Numerical study on scale effect of nominal wake of single screw ship

Ocean Engineering 104 (2015) 437–451

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Numerical study on scale effect of nominal wake of single screw ship Zhan-Zhi Wang a, Ying Xiong a,n, Rui Wang a, Xing-Rong Shen b, Chen-Hua Zhong a a b

Department of Naval Architecture, Naval University of Engineering, Wuhan, China Marine Design and Research Institute of China, Shanghai, China

art ic l e i nf o

a b s t r a c t

Article history: Received 26 May 2014 Accepted 20 May 2015 Available online 12 June 2015

A 4000TEU container ship was studied without considering free surface effect, and the viscous flow fields of ship at different scales were solved numerically by RANS method, numerical uncertainty analysis was employed according to factors of safety method for Richardson extrapolation, scale effect of axial nominal wake field was analyzed in details. It shows that the reciprocal of mean axial wake fraction of propeller disc exhibits a near linear dependence on Reynolds number in logarithmic scale. For the single screw ship without bilge vortex, linear function is fit perfectly for the relationship between the reciprocal of mean axial wake fraction at each radius, reciprocal of amplitude of wake peak right above propeller disc and Reynolds number in logarithmic scales. In inner area of propeller disc, the reciprocal of amplitude of wake valley and wake peak right down propeller disc reveal nearly linear dependence on Reynolds number in logarithmic scales. While in outer area, the amplitude of wake peak and valley decline rapidly to potential wake fraction, and the wake width reveals negative exponent power function dependence on Reynolds number in logarithmic scales. On this basis, an extrapolated wake field scaling method is proposed. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Nominal wake field Scale effect Single screw ship RANS

1. Introduction Wake is the flow field caused by the relative motion between a hull and the incident flow. Nominal wake is the wake field at propeller disc without the presence of the propeller. According to the wake causation (see Sheng and Liu 2004), nominal wake can be expressed as

ω ¼ ωp þ ωf þ ωw

ð1Þ

where, ωp is the potential wake field, which would arise if the hull were in an ideal fluid without viscous effect. ωf is the frictional wake field which arises from the viscous nature of the fluid passing over a hull. ωw is the wave wake field which is caused by the movement of water particles in the gravity waves set up by the hull. Since the model tests are usually carried out at equal Froude numbers between model and full scale ship (the potential wake fraction can considered to be equal between model and full scale ship), a disparity in Reynolds number exists. The Reynolds number for model tests are typically in the range 106–107, while full scale ships are mainly working at Reynolds number of 109. Increasing Reynolds number leads to a relative reduction of the boundary layer thickness and an altering of the velocity profile in the near n

Corresponding author. Tel.: þ 86 15807182543; fax: þ86 02765461615. E-mail address: [email protected] (Y. Xiong).

http://dx.doi.org/10.1016/j.oceaneng.2015.05.029 0029-8018/& 2015 Elsevier Ltd. All rights reserved.

wall region, arousing the difference of wake field between the model and full scale ship. This phenomenon is called scale effect of wake field. At past, scale effect of wake field was not obvious due to the shorter and thinner ship form. But with the appearance of high block coefficient ship and super tanker, scale effect of wake field makes a great difference in cavitation, excitation force, noise and propulsive performance between model and full scale ship. It often happens that model test does not have cavitation and vibration in the cavitation tunnel according to geometric similarity and equal cavitation number, but full scale ship happens seriously, or model test happens but full scale ship does not, it is mainly caused by the great difference of wake field between model and full scale ship (see He and Wang 1987). From the propeller design point of view, nominal wake field determines the propeller operating environment. Mean wake fraction along with other parameters determine the overall design dimensions of the propeller, while the wake field influences the propeller blade section design and pitch. It is very important and necessary to consider scale effect on nominal wake field, all previous ITTC meetings are very concerned about this issue. At present, many wake field scaling methods can be utilized. Sasajima and Tanaka (1966) considered the wake divided into two components, potential wake and frictional wake, ignoring the wave wake. Potential wake was determined by towing the model astern. Potential wake was supposed to be independent of scale but the frictional wake suffered scale effect and should be contracted toward the center plane. Dyne (1974) determined scale

