance of PJP, the open water efficiencies of NPM and SDPM are compared in Fig. 12. In addition, the hydrodynamic coefficients of both models are shown in Table 4. K’Tt , K’Ts , K’T , K’Mt , K’Ms and η’ are the thrust coefficient of rotor, the thrust coefficient of stator-duct, the total thrust coefficient, the torque coefficient of rotor, the torque coefficient of stator-duct and the open water efficiency of SDPM, respectively, while KTt , KTs , KT , KMt , KMs and η are the according physical quantities of NPM. ΔKT is defined as

ΔKTs ð%Þ

ΔKT ð%Þ

ΔKMt ð%Þ

ΔKMs ð%Þ

Δη ð%Þ

1.54 0.80 0.29 0.07 0.41 0.69 0.95 1.12 1.28 1.44

10.21 2.38 28.80 70.07 31.34 23.62 19.81 17.09 14.81 12.68

4.46 1.18 0.70 1.81 2.59 3.15 3.55 3.71 3.75 3.69

1.07 1.04 0.77 0.37 0.04 0.46 0.92 1.40 1.92 2.47

0.99 1.45 1.70 1.80 1.81 1.76 1.52 1.14 0.59 0.14

3.39 1.47 0.04 1.00 1.84 2.52 3.10 3.51 3.83 4.09

5.2. The effects of sawtooth duct on the flow of PJP To analysis the effects of sawtooth duct on the flow of PJP, the steady RANS flow results of NPM and SDPM are contrasted in the case J ¼ 1.90, which is the design point of PJP. Fig. 13 (a) and (b) visualize the pressure contours of hub and blades of NPM contrast to SDPM, respectively. And Fig. 13 (c) and (d) show the duct pressure contours of NPM and SDPM. As shown, the pressure contours of blades of NPM are almost the same as those of SDPM. However, the pressure distributions in the inside and outside surface of sawtooth duct are obviously different with the duct of NPM (see zoom-in view in Fig. 13 (c) and (d)). There is a circle low pressure zone in the inner surface of duct trailing edge for NPM, but low pressure area appears in the root of sawtooth for SDPM. The sawtooth completely changes the flow around the trailing edge of duct, while has relatively little effect on the pressure distributions of rotor and stator blades. The 3D streamlines around the normal duct and one sawtooth channel are illustrated in Fig. 14. As shown, for NPM, the flow is sucked from the outer surface of duct to the duct trailing edge. A circle of vortex called “duct induced vortex” is formed behind the duct trailing edge. As for SDPM, due to the pressure difference of side surfaces and top-bottom surfaces, the flow is sucked into the sawtooth channel from the topbottom surface of duct and a pair of vortexes is generated in every sawtooth channel. Furthermore, the effect of sawtooth on the velocity field is discussed. Fig. 15 shows the zoom-in overviews of tangential and axial velocity component velocity contours around duct for NPM (left) and SDPM (right), respectively. There appear two low velocity zones near the root in every sawtooth channel for SDPM (see in Fig. 15 (d)), in contrast to a circular low velocity zone behind the trailing edge of duct

Also ΔKTt , ΔKTs , ΔKMt , ΔKMs , Δη are defined in

the same way (e.g. Δη ¼

ΔKTt ð%Þ

decreases with the increasing of J. However, the η of SDPM is smaller than that of NPM at J ¼ 1:45 � 2:18, but greater than that at J ¼ 2:18 � 2:54. The η of NPM reaches its maximum 70:17% at J ¼ 1:79, but the η of SDPM has a peak about 68:03% at J ¼ 2:03. Hence, it means that the maximum water efficiency has been reduced about 2% by sawtooth duct. In contrast, the sawtooth duct improves the η of PJP at high J range, about 3:4% at J ¼ 2:54. As shown in Table 4, the relative error of total thrust ΔKT varies form 3:69% to 0:7% in the case J from 1.45 to 2.18. Moreover, ΔKT cor­ responds to 4:46%, which resulting to the open water efficiency improvement about 3:39% at J ¼ 2:54. Specifically, the torque coeffi­ cient relative error of stator-duct, ΔKTs , is obviously bigger than ΔKTt . Similarly, the ΔKMs is relatively bigger than ΔKMt at most cases. This means that the sawtooth duct is mainly affect the flow of stator-duct domain rather than rotor domain and leads to the thrust and torque change of propulsor. Consequently, the sawtooth duct results in water efficiency loss at J ¼ 1:45 � 2:18, but improves the open water efficiency at J ¼ 2:18 � 2:54. The maximum water efficiency has been reduced about 2% by sawtooth duct. The sawtooth duct mainly affects the thrust and tor­ que of stator-duct, resulting in the efficiency change.

