Wake-adapted ducted propeller for full-scale generic underwater vehicle: parametric study on blade skew for unsteady propeller thrust Kenshiro Takahashi 1 Naval Systems Research Centre, Acquisition, Technology &Logistics Agency, Ministry of Defense 2-2-1 Nakameguro, Meguro-ku, Tokyo, Japan Chris Gargan-Shingles Maritime Division, Defence Science and Technology Group 506 Lorimer St, Port Melbourne, Victoria 3207, Melbourne, Australia

ABSTRACT A parametric study of blade skew for unsteady propeller thrust reduction in the wake of an underwa- ter vehicle (BB2) is presented. Unsteady thrust forces transmit via the propeller shaft and produce undesirable noise and vibration, particularly at frequencies corresponding to multiples of the blade pass frequency. The effectiveness of blade skew in reducing noise and vibration has been reported in previous studies, though not for underwater vehicles. The amplitudes of the dominant harmonics of the unsteady propeller thrust are determined using a Fourier series. Balanced skew designs reduce the dominant fifth harmonic more significantly than biased skew designs at identical total-skew an- gles. The total skew angle strongly influences the unsteady propeller thrust magnitude for balanced skew designs, whereas the effect of the maximum forward-skew angle is negligible. Significant fluc- tuations in single-blade loads are observed when the blade passes through the downstream wake of the vehicle’s sail and casing. Increasing the blade total-skew reduces the blade load fluctuation when the blade approaches the top-dead centre position, but has limited influence on the fluctuations when the blade moves away from this position.

1. INTRODUCTION

Propellers on marine vehicles experience an unsteady thrust force under standard operating condi- tions. These unsteady forces transmit via the propeller shaft and produce undesirable noise and vi- bration, particularly at frequencies corresponding to multiples of the blade pass frequency, known as the blade rate (BR) noise. Reducing the propeller thrust fluctuations should lessen structural vibra- tions and improve hydroacoustic radiation. Hence, this study focuses on the effect of blade skew on the unsteady propeller thrust of a full-scale BB2 underwater vehicle. The effectiveness of blade skew in reducing noise and vibration has been reported in previous studies. Boswell and Miller [1] demon- strated that the components of propeller thrust of three-bladed propellers at BR frequencies in three- and four-cycle wakes were effectively reduced by skewing the propeller blades. Nelka [2] evaluated the unsteady propeller thrusts of skewed five-bladed propeller series and reported that blade skew significantly reduced the blade rate forces and moments in a five-cycle wake. Valentine and Chase [3] presented a skewed propeller that minimised propeller vibration without adversely affecting the efficiency of a 40-177 class fleet oiler. Hammer and McGinn [4] concluded that highly skewed pro- pellers significantly reduced overall ship vibration levels based on full-scale tests for merchant ships.

1 kenshiro.takahashi@defence.gov.au

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Jadmiko et al. [5] numerically investigated the effect of varied skew angles on efficiency and cavita- tion using Wageningen B-series propellers. They demonstrated that increasing the skew angle re- duced the cavitation risk without deteriorating the efficiency.

Additional studies are necessary to analyse the effect of blade skew on the unsteady propeller thrust of underwater vehicles. The wakes of such vehicles are more complicated than those used in previous studies. A non-uniform propeller inflow wake develops due to upstream appendages such as the forward hydroplanes, the sail, the casing, and the aft control surfaces. Additionally, boundary layers and vortices are generated by the appended hull. By contrast, cycle wakes merely impose al- ternating higher and lower velocity regions. In addition, the skewed propellers used in the experi- ments had a simple constant skew rate (biased skew designs), whereas the propellers in underwater vehicles typically have several variables for balanced skew designs. Furthermore, inflow wakes gen- erated by underwater bodies have significant scaling effects, and evaluations of full-scale wakes are necessary to analyse the phenomena. Hence, the present study aims to clarify the effect of blade skew on unsteady propeller thrust in a full-scale wake by systematically varying the skew distributions. Unsteady propeller thrust in the wake of a full-scale BB2 underwater vehicle was computed using computational fluid dynamics (CFD), with the dominant harmonics analysed using Fourier series. In addition to this, load fluctuations of a single blade were computed to analyse the mechanism of un- steady propeller thrust. The nominal wake in the propeller plane was also analysed to correlate the velocity profiles and blade load fluctuations.

2. TEST CASES

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The BB2 underwater vehicle [6] was used as a benchmark. The model comprised a hull, sail, sail planes, casing, and aft control surfaces, as shown in Figure 1. Table 1 lists the principal parameters used for the BB2 underwater vehicle. The design parameters of the ducted propeller, which were developed previously [7], are listed in Table 1. Based on an earlier study [8], the ratio of the duct length to the propeller diameter, 𝐿 ௡ 𝐷 ௉ ⁄ (i.e., 0.8), accounts for the space of the struts to mount the duct on to the hull.