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Nomendature

ωp [dimensionless] ωf [dimensionless] ωp [dimensionless] ωx [dimensionless]

potential wake fraction frictional wake fraction wave wake fraction Axial wake fraction LWL [m] length of waterline T [m] daft Δ [m3] Displacement

effect on nominal wake consists in carrying out wake measurements on a geosim series of models covering a large range of Reynolds numbers and studying the variation of the wake with ship frictional coefficient. In Dyne's method, the potential wake was extrapolated from the local wake at different serving speed, and the full scale ship wake was obtained through mean wake using the least square method. Hoekstra (1975) based on the assumption that the wake components, potential wake, frictional wake and wave wake were not independent of each other. Scale effect between model and full scale ship not only affected the frictional wake but also the total wake. The total contraction of nominal wake was threefold, namely contraction to the center plane, contraction to the hull above the propeller and a concentric contraction around the propeller axis. Unfortunately the full scale ship wakes with this method from the results of the geosim models differed largely from one another. Tanaka (1979) considered scale effect by utilizing the interpolation of the flow characteristics between upstream and downstream propeller disc, ignoring the wave wake. It was believed that the hull boundary layer could be considered by the flow field upstream propeller disc. This method was easy to apply but the wake prediction still depends upon the model scale. Garcia-Gomez (1989) developed a new method, based on Tanaka's method but introducing the dependency of the form factor on the model scale and its repercussion on the friction wake component. Garcia's method could avoid or at least mitigate the dependency of the wake extrapolation on the model scale and obtain a unique ship prediction no matter which model was used. Nagamatsu (1979) presented a method to predict ship wake from measured model wake by applying the theory of two-dimensional turbulent wake. Tanaka (1983) extended the study on the wake scale effect of ships with bilge vortices. It is assumed that the flow consists of the ordinary wake portion without bilge vortices and the vortex wake. The scaling method is proposed by investigating the Reynolds number effects on the location of the vortex center, circulation, velocity and vortices distributions. It is possible to apply numerical simulation of full scale nominal wake with the rapid development of CFD software and hardware, but it cannot be validated due to the lack of full scale nominal wake data (Choi, et al., 2009, 2010; Eca and Hoekstra, 2009; Min and Kang, 2010; Castro et al., 2011). CFD method provides a new way to study scale effect of nominal wake field. RANS method is dominant in numerical simulation of wake fields. Turbulence model is the closure equation which combines the fluctuating and time average in RANS method. Larsson et al. (2003) pointed out that turbulence model is crucial for the quality of the wake prediction. Visonneau (2005) drew the conclusion that turbulence model plays the decisive role in the proper modeling of some wake features such as the wake deficit and so-called “hooklike” flow patterns. Tokyo 2005 and Gothenburg 2010 workshops on Numerical Ship Hydrodynamics which represent the highest level of CFD simulation gave the same conclusion. Tahara et al. (2003) presented the numerical simulation of viscous flow field at full scale Reynolds number taking the surface

Sw [m2] wetted surface λ [dimensionless] scale factor Re [dimensionless] Reynolds number y þ [dimensionless] nondimensional spaing V0 [m/s] the inflow velocity Vx [m/s] axial velocity r h [m] the most inner radius of measuring point R [m] propeller radius