Fig. 11. Hydrodynamic coefficients and open water efficiency curves of NPM.

K0 T KT KT .

Relative Errors

η’ η η ).

As shown in Fig. 12, the open water efficiency curve of SDPM also shows the same change law of NPM, which first increases and then

Fig. 12. Open water efficiency curve of NPM compared with that of SDPM. 7

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Fig. 13. The overview pressure contours of blades and duct for NPM contrast to SDPM (J ¼ 1.90).

Fig. 14. The 3D streamlines around duct which color is defined by axial velocity for NPM and SDPM. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 15. The zoom-in overview of v (tangential velocity component) and w (axial velocity component) velocity contours near trailing edge of duct for NPM and SDPM.

for NPM (Fig. 15 (c)). Additionally, it is clearly observed that the tangential velocity components of the two vortexes are opposite (the red zone and the blue zone in Fig. 15 (b)). Therefore, there is a pair of counter rotating vortexes in every sawtooth channel, and the two vor­ texes are formed in the root of sawtooth and develop following two side surfaces of sawtooth along the streamwise direction. To further study the effect of sawtooth on the wake flow of PJP, the velocity and turbulence kinetic energy (TKE) of NPM and SDPM are discussed. Fig. 16 has visualized the velocity magnitude contours of center plane for NPM and SDPM, respectively. And Fig. 17 illustrates the

turbulence kinetic energy (TKE) contours of center plane for NPM and SDPM, respectively. As shown in the Fig. 16, the velocity in the wake region of SDPM is nearly same with that of NPM. Furthermore, the axis velocity contours at six different z locations (z/L ¼ 0.45; 0.5; 0.75; 1; 2; 5) for NPM and SDPM are compared in Fig. 18 (a). Firstly, at z/L ¼ 0.45, there are both 11 high velocity regions (the red zones) in the propulsor outflow, which is caused by the high velocity flows from 11 rotor blade channels (some areas are not obvious due to the effect of 9 stator blades), while, for SDPM, there exists low velocity zone in every sawtooth channel due to 8

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the sawtooth vortexes discussed before. Similar features are also observed in contours at z/L ¼ 0.5, and there is a petal shaped low ve­ locity zone whose petal number is associated with sawtooth number N ¼ 18 in the wake region. Then, from z/L ¼ 0.75 to z/L ¼ 2, the high velocity zone in the propulsor outflow is becoming more and more uniform for both models, which means the propulsor outflow is gradu­ ally weakening. However, for NPM, the shape of outer boundary of high velocity propulsor outflow (the red area in Fig. 18 (a)) is changing from nearly round to petal shape which number is 9, which means the stator channel flows began to dominate the outflow. In contrast, for SDPM, the shape of outflow outer boundary is always maintain as uniform circles, and the axial velocity distribution of wake region is more uniform than that of NPM. The reason can be also found from Fig. 15, the tangential velocity of sawtooth wake region is obviously complex and there are more positive and negative regions than the NPM (see Fig. 15 (a)). The pair of counter rotating vortexes in every sawtooth channel, similar like fluid mixer, strongly exchange the fluid of high velocity outflow with the surrounding fluid. This results in that the shape of high velocity outflow outer boundary for SDPM is nearly circle. In addition, the high axis velocity fluid is mixed fully enough by the vortexes and so the distri­ bution of axis velocity is more uniform for SDPM compared with NPM (see Fig. 15 (d)). Finally, at z/L ¼ 5, the axial velocity distributions of NPM and SDPM are all uniform, though the velocity of NPM still is slightly affected by stator passage outflow. Moreover, based on the results in Figs. 17 and 18 (b), there exist significant differences between the TKE distributions of NPM and SDPM. As shown in Fig. 17, the high TKE area in the wake region of SDPM is obviously less than that of NPM, which means that the sawtooth duct significantly reduces the TKE in the most wake region of PJP. The more detailed information of the TKE reduction is shown in Fig. 18. (b), and the TKE contours at six different z locations (z/L ¼ 0.45; 0.5; 0.75; 1; 2; 5) of NPM and SDPM are compared. Firstly, at z/L ¼ 0.45, the high TKE region of NPM is shown as a circle, coinciding with the shape of circle duct. This is due to the periodical duct shedding vortices. In comparison, there are 18 high TKE regions in the channel of sawtooth for SDPM, which is caused by the formed pair of counter rotating vor­ texes in every sawtooth channel. Then, at z/L ¼ 0.5, there are petal shaped high TKE regions behind the sawtooth for SDPM. This also resulting that the TKE of SDPM in the area behind the duct is greater than NPM (the detail will be discussed in Fig. 19). Additionally, from z/ L ¼ 0.75 to z/L ¼ 5, there is a circle of petal-shaped high TKE region for NPM. Conversely, the high TKE region behind the duct for SDPM is circumferentially uniformly distributed and the magnitude is signifi­ cantly less than NPM. The reason has been discussed before and it is caused by the formation and the development of sawtooth vortexes. The pair of counter rotating vortexes in every sawtooth channel increases the