Figure 1: Geometry of BB2 underwater vehicle and ducted propeller. Two skew designs were investigated, i.e., biased and balanced skew, as shown in Figure 2 A and B, respectively. The biased skew distributions were determined based on DTNSRDC propeller 4382 [9], which had a total skew angle 𝜃 ௌ௉ = 36 ° . The balanced skew design was characterised by 𝜃 ௌ௉ , the

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maximum forward-skew angle 𝜃 ௌ௙ [ ° ], the radial location of the maximum forward-skew angle 𝑟 ఏ ೄ೑ [m], and the tip-skew angle 𝜃 ௌ௧ [ ° ], as shown in Figure 2 C.

Table 1: Principal parameters of BB2 underwater vehicle and ducted propeller. - Parameters Full Scale Model Scale

Overall length, 𝐿 ை஺ [m] 70.2 3.826 Beam, 𝐵 [m] 9.6 0.5232 Depth to deck, 𝐷 [m] 10.6 0.5777 Depth to top of sail, 𝐷 ௦௔௜௟ [m] 16.2 0.8829

BB2 underwater vehicle

Number of blades, 𝑍 5 Propeller diameter, 𝐷 ௉ [m] 4.33 0.236 Boss ratio, 𝑑𝐷 ௉ ⁄ 0.215 Pitch ratio, 𝑃𝐷 ௉ ⁄ 1.36 Expanded area ratio, 𝐴 ா 𝐴 ை ⁄ 0.75 Nozzle type No. 19A Duct length to propeller diameter, 𝐿 ௡ 𝐷 ௉ ⁄ 0.8

Ducted propeller

A: Biased skew [10] Figure 2: Biased (A) and balanced

A: Biased skew [10]

B: Balanced skew [10]

C: Skew characterisation Figure 2: Biased (A) and balanced (B) blade skew designs and skew-design characterisation (C).

M #0 M #3 M #6 M #9 M #12 M #15 M #18 M #21

M #1 M #4 M #7 M #10 M #13 M #16 M #19 M #22

M #5 M #8 M #11 M #14 M #17 M #20 M #23 Figure 3: Plan views of propeller blade geometries.

M #2

Table 2: Principal parameters of blade skew distributions. Model # Category Skew 𝜽 𝑺𝑷 [ ° ] 𝜽 𝑺𝒕 [ ° ] 𝜽 𝑺𝒇 [ ° ] 𝒓 𝜽 𝑺𝒇 𝑹 ⁄ 0 Benchmark - 0 - - 1

12 12 - - 2 24 24 - - 3 36 36 - - 4 48 48 - - 5

A Biased

12 7 5 0.3 6 24 19 5 0.3 7 36 31 5 0.3 8 48 43 5 0.3 9

B Balanced

36 31 5 0.4 10 36 31 5 0.5 11 36 26 10 0.3 12 36 26 10 0.4 13 36 26 10 0.5 14 36 21 15 0.3 15 36 21 15 0.4 16

C Balanced

41 36 5 0.3 17 41 36 5 0.4 18 41 36 5 0.5 19 46 36 10 0.3 20 46 36 10 0.4 21 46 36 10 0.5 22 51 36 15 0.3 23 51 36 15 0.4 The distribution of the balanced skew was modelled using a quadratic regression curve expressed as,

D Balanced

𝜃 ௌ = 𝛼∙(𝑟𝑅 ⁄ ) ସ + 𝛽∙(𝑟𝑅 ⁄ ) ଷ + 𝛾∙(𝑟𝑅 ⁄ ) ଶ + 𝛿∙(𝑟𝑅 ⁄ ), (1) where 𝛼 , 𝛽 , 𝛾 , and 𝛿 are coefficients [°], 𝑟 is the radial coordinate [m], and 𝑅 is the propeller radius [m]. The blade rake was adjusted to negate the skew-induced rake and achieve a total rake of zero. Table 2 shows the principal parameters of the skew distributions of the simulated blade skew models. Model #0 had a benchmark skew distribution for 𝜃 ௦௧ = 0 ° , which was used in earlier studies [7][8]. Category A designs, corresponding to Models #1–4, had biased skew distributions for 𝜃 ௌ௧ = 12–48 ° . Category B designs, corresponding to Models #5–8, had a balanced skew with 𝜃 ௦௙ = 5 ° at 𝑟 ఏ ೄ೑ 𝑅 ⁄ = 0.3 for 𝜃 ௌ௧ = 12 ° –48 ° , which enables a direct comparison of results with the designs in category A. In category C, corresponding to Models #9-15, 𝜃 ௌ௙ and 𝑟 ఏ ೄ೑ were systematically varied at a constant total skew angle 𝜃 ௌ௉ = 36 ° . By contrast, 𝜃 ௌ௙ and 𝑟 ఏ ೄ೑ were varied at a constant tip-skew angle 𝜃 ௌ௧ =