roughness effect into consideration, and a comparison was made with the extrapolated wake by Tanaka method. Schweighofer et al. (2005) carried out the numerical simulation of viscous flow around two existing vessels at model and full scale Reynolds numbers with and without acting propeller. The free surface was taken into account. The boundary layer was either resolved till the wall or wall functions were used. Verkuyl et al. (2006) presented an overview of the EFFORT project results and shows the numerical developments that have been implemented by six CFD European groups. It also discussed the full scale LDV measurements and the model scale experiments that are carried out on two vessels. He also showed some of the validation results based on these experiments. Starke et al. (2006) calculated and validated nominal wake of a ship in EFFORT project (European Full-scale Flow Research and Technology) using the RANS method. It was shown that the numerical results showed good correlation with experimental data measured during sea trials with ship mounted LDV systems, the longitudinal vortices, well-known at model scale, occurred at full scale as well, the turbulence model which performed well at model scale was also found to perform well at full scale. Visonneau and Deng (2006) investigated the model and full scale flow field of two appended hull in EFFORT project using unstructured grids composed exclusively of hexahedrons with local refinement. It was found that the thickness of boundary layer declined rapidly in full scale ship, the vortex which was convicted between the V brackets at model scale had somewhat disappeared at full scale and the longitudinal velocity component seemed to be more homogeneous below the propeller shaft at full scale. At model scale, the strong longitudinal vortex emanating from the junction between the propeller shaft and the hull which was clearly visible, while there did not show any longitudinal vortex near the propeller shaft at full scale. Wang (2010) performed numerical simulation of the viscous wake field and predicted the full scale ship resistance of KVLCC2. The influences of grid topology and turbulence model were also investigated for a better prediction of wake field. The method developed for the KVLCC2 was also used for a VLCC oil tanker, and a comparison between a double-model and free surface was made. Compared with the experimental data, it is found that structured grids are better than unstructured ones and the RSM model is the best turbulence model to capture wake characteristics such as the “hook-like” pattern, and free surface effect has little influence on nominal wake field. Unfortunately, the dominant above researches have just carried out two kinds of numerical simulation, one is in model scale and the other in full scale, and analyzed the difference of flow field between model scale and full scale, but do not perform a comprehensive study of flow field ranging a broad Reynolds number and propose a new extrapolated scaling method of wake field. The Specialist Committee on Scaling of Wake Field of 26th ITTC (2011) performed the comparisons between full scale CFD results and extrapolation full scale wake field from Mehlhorn method, Hoekstra method, Sasajima method, Tanaka method and Garcia

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Gomez method, using the same input experimental data of HTC form (container vessel), KCS form (container vessel) and ITU form (tanker). It was found that none of the extrapolation methods gave perfect agreement with the full scale CFD results. Some methods (Mehlhorn, Hoekstra) gave somewhat unrealistic wake pattern. Some methods tended to underestimate the wake peak (Tanaka, Garcia Gomez). Sasajima method tended to give the best approximation of the upper part of the wake field in terms of the wake deficit depth and width. For the bottom part of wake, the discrepancies were large but these did not influence the cavitation performance in major way. It was believed that, up to the present, the best prediction of full scale nominal wake field could be obtained using high resolution CFD calculations. A 4000TEU container ship was studied without considering free surface effect, and the viscous flow fields of ship in different scales including full scale were solved numerically by RANS method and SST k  ω turbulence model, scale effect of nominal wake was analyzed in detailed. On this basis, an extrapolated scaling method of wake field is proposed from model scale to full scale ship.

439

Fig. 1. The outline of 4000TEU container ship.

Table 2 Numerical simulation parameters of 4000TEU container ship in different scales λ

LPP

V

Re

yþ

Cell number

63 32 16 8 4 2 1

4.1951 8.2591 16.5181 33.0363 66.0725 132.1450 264.2900

1.6202 2.2733 3.2150 4.5467 6.4300 9.0934 12.8600

6,763,017 18,682,128 52,841,037 149,457,023 422,728,298 1,195,656,184 3,381,826,382

120 150 200 300 300 300 300

3,347,180 4,201,380 5,465,070 7,963,598 11,796,872 22,497,728 42,408,214

2. Numerical model 2.1. Geometry

Fig. 2. Grid of 4000TEU container ship.

A 4000TEU container ship driven by a single screw is studied. Marine Design and Research Institute of China has made detailed experimental research on the model with the scale factor of 63.The main characteristics are summarized in Table 1 and the hull's outline is shown in Fig. 1, where λ is the scale factor, LWL is the length of waterline, T is daft, Δ is the displacement, and Sw is wetted surface. In order to study scale effect of nominal wake, the flow fields of a series of scale models were solved numerically without considering the free surface effect, sinkage and trim, the Froude number is 0.2525, and the corresponding speed is 25 kn. The numerical simulation parameters in different scales are shown in Table 2, where Re is Reynolds number, y þ is the nondimensional spacing near wall.