Fig. 16. The overview velocity contours of center plane for NPM and SDPM.

Fig. 17. The overview TKE contours of center plane for NPM and SDPM.

Fig. 18. The axial velocity and TKE contours at different z locations for NPM and SDPM (Left: NPM; Right: SDPM).

Fig. 19. The mean TKE values at different z locations for NPM and SDPM. 9

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contact area of low velocity surrounding flow and high velocity outflow and helps the mixing of them, prompting the dissipation of TKE in the wake region. Furthermore, in order to quantitatively reveal the effect of sawtooth duct on the TKE, the mean TKE values at different z locations for NPM and SDPM are compared in Fig. 19. Based on the results, at the region behind the duct (from z/L ¼ 0.45 to 0.75), the mean TKE value of SDPM (the red curve) is a little higher than that of NPM (black curve). This is because of that the formation and development of sawtooth vortexes obviously increase the turbulent energy. However, at the most of the wake region (from z/L ¼ 0.75–5), the mean TKE value of SDPM is significantly smaller than that of NPM. Fig. 17 (b) also clearly visualized the difference. The maximum mean TKE value reduction caused by sawtooth duct is about 38% at z/L ¼ 2. Fig. 21. The spatial structures of the instantaneous propeller vortices for NPM (using the Q-criterion and colored by TKE) (J ¼ 1.90).

5.3. The effects of sawtooth duct on the vortices structure of PJP

mainly “impact-separation vortex”, flow towards downstream direction of the propeller. The spatial structures of the instantaneous vortex in the wake region of NPM and SDPM are illustrated in Fig. 22 and Fig. 23, respectively. As shown, the fine vortex structures in the wake region are well captured. For NPM, the wake vortexes are mainly composed by the impactseparation vortex; hub vortex and duct-induced vortex. As discussed before, the duct induced vortex of NPM is mainly periodical shedding small vortex groups with uniform circumferential distribution. But for SDPM, the duct-induced vortex is shown as the shape of sawtooth ac­ cording with the sawtooth number. Both duct induced vortexes gradu­ ally break down with the flow of fluid. By contrast the Figs. 22 and 23, we can see that the sawtooth duct-induced vortexes are more finely broken and the TKE value is significantly smaller than normal duct

With the aim of exploration of effects of sawtooth duct on the vertical structure of PJP, the DES simulations of NPM and SDPM in the case J ¼ 1.90 are further analyzed and compared. Fig. 20 (a) has visualized the local pressure contours at different streamwise locations in the rotor blade passage. It is clearly observed that two low pressure zones first appear in the suction-tip line(the boundary line of suction surface and rotor tip sur­ face) and pressure-tip line of rotor blade (see zoom-in view in Fig. 20 (a)). This is also why the two tip vortexes, “Tip-separation vortex” and “Tip-leakage vortex” (see Fig. 20 (b)), are formed. The flow is sucked into the suction surface from the pressure side of rotor blades. Then, by comparing the local pressure contours at different streamwise locations, it is clearly observed that the low pressure zone (resulting in tip-leakage vortex) is gradually moving towards the middle of the rotor passage along with the fluid flowing. Thus the tip-leakage vortex is formed in the leading edge of rotor tip and develops gradually towards the middle of the rotor passage along the streamwise direction. With respect to tipseparation vortex, it is also formed in same location but develops gradually following the boundary line of pressure surface and rotor tip surface. Furthermore, it should be noted that those two vortexes will exchange fluid and some fluid is sucked to tip-leakage vortex from tipseparation vortex (see Fig. 20 (b)). Detailed spatial structures of those vortices can be visualized in Fig. 21. The large amount of rotation energy in the tip vortex flow causes severe flow loss of propeller. Additionally, the spatial structure of the instantaneous vortex in the rotor and stator domain of NPM constructed by using Q-criterion is shown in Fig. 21. There are mainly rotor tip-leakage vortexes propa­ gating downstream with the fluid flowing. Due to the number of rotor and stator blades is different, some rotor tip-leakage vortexes flow directly into the stator channel, called “channel vortex”, and then hit the suction surface of the stator blade. In addition, other vortexes impact on the leading edge of stator blades directly, called “impact-separation vortex” and then broking into 2 strands of vortexes which mainly focus on the pressure side of stator blade (see Fig. 21). Finally, those vortexes,