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36 ° in category D, corresponding to Models #16-23. Figure 3 shows the plan views of the blade mod- els tested. 3. METHODOLOGY

The CFD package STAR-CCM+2020.2.1 was used for the computations. STAR-CCM+ utilises a finite volume method solver, and the governing equations are the continuity and Reynolds Averaged Navier–Stokes equations. The k–ω shear stress transport turbulence model was employed, and a sec- ond-order scheme was used in both space and time. Figure 4 illustrates the computational domain used in the simulations. An unstructured trimmed mesh was employed, and prism layer meshes were generated on the hull and appendage surfaces to ensure that the non-dimensional initial layer thick- ness 𝑦 ௠௜௡

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ା ≈ 30 with 40 layers at an expansion rate of 1.15. Herein, 𝑦 ௠௜௡

ା was estimated using the equations presented in the literature [11]. A sliding mesh method was used to rotate the propeller blades through the incoming wake. The simulations were performed for the velocity 𝑉 = 10 knots ( 𝑅 ௘ = 2.66 × 10 8 ) at the propeller rotational speed 𝑛 = 1.067 rps for the self-propulsive condition deter- mined in [8], where 𝑅 ௘ = 𝑉∙𝐿 ை஺ 𝜈 ⁄ is the Reynolds number and 𝜈 is the kinematic viscosity [m 2 /s]. A time-step size was chosen to ensure that the propeller rotates 1.0 ° each time step with five inner iterations used to achieve convergence of the solution at each time step. The simulation was run for three complete rotations of the propeller, with time histories for the third rotation used for the har- monic analysis of unsteady propeller thrusts. The computed unsteady propeller thrusts were analysed using the Fourier series presented in [12]:

ଷହ

, (2)

𝑇 ௕ (𝜃) = (𝑇 ௕ ) ேୀ଴ + ෍(𝑇 ௕ ) ே cos[𝑁𝜃−(𝜙 ் ) ே ]

ேୀଵ

where (𝑇 ௕ ) ேୀ଴ is the circumferential average value of the propeller thrust [N], (𝑇) ே is the amplitude of the N th harmonic of the propeller thrust [N], and (𝜙 ் ) ே is the phase angle of the N th harmonic of the propeller thrust [°]. The coefficients were obtained by fitting a curve to the data using the least- squares method.

A: Computational domain B: Volume mesh of the ducted BB2 (36-million-cells) Figure 4: Computational domain (A); volume mesh around underwater vehicle and propeller (B). 5. RESULTS

Figure 5 shows the fifth harmonic of the unsteady propeller thrust for categories A and B (Models #0–8). The amplitude decreased with an increase in the total skew angle 𝜃 ௌ௉ for both the biased and balanced skew designs, and those at 𝜃 ௌ௉ = 48 ° almost halved when compared with that at 𝜃 ௌ௉ = 0 ° . The amplitudes of the fifth harmonic of the unsteady propeller thrust for the balanced skew designs

were generally smaller than those of the biased skew designs, and the differences were more signif- icant at 𝜃 ௌ௉ = 36 ° and 48 ° . This result suggests that the balanced skew design can reduce unsteady propeller thrust more effectively than the biased skew design at an identical total-skew angle. Figures 6 and 7 show the fifth harmonic of the unsteady propeller thrust for categories C and D, respectively. Figure 6 shows that at a constant total-skew angle 𝜃 ௌ௉ of 36 ° (Models #7 and #9–15), little difference was observed in the dominant fifth harmonic (𝑇 ௕ ) ହ , although (𝑇 ௕ ) ହ was slightly higher when the ra- dial location of the maximum forward-skew angle 𝑟 ఏ ೄ೑ was 0.5 𝑅 . By contrast, Figure 7 shows that (𝑇 ௕ ) ହ decreased with increasing 𝜃 ௌ௉ at a constant tip-skew angle 𝜃 ௌ௧ = 36 ° (Models #16–23), with a greater reduction in (𝑇 ௕ ) ହ achieved with smaller values of 𝑟 ఏ ೄ೑ .

Benchmark Biased Balanced

6

Model #0

M #1

M #5

5

M #2

M #6

4

M #3

M #4

3

M #7

M #8

2

1

0

0 12 24 36 48

[ ]

Figure 5: Fifth harmonic of unsteady propeller thrust for test categories A and B (Model #0–8).