inlet boundary condition. The outflow boundary, with 2.5 times the hull length downstream distance from the stern of the model, is set as the undisturbed pressure-outlet boundary condition. The outer boundary, with one hull length distance from the central axis of the model, is set as the undisturbed velocity boundary condition. The hull surface is set as the no-slip boundary condition. The turbulence model is SST k  ω model. The governing equations and turbulence model are discretized by finite volume method with a second order upwind and the pressure-velocity coupling is SIMPLEC method. Numerical simulations are realized by ANSYS FLUENT14.0 code, and full scale computation is taken parallel processing in 64 cores (Intel Xeon E52670, 2.6 GHz) of Dawning TC4600 high performance computer.

3. Results and discussion 2.2. Grid system and boundary condition The multi-block structured grids are constructed to discrete the computational domain. The grid, of HO topology, is refined longitudinally towards bow and stern to resolve the large velocity gradients. Grid topology of each scale model is completely the same, except the absolute near-wall spacing in which the wall function can be employed. The grids are also refined in three directions according to the increase of scale, ensuring that the aspect ratios of numerical cells are smaller than 300 at the hull middle and 100 at the bow. The computational grid of bare hull is shown in Fig. 2. The right-handed global coordinate system (x, y, z) is defined as positive x opposite to the flow direction, positive y larboard, and positive z upward. The inflow boundary, with one hull length upstream distance from the bow of model, is set as the velocityTable 1 Principal particular of 4000TEU container ship Parameter

Full scale

Model scale

λ LWL (m) T(m) Δ (m3) Sw (m2)

1 264.29 12 64,387.1 11,355.31

63 4.195 0.19 0.2575 2.861

3.1. Comparisons between numerical results and experimental data in model scale In order to investigate effect of free surface on nominal wake field, comparative simulations of ‘double-body flow’ and free surface flow are carried out firstly, the grids are similar and the Volume-of-Fluid (VOF) method is used for the free surface modeling. Comparisons of total resistance at λ ¼63 between numerical results and experimental data are shown in Fig. 3. It can be seen from Fig. 3 that numerical results agree quite well with experimental data. The maximum error is below 4%. Comparisons of nondimensional axial velocity on propeller disc at λ ¼63 between experimental data, numerical results with and without free surface are shown in Fig. 4. In Fig. 4, EFD represents experimental data, 01 is the angular position of 12 o’clock and positive direction is clockwise from the stern to bow. It can be seen from Fig. 4 that nondimensional axial velocity agrees well with experimental data, especially in the inner radius. The maximum error of velocity peak amplitude is less than 10%. Experimental data can hardly keep symmetry due to the circumferential measurements on the whole propeller disc, which is clearly reflected in Fig. 4 (r/R¼0.3–0.7). While numerical results are completely symmetry to the centerplane because a half of the

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as SGCi ,SGMi ,and SGFi ði ¼ 1; …; 37Þ respectively. The Grid changes ε for coarse-medium and medium-fine grids at every circumferential angle are defined by

εGMCi ¼ SGMi  SGCi ði ¼ 1; …; 37Þ

ð2Þ

εGFMi ¼ SGFi  SGMi ði ¼ 1; …; 37Þ

ð3Þ

For point variables, εGMCi and εGFMi should be dealt with based qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pi ¼ 91 2 on the L2 norm, i.e., j j εG j j 2 ¼ i ¼ 1 εG . So the grid converge ratio hRG i is calculated by hRG i ¼ ‖εGFM ‖2 =‖εGMC ‖2

ð4Þ

The estimated order of accuracy hP G iRE is calculated by   hP G iRE ¼ ln ‖εGMC ‖2 =‖εGFM ‖2 =lnðrÞ

ð5Þ

The distance metric to the asymptotic range hP G i is defined as hP G i ¼ hP G iRE =hP G ith Fig. 3. Comparison experimental data.