Fig. 22. The spatial structures of the instantaneous vortices (using the Q-cri­ terion and colored by TKE) for NPM.

Fig. 20. The local pressure contours and flow streamlines around the rotor blade (J ¼ 1.90). 10

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Ocean Engineering 188 (2019) 106228

Fig. 23. The spatial structures of the instantaneous vortices (using the Q-cri­ terion and colored by TKE) for SDPM.

Fig. 25. The PSD of kinetic energy for point P1 and P2 (J ¼ 1.9).

induced vortexes. Additionally, another big difference is the impactseparation vortex. For NPM, the impact-separation vortexes sustain development until far away from the propeller. But the impactseparation vortexes of SDPM are quickly breaking down and almost disappear in the region far away from the PJP. Consequently, the sawtooth channel has significantly changed the vortex structures of PJP wake region. The vortexes in the sawtooth channel help the mixing of the high speed jet of propeller and surrounding flow, and promoting dissipation of vortexes. And the TKE value of SDPM wake vortexes is remarkablely less than that of NPM. In order to explore the difference of instantaneous vortices between NPM and SDPM quantitatively, four probing points (P1, P2, P3, P4) are chose and the spectral characteristics of the TKE for those probing points are analyzed and compared. The layout sketch of those probing points are shown in Fig. 24. P1 is in front of rotor blade, and P2 is in the middle of the rotor blade and the stator blade. P3 is behind of the stator blade. P1, P2 and P3 are all located around the tip of blades which is aimed at exploring the evolution of the rotor tip vortices. P4 is behind the trailing edge of duct for NPM and in the middle of sawtooth passage for SDPM, which is aimed at analyzing the spectrum characteristics of duct-induced vortexes. For both modes, the 3 dimensional coordinates of those probing points is the same. Figs. 25–27 show the power spectral density (PSD) of kinetic energy for P1, P2, P3, P4, respectively. The k is defined as k ¼ f= fBPF , and fBPF is the rotor blade passage frequency (BPF). For J ¼ 1.9 (N ¼ 3200 rpm), fBPF ¼ 586:67Hz. As shown in Fig. 25, the peaks appear at the harmonics of fBPF (k ¼ 1,2,3, …). It shows that the periodicity of rotor blades

Fig. 26. The PSD of kinetic energy for point P3.

Fig. 27. The PSD of kinetic energy for point P4.

rotation is well simulated, and the shedding frequency of rotor tipleakage vortexes are corresponding to the blade passage frequency. For P1 and P2, the PSD of NPM and SDPM are similar. In the lowfrequency range, PSD decreases with the increasing of frequency and the PSD tends to stabilize in the high-frequency range. Furthermore, it should be noted that the kinetic energy PSD magnitude of P2 is obvi­ ously higher than that of P1, which means that the kinetic energy fluc­ tuation increases sharply in the area between the rotor and stator blades.