= 5 = 1 = 1

4

Model #7 M #9 M #10 M #11 M #12 M #13 M #14 M #15

3

2

1

0

0.4 0.5 0.3 0.4 0.5 0.3 0.4

= 0.3

Figure 6: Fifth harmonic of unsteady propeller thrust for test category C (Model #7, #9–15).

= 4 = 4 = 5

4

M #17 M #18 M #19 M #20 M #21 M #22 M #23

Model #16

3

2

1

0

= 0.3 0.4 0.5 0.3 0.4 0.5 0.3 0.4

Figure 7: Fifth harmonic of unsteady propeller thrust for test category D (Model #16–23).

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The single-blade load was computed to analyse the mechanism of the propeller thrust fluctuations. Figure 8 shows the time histories of the single-blade loads for models #0–4. Two positive peaks and a negative peak were observed when the blade passed downstream of the sail and casing (0 ° ≤𝜃≤ 90 ° and 270° ≤𝜃≤ 360 ° ). The necklace vortices produced at the sail, combined with the wake from the sail and aft control surfaces produce regions of low velocity in the propeller plane wake, causing blade load fluctuations [7]. The amplitudes of unsteady blade load decreased significantly from 270° ≤𝜃≤ 360 ° (from the port side to the top) as the total skew angle increased. The angle ( 𝜃 ) at which the peak blade load occurred shifted towards top-dead centre (360 °) as the total skew angle increased. By contrast, the reduction in the amplitudes was smaller for 0 ° ≤𝜃≤ 90 ° (from the top to the star- board side), shifting further away from top-dead centre as the total skew angle increased. Figure 9 shows the distribution of the axial velocity at the propeller plane (0.99 𝐿 ை஺ ). The axial velocity fluc- tuations shifted forward from the root to the tip when 270° ≤𝜃≤ 360 ° . This forward shift of the troughs and crests is in the opposite direction to the skew, distributing the fluctuating load over a broader angle and thereby reducing the peak level. By contrast, the axial velocity fluctuations shifted backward when 0° ≤𝜃≤ 90 ° , i.e., in the same direction as the blade skew, resulting in a lower re- duction in peak unsteady blade load.

0° (M #0) 12° (M #1) 24° (M #2) 36° (M #3) 48° (M #4)

1.30

(single blade)

1.20

1.10

1.00

0.90

0.80

0 45 90 135 180 225 270 315 360

[ ]

Figure 8: Time histories of single-blade loads for Model #0–4.

Propeller rotational direction

1.1

backward forward

1.0

0.9

0.8

0.7

0.6

0.5

0 45 90 135 180 225 270 315 360 [ ]

Figure 9: Axial velocity distributions at propeller plane for 0.3 ≤𝑟𝑅 ⁄ ≤ 0.9. 5. CONCLUSION

A parametric study on the effect of blade skew on the unsteady propeller thrust of ducted propellers was presented. Blade skew reduced the propeller thrust fluctuations in the wake of a full-scale BB2

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underwater vehicle. Balanced skew designs reduced the dominant fifth harmonic more significantly than biased skew designs at an identical total-skew angle. The total-skew angle strongly influenced the unsteady propeller thrust magnitude for balanced skew designs, whereas the effect of the maxi- mum forward-skew angle was negligible. However, the radial location of the maximum forward skew angle was an important parameter and should be kept small to decrease thrust fluctuations. The blade skew significantly decreased the blade load fluctuation when the blade passed between 270° ≤ 𝜃≤ 360 ° , whereas the reduction in the blade load fluctuations by the blade skew was less significant when 0° ≤𝜃≤9 0 ° . In future studies, the number of blades will be increased to further reduce un- steady propeller thrusts. 5. ACKNOWLEDGEMENTS

The authors wish to express their gratitude to their colleagues in the Defence Science and Technology Group (DSTG) and the University of Tasmania (UTAS) for the support provided to carry out this work. 6. REFERENCES

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water performance, cavitation performance, field-point pressures, and unsteady propeller loading. R&D Report No. 4113 , 1974. 3. Valentine, D. T. & Chase, A. Highly skewed propeller design for a naval auxiliary oiler (AO 177).

SPD-544-12 , 1976. 4. Hammer, N. O. & McGinn, R. F. Highly skewed propellers—full scale vibration test results and

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variation skew angle B-series propeller on performance and cavitation. International Journal of Mechanical Engineering and Technology , 10(5) , 219–234 (2019). 6. Overpelt, B., Nienhuis, B. & Anderson, B. Free running manoeuvring model tests on a modern

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