of

total

resistance

between

numerical

result

and

model is calculated in the simulation. The dominant difference between numerical results with and without free surface is less than 4%.That is to say, free surface effect is relatively small on nominal wake field at propeller disc, this conclusion is similar to reference (Wang, 2010). Comparison of axial nominal wake fraction contour at λ ¼ 63 between numerical result without free surface and experimental data is shown in Fig. 5, that of the tangential and radial flow vectors are shown in Fig. 6. In Figs. 5 and 6, the experimental result is in the left part and the numerical result is in the right part, the red dashed line shows propeller disc position with 1.2 times of propeller diameter. It can be seen from Figs. 5 and 6 that there is no bilge vortex in this ship. Numerical results agree well with the experimental data, which captured the “V” pattern of the transverse flow in single screw ship well without bilge vortex. In a word, the grid topology and numerical method are suitable for nominal wake study of this kind of ship, and the double model calculation without considering free surface effect are performed for scale effect study of wake field. 3.2. Uncertainty analysis of numerical results In numerical simulations on viscous flow field of 4000TEU container ship in different scales, uncertainty analysis must be carried out for the solution and computational grid. Uncertainty analysis consists of verification and validation. Verification is defined as a process for assessing numerical simulation uncertainty, U SN . Validation is defined as a process for assessing modeling uncertainty U SM by using benchmark experimental data. RANS method is a steady simulation method, thus the simulation uncertainty U SN is composed of grid uncertainty U G , iterative uncertainty U I , that is U 2SN ¼ U 2G þ U 2I . Uncertainty analysis is employed for nondimensional axial velocity at scale factor λ ¼63, 8, 1. According to factors of safety method for Richard extrapolation (Xing and Stern, 2010), three sets of computational grids (i.e., fine, medium, and coarse grids) are built for the numerical simulation of viscous flow field. The refinement ratio,r G , is 1.2 in each direction of the coordinate. Fig. 7 shows the calculated nondimensional axial velocity for three grids at r/R ¼0.5 and λ ¼63, the numerical uncertainty analysis procedure is illustrated below taking r/R¼0.5 for example. The calculated values of nondimensional axial velocity from θ ¼01 to θ ¼ 3601 are drawn out at intervals of 101. The values calculated by coarse grids, medium grids and fine grids are noted

ð6Þ

where, hP G ith is the theoretical order of accuracy. The grid uncertainty at every circumferential angle hU Gi i is calculated by   8   > > ð2:45  0:85hP G iÞ εGFMi ; 0 o hP G i r1 > rhPG iRE  1 <   hU Gi i ¼ ð7Þ  ε  > >  GFMi ; hP G i 4 1 > : ð16:4hP G i  14:8Þ hP i  r

G RE

1

The global uncertainty,‖U G ‖2 , at each radius is dealt with hU Gi i based on the L2 norm. The assessment of iterative uncertainty U I was carried out by observing the oscillatory amplitude of the calculated value, as calculation time increases after iteration becomes convergent. According to the work of Zhang et al. (2008), iterative uncertainty is smaller than the grid uncertainty by 2 orders of magnitude, and the values can be neglected. Thus, the numerical simulation uncertainty,U SN , was equal to grid uncertainty, U G . The parameters of the uncertainty verification and validation about nondimensional axial velocity on propeller disc at scale factor λ ¼63, 8, and 1 are shown in Tables 3, 4, and 5 respectively. As shown in Tables 3–5, the grid converge ratios hRG i at each radius is less than 1, which suggests that the computational grids exhibit monotonic convergence. The circumferential distributions of numerical simulation uncertainty,U SNi ,of nondimensional axial velocity at λ ¼63, 8, and 1 are shown in Figs. 8–10. Because of the lack of uncertainty for the velocity test at model scale and absence of full scale wake field experiment, the validation process cannot be done. 3.3. Scale effect of the mean axial wake fraction of propeller disc The mean axial wake fraction of propeller disc determines the propeller design and ship-speed performance prediction. The volumeintegration is used to calculate the mean axial wake fraction after obtaining each measuring point value at different radius and angular position. It can be expressed as: R 2π R R ωx ðr; θÞr dr dθ ωx ¼ 0 R r2hπ R R ð8Þ 0 r h r dr dθ where, r h is the most inner radius, R is the propeller radius. The mean axial wake fractions of propeller disc in different scales are summarized in Table 6. The variation of mean axial wake fractions of propeller disc with Reynolds number is shown in Fig. 11. It can be seen from Fig. 11 that the mean axial wake faction decreases with the increase of scale and Reynolds number, and a nonlinear relationship is found between them. The curve's slope

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441

Fig. 4. Comparisons of the nondimensional axial velocity at each radius between numerical results and experimental data.

also decreases with the increase of Reynolds number. The mean axial wake in full scale ship is 55.5% of that of the model at λ ¼63. The variation of reciprocal of mean axial wake fraction with Reynolds number is shown in Fig. 12.