Fig. 24. The layout sketch of probing points. 11

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As discussed before, the rotor tip-leakage vortexes flow directly into the stator channel, impacting on the leading edge of stator blades directly, causing the increase of kinetic energy fluctuation. However, significant differences exist between the kinetic energy PSD spectra of SDPM and NPM for P3 and P4. As shown in Figs. 26 and 27, the PSD characteristics of P3 and P4 show a few similarities. For NPM, the peaks corresponding to BPF and its multiples are still clearly observed in the low-frequency range. But for SDPM, significant peaks at k ¼ 1, 2, 3 are not observed, while, several other peaks are clear observed in the low-frequency range, for example, k ¼ 1.4 (see Fig. 27). In addition, the PSD magnitude of SDPM is significantly smaller than the signal level of NPM in the middle to high frequency range. As discussed before, those differences are mainly caused by the duct induced vortex. For NPM, the periodically shedding frequency of duct induced vortex is corresponding to the BPF and their harmonics (see black line in Fig. 27). But for SDPM, the duct induced vortex is shown as the shape of sawtooth according with the sawtooth number. The formation and evolution of sawtooth duct induced vortexes have seriously impact the periodicity and spectrum characteristics in the flow field around the sawtooth. It seem like the periodically shedding frequency of SDPM duct induced vortex is about 800HZ (k ¼ 1.4). In general, the sawtooth has significantly change the kinetic energy PSD spectrum characteristics in behind of the stator blade and the area around the sawtooth. The kinetic energy PSD of magnitude of SDPM is slightly lower than that of NPM.

following analyze, the SPL in dB, Pref ¼ 1μPa, 1 Hz and 1 m is presented. Additionally, to make the display of the difference between the noises of NPM and SDPM more clearer, some presentations of SPL curves are using 1/3 octave band. Hanning filter is used. 6.2. The noise results discussion The underwater radiated noise predictions at four receivers for NPM and SDPM in the case J ¼ 1.52; 1.90 and 2.18 are carried out. Fig. 29 has visualized the SPL results at varying receiver locations of NPM compared to SDPM in the case J ¼ 1.90 (N ¼ 3200 RPM). In this case, the shaft frequency (SF) is fh ¼ 3200=60 ¼ 53:33Hz, and the rotor blade passage frequency (BPF) is fBPF ¼ fh � 11 ¼ 586:67Hz (the number of rotor blades is 11). A comparison of the SPL curves of both models indicates that the noise of SDPM has the same noise spectrum characteristic as NPM. The noise frequency spectrums of both models are mainly composed by two parts: the discrete tonal noise and the broadband noise. The discrete tonal noise is clearly observed that the peaks appearing at the harmonics of SF and BPF (see Fig. 29). The first obvious peak is located at about f ¼ 160Hz (3fh ), which shows that the influence of hub rotating fre­ quency on noise spectrum of propeller. The high peak at 2fh can also be observed, but reduces in the amplitude. The peak at fh is not so promi­ nent maybe because of the resolution of noise signal is not high enough. Furthermore, the second obvious peak appears at about f ¼ 585Hz (at about 1 BPF). Also, the clear peaks can also be observed in multiples BPF (coinciding with nBPF, n ¼ 2, 3, 4, 5…), but reduced in the amplitude. Furthermore, it should be noted that the SPL of broadband noise grad­ ually undergoes intense attenuation at high frequency range of 700–5000 Hz. Significantly, by contrast the SPL of SDPM (the red line) to that of NPM (the black line), it is clearly observed that the noise level of SDPM is almost less than the level of NPM throughout the entire frequency range 10–5000 Hz. A more clear differences are illustrated in Fig. 30, in which the SPL is graphically presented in 1/3 octave band. In lowfrequency range 10–700 Hz, the noise reduction of SDPM is signifi­ cantly obvious with about 5 dB for all the receivers at J ¼ 1.90. The maximum noise reduction of SDPM is about 10 dB at f about 60 Hz. As shown in Figs. 22 and 23, it is easy to understand why the noise is reduced. The duct-induced vortexes of SDPM help the mixing of the high speed jet of propeller and surrounding flow, and promote the dissipation of wake vortexes. The large eddies of SDPM break up into small eddies faster than NPM. Benefitting from the sawtooth, the vortex intensity and turbulent energy in the wake region of SDPM are remarkablely less than that of NPM. Therefore, the SPL of SDPM is obviously smaller than NPM in low-frequency range. In addition, at high-frequency range, the SPL of SDPM is only slightly less than that of NPM, or even a little higher than SPL of NPM at frequency range 700–1000 Hz (see Fig. 30 (c)). It should be noted that the peak at 1 BPF of SDPM is shifting to about 800 Hz, which also can be observed in Fig. 29 (c). As discussed before, there are a counter rotating vortexes in every sawtooth channel, which resulting the sawtooth duct induced vortices. The periodically shedding frequency of SDPM duct induced vortex is about 800HZ (k ¼ 1.4). The frequency shifting is caused by the periodic shedding of sawtooth duct induced vortices. Furthermore, the SPL curves of receivers for NPM and SDPM with different rotation speeds are shown in Fig. 31. Based on the results, the noise results at 3 different working conditions, N ¼ 2800; 3200; 4000 RPM at receiver 1 and 3 are compared. It is clearly observed that, for both models, the SPL curve of N ¼ 4000 RPM with red color is obviously greater than the curve of N ¼ 3200 RPM with blue color throughout the frequency range, also the same features are also observed in SPL of N ¼ 3200 RPM compared to that of 2800 RPM, though the noise level of high N is a little lower than SPL of lower N case at some low frequency range (see Fig. 31 (b)). Hence, the SPL of both models basically show the

6. Underwater radiated noise results and discussion 6.1. The set-up of noise calculation Based on the DES results, the underwater radiated noises of NPM and SDPM are predicted based on FW-H equation. The noise sampling time step is the same as flow time iteration step (5 � 10 5 s for the case N ¼ 3200 RPM). 10000 time steps CFD results are sampled. As discussed before, results show that using porous surface around the propeller shows better noise prediction value compared with using the blades and hub. In this study, the PJP is placed inside a cylinder domain with 1.5D diameter and 4D length. The outer wall of this domain is defined as the noise source (see Fig. 28). As requested by ITTC, the sound pressure levels should be corrected to a standard measuring distance of 1 m (in dB and Pref ¼ 1μPa). Hence, four receivers are selected with 100D away from the propulsor center as shown in Fig. 28. And then underwater radiated noise results are corrected to 1 m as follow: SPL ¼ SPL1 þ 20logðrÞ

(15)

where SPL is the value at 1 m distance and SPL1 is the SPL in the receiver located r from the propeller center (as r ¼ 100D in the study). So in the

Fig. 28. The receiver locations. 12

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Fig. 29. Underwater noise predictions of receivers for NPM and SDPM (in 1 Hz frequency band) in the case J ¼ 1.90 (N ¼ 3200 RPM).

Fig. 30. Underwater noise predictions of receivers for NPM and SDPM (1/3 octave band) in the case J ¼ 1.90 (N ¼ 3200 RPM).

same variation tendency, which increases with the increase of rotor rotation speed. Moreover, the underwater radiated noise predictions of receivers for NPM and SDPM in the case J ¼ 1.52 and 2.18 are illustrated in Fig. 32 and Fig. 33. At J ¼ 1.52, the SPL of all the four receivers of SDPM are still less than the noise level of NPM, but the noise reduction is so obvious in contrast to the case J ¼ 1.90. Also the same frequency shifting at 1 BPF is observed. With respect to J ¼ 2.18, the noise level of SDPM at high frequency 1000–5000 Hz is similar with that of NPM, or even a little bigger than that of NPM (see Fig. 33 (d)). However, the noise reduction of SDPM can still be obvious observed at low frequency. In addition, the overall sound pressure level (OASPL) results of receivers for NPM and SDPM are listed in Table 5. At J ¼ 1.90, the noise reduction of SDPM is the most great, with the maximum noise reduction 4.88 dB at receiver 2. As to J ¼ 1.52, the SPL of SDPM is also less than SPL of NPM, with the maximum noise reduction 3.43 dB at receiver 2. While the noise reduction of SDPM at J ¼ 2.18 is the smallest, with only 1.44 dB at receiver 1. Furthermore, it should be noted that, at all the studied work conditions, the noise reductions at receiver 1 and 2 (in the front and behind of the propeller) are greater than receiver 3 and 4 (in the circumference of the propeller). Also the same differences can be seen at Figs. 30, Figs. 32 and 33. So the sawtooth has a better noise effect at the axial direction than the circumference direction of propeller.

7. Conclusions In this paper, in order to investigate the effect of the sawtooth duct on hydrodynamic and noise performances of pumpjet propulsor (PJP), a normal pumpjet propulsor model (NPM) and the sawtooth duct pumpjet propulsor model (SDPM) have been presented. Firstly, the flow of PJP was simulated using RANS method with SST k ω turbulence model. Then, the underwater radiated noise of PJP is predicted based on the DES simulation and FW-H equations. Numerical simulations were car­ ried out to analyze the effect of the sawtooth duct on the hydrodynamic performance, flow field, 3-D structure of vortices and noise perfor­ mance. The hydrodynamic coefficients of NPM and SDPM at different working conditions are compared. Additionally, the effects of sawtooth duct on the pressure field, velocity field and TKE of PJP are discussed. Furthermore, the instantaneous vortex structures was analysed. Finally, the underwater radiated noise predictions at receivers of NPM and SDPM were carried out. The numerical results revealed that: 1) The sawtooth duct results in water efficiency loss at J ¼ 1.45–2.18, but improves the open water efficiency at J ¼ 2.18–2.54. The maximum water efficiency has been reduced about 2% by sawtooth duct. 13

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Fig. 31. Underwater noise predictions of receivers for NPM and SDPM (in 1 Hz frequency band) with different N.