It can be seen from Fig. 12 that there is nearly a linear relationship between the reciprocal of mean axial wake fraction of propeller disc and Reynolds number in logarithmic scale.

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Fig. 5. Comparison of axial nominal wake fraction contour between numerical result and experimental data.

Fig. 6. Comparison of tangential and radial flow vectors between numerical result and experimental data

The regressive analysis is carried out using linear formula, and it is expressed as 1

ω

¼ alog 10 ðReÞ þ b

ð9Þ

The parameters, a, b, are estimated using a weighed least square method. For this ship, a ¼1.09036, b¼  3.67584, so the linear formula is as 1

ω

¼ 1:09036log 10 ðReÞ−3:67584

ð10Þ

Comparison of mean axial wake fraction of propeller disc between original numerical results and regressive data is shown in Fig. 13, where the solid dot is the original numerical result and the open dot is regressive data. Fig. 13 shows that the regressive data agree excellent well with the original numerical result.

3.4. Scale effect of axial nominal wake field Variation of the axial nominal wake field with Reynolds number is further analyzed. The nondimensional axial velocity contours on propeller plane in different scales are shown in Fig. 14. It can be seen from Fig. 14 that there are two wake peaks for this kind of ship without bilge vortex, one is right above propeller disc corresponding to the angular position of 12o’clock and the other is right down propeller disc corresponding to the angular position of 6 o’clock, where large velocity gradient exits. The boundary layer thickness becomes thinner and the axial velocity contour contracts toward the center plane with the increase of Reynolds number. The circumferential distribution of the axial wake fraction on propeller disc in different scales is shown in Fig. 15, where 01 is the angular position of 12 o’clock and positive direction is clockwise from the stern to bow. The wake distribution is periodically extended for a clear display of the wake pattern.

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Fig. 7. Calculated results for three grids at r/R ¼0.5 and λ ¼ 63.

443

Fig. 8. Circumferential distributions of numerical simulation uncertainty at λ ¼63.

Table 3 Parameters of the uncertainty verification at λ¼ 63 r/R

hRG i

hP G i

‖U SN ‖2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.6606 0.6758 0.6932 0.8097 0.7706 0.6127 0.5702 0.5254

1.1372 1.0745 1.0048 0.5790 0.7146 1.3435 1.5407 1.7650

0.1245 0.1103 0.0669 0.0540 0.0528 0.0735 0.0714 0.0945

Table 4 Parameters of the uncertainty verification at λ¼ 8 r/R

hRG i

hP G i

‖U SN ‖2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.6869 0.6702 0.5717 0.7172 0.6031 0.5173 0.5148 0.3499

1.0301 1.0976 1.5333 0.9116 1.3869 1.8076 1.8209 2.8795

0.0416 0.0464 0.0446 0.0175 0.0498 0.0365 0.0383 0.0292

Fig. 9. Circumferential distributions of numerical simulation uncertainty at λ¼ 8.

Table 5 Parameters of the uncertainty verification at λ¼ 1 r/R

hRG i

hP G i

‖U SN ‖2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.6459 0.6126 0.5964 0.5589 0.7401 0.7069 0.5439 0.4947

1.1989 1.3437 1.4176 1.5954 0.8253 0.9511 1.6703 1.9300

0.0389 0.0328 0.0335 0.0228 0.0094 0.0086 0.0283 0.0315

It can be seen from Fig. 15 that circumferential distribution of axial wake fraction on propeller disc shows “W” pattern for the single screw ship without bilge vortex. There are two wake peaks during a circle, one is in angular position of 01and the other is in angular position of 1801, which are caused by the stern wake field. Different from the twin screw ship, the corresponding phase angle does not change with the variation of Reynolds number. The axial wake field suffers great scale effect especially in the medium and