Fig. 32. Underwater noise predictions of receivers for NPM and SDPM (1/3 octave band) in the case J ¼ 2.18 (N ¼ 2800 RPM).

Fig. 33. Underwater noise predictions of receivers for NPM and SDPM (1/3 octave band) in the case J ¼ 1.52 (N ¼ 4000 RPM).

2) There is a pair of counter-rotating vortexes in every one of sawtooth channels, and the vortexes are formed in the root of sawtooth and develop following two sides of sawtooth surfaces along the stream­ wise direction. Additionally, it promotes the dissipation of TKE in the wake region; leading to significantly reduction of mean TKE value at the most of the wake region of SDPM, with maximum reduction about 38%, though increasing it at the little region behind the duct. 3) The PJP’s wake vortex system is comprised of impact-separation vortices, hub vortex and duct induced vortices. The sawtooth duct induced vortexes which show as the shape of sawtooth are more finely broken than NPM, promoting the energy redistribution in the

wake flow fields, leading to the rapid breaking down of impactseparation vortices. 4) Benefiting from the sawtooth duct, the noise levels of PJP are reduced over almost the entire frequency range 10–5000 Hz at all the studied work condition, though slightly increase it at 700–1000 Hz due to the sawtooth duct induced vortexes shedding frequency. Specifically, the noise level is reduced about 5 dB in low-frequency range below 1000 Hz. Significantly, when J ¼ 1.90 (the design point of PJP), the overall SPL reduction of SDPM varies from 2.5 dB to 4.88 dB, with the maximum value 4.88 dB.

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Table 5 The Overall SPL of different receivers for NPM and SDPM. J

Receivers

NPM SPL (dB)

SDPM SPL (dB)

Reduction (dB)

1.52

RE1 RE2 RE3 RE4

146.44 145.83 146.26 146.27

143.13 142.41 143.76 143.49

3.31 3.43 2.50 2.78

1.90

RE1 RE2 RE3 RE4

143.30 142.89 143.41 143.37

138.55 138.00 140.65 140.76

4.75 4.88 2.76 2.61

2.18

RE1 RE2 RE3 RE4

141.65 140.92 141.80 141.67

140.22 139.90 141.09 141.59

1.44 1.02 0.71 0.08

Consequently, a propeller noise reduction technology using sawtooth duct is proposed and applied for the first time to PJP. This technology is very easy to apply to pumpjet propeller. It only adds the sawtooth in the trailing edge of duct, without changing the geometry of propeller and the basic shape of duct. Therefore, this technology has small impact on the weight of propeller. Most importantly, this technology has excellent noise reduction at low frequencies which has been known to propagate over long distances since low-frequency sound experiences little atten­ uation. Generally speaking, the sawtooth duct is an effective mean to reduce underwater radiated noise of PJP with minimal impact on the geometry, operability, weight, cost and hydrodynamic performance. Acknowledgments This work was supported by the Fundamental Research Funds for the Central Universities (Grant No.3102019HHZY030019), the National Natural Science Foundation of China (Grant No. 51479170; No. 51709229; 51879220 and 61803306); Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2018JQ5092), and we are grateful for that. References Bagheri, M.R., Seif, M.S., Mehdigholi, H., Yaakob, O., 2015. Analysis of noise behaviour for marine propellers under cavitating and non-cavitating conditions. Ships Offshore Struct. 12 (1), 1–8. Baltazar, J., Rijpkema, D., Falc~ ao de Campos, J., Bosschers, J., 2018. Prediction of the open-water performance of ducted propellers with a panel method. J. Mar. Sci. Eng. 6 (1), 27. Bass, A.H., Mckibben, J.R., 2003. Neural mechanisms and behaviors for acoustic communication in teleost fish. Prog. Neurobiol. 69 (1), 1–26. Callender, B., Gutmark, E.J., Martens, S., 2005. Far-field acoustic investigation into chevron nozzle mechanisms and trends. AIAA J. 43 (1), 87–95.

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