Fig. 10. Circumferential distributions of numerical simulation uncertainty at λ ¼1.

inner radius area, and upper part of the outer radius area of propeller disc. The nonuniformity of wake field makes the advance angle and resultant velocity flowing to the propeller blade fluctuant, resulting in cavitation and vibration. It is found that the fluctuation develops most strongly when the propeller blade passes to the

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Table 6 The mean axial wake fractions of propeller disc in different scales λ

V

Re

ωx

63 32 16 8 4 2 1

1.620 2.273 3.215 4.547 6.430 9.093 12.860

6,764,372.384 18,685,871.78 52,851,626.61 149,486,974.3 422,813,012.8 1,195,895,794 3,382,504,103

0.27174 0.23497 0.20823 0.18847 0.17297 0.16086 0.15080

Fig. 13. Comparison of the mean axial wake fraction of propeller disc between original numerical results and regressive data

Fig. 11. Variation of mean axial wake fractions of propeller disc with Reynolds number.

Fig. 12. Variation of the reciprocal of mean axial wake fraction with Reynolds number.

wake peak. In this paper, we focus on the variation of characteristic quantity of wake pattern with Reynolds number. Fig. 16 shows the axial wake pattern at a characteristic radius of a single screw ship, where the solid line is the actual distribution and the dash line is the theoretical analysis. 2b is the width of wake peak, 2e is the width of wake waist, 2 m is the width of mean wake, ωh is the amplitude of wake peak, ωl is the amplitude of wake valley, ωm is the mean wake at a radius. When the propeller blade passes to the wake peak, it is just like that the airfoil encounters a sine gust. ωh =b  a is the wake rate, corresponding to the cavitation collapsing strength which is the determining factor for the excitation force. 2e influences the

cavitation inception and ωh influences the cavitation area of propeller blade. Here, we name the wake peak at the angular position of 01wake peak 1 and that of 1801 wake peak 2. The amplitude of wake peaks 1 and 2 in different scales is summarized in Tables 7 and 8, respectively. The mean axial wake fraction at each radius is summarized in Table 9. The variation of the amplitude of wake peaks 1 and 2 with Reynolds number is shown in Figs. 17 and 18, respectively, and the variation of mean axial wake fraction at each radius with Reynolds number is shown in Fig. 19. It can be seen from Fig. 17–19, the amplitude of wake peak 2 corresponding to the angular position of 1801 declines rapidly with the increase of radius, it almost vanishes at r/R¼1.0, while the amplitude of wake peak 1 corresponding to the angular position of 01 declines slightly. At the same radius, the amplitude of wake peak decreases with the increase of Reynolds number. The curve's slope does not vary obviously for wake peak 1, but it decreases gradually to 0 with the increase of Reynolds number for wake peak 2. It can be explained that wake peak 2 is located right down propeller disc, the thickness of stern wake field becomes thinner and thinner with the increase of Reynolds number, so it gradually deviates from the stern wake field. The mean axial wake fraction at each radius also decreases with the increase of Reynolds number, the mean wake curve's slope declines too, a nonlinear relationship is found between it and Reynolds number in logarithmic scales. The variations of reciprocal of the amplitude of wake peak, the mean axial wake at each radius are shown from Figs. 20–22. It can be seen from Fig. 20 to 22 that a nearly linear relationship is found between the reciprocal of the amplitude of wake peak 1, the mean axial wake fraction at each radius and Reynolds number in logarithmic scales respectively. For wake peak 2, the linear relationship still works well in medium and inner radius area of propeller disc (0:3 r r=R r 0:7), while it does not satisfy and keeps constant when Reynolds number reaches to a critical value in the outer radius area (0:7 or=R). The amplitude of wake valley in different scales is summarized in Table 10, and the variation of amplitude of wake valley with Reynolds number is shown in Fig. 23 It can be seen from Fig. 23 that the amplitude of wake valley at each radius decreases with the increase of Reynolds number, and a nonlinear relationship is also found between them. The curve's slope gradually decreases to 0 with the increase of Reynolds number. In the outer radius area of propeller disc (0:7 r r=R), the

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445

Fig. 14. Nondimensional axial velocity contours on propeller plane in different scales.

amplitude of wake valley almost keeps constant when Reynolds number reaches to a critical value (about 108), but it varies much in the medium and inner radius area (0:3 r r=R r0:6). The variation of reciprocal of amplitude of wake valley with Reynolds number is shown in Fig. 24.

It can be seen from Fig. 24 that the reciprocal of amplitude of axial wake valley almost keeps constant when Reynolds number reaches to a critical value in medium and outer radius area (0:6 rr=R), and the critical Reynolds number decreases with the increase of radius. The linear relationship also works well when

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Fig. 15. Circumferential distribution of the axial wake fraction on propeller disc in different scales.

Reynolds number is lower than the critical value. In the inner radius area (0:3 r r=Rr 0:5), it is always the linear relationship between them.

In Fig. 15, two points can be obtained when drawing the contour line ωx ¼ ωm which respects to the mean axial wake fraction at each radius at λ ¼63, respectively. The width called 2m

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is the corresponding angle of the mean axial wake fraction at λ ¼63. The widths at some typical radius are summarized in Table 11, where the width is in radian. The variation of width with Reynolds number is shown in Fig. 25. It can be seen from Fig. 25 that the width gradually declines with the increase of Reynolds number, a nonlinear relationship is found between them, and the curve's slope also declines. The regressive analysis is carried out using the negative exponent power function on the data in Table 8. The function defined as 2m ¼ alog 10 ðReÞ  b

(3)

(4)

ð11Þ

The parameters,a and b, are estimated using a weighed least square method. At r=R ¼ 0:5 : 2 m ¼ 2186 log 10 ðReÞ  3:64 ; At r=R ¼ 0:7 : 2 m ¼ 343 log 10 ðReÞ  2:64 ; At r=R ¼ 0:9 : 2 m ¼ 135 log 10 ðReÞ  2:09 : The comparison of width at typical radius between original numerical results and regressive data is shown in Fig. 26, where the solid dot is the original numerical result and the open dot is regressive data. It can be seen from Fig. 26 that the regressive data agree excellently well with the original numerical results. On the basis, a wake field scaling method from model scale to full scale is proposed for the single screw ship without bilge vortex using experimental data combined with the RANS method. The extrapolation method is expressed as follows: (1) Conduct nominal wake field measurement experiment at the model scale. (2) Carrying out the numerical simulation at different scales in order to achieve a large range of Reynolds numbers. Calculate the characteristic quantity of the circumferential distribution

(5)

(6)

447

of the axial wake fraction at each radius, for instance, the mean axial wake at each radius, the amplitudes of axial wake peak 1, peak 2, axial wake valley, and the width corresponding to the mean axial wake at model scale. Calculate the circumferential distribution of axial potential wake fraction using potential theory, so as to define the trend of regression. Obtain the characteristic quantity using the extrapolation method and it is divided into three conditions: (a) Adopt linear function (9) to conduct the regressive analysis on the mean axial wake at each radius, reciprocal of the amplitude of wake peak 1 with Reynolds number in logarithmic scales, and use weighed least square method to estimate the undetermined coefficient. (b) Adopt linear function (9) to conduct the regressive analysis on reciprocal of the amplitude of wake peak 2 and wake valley in the inner radius area with Reynolds number in logarithmic scales, and use weighed least square method to estimate the undetermined coefficient. Introduce the potential wake fraction to replace the axial wake fraction in the medium and outer radius area at full scale ship. (c) Adopt negative exponent power function (11) to conduct the regressive analysis on the width corresponding to the mean axial wake fraction at model scale with Reynolds number in logarithmic scales, and use the weighed least square method to estimate the undetermined coefficient. Calculate the characteristic quantity of the circumferential distribution of axial wake fraction at each radius in full scale ship using the regressive formula and experimental wake field data in model scale. According to the characteristic quantity in full scale ship, obtain the full scale ship wake field by smoothing the circumferential distribution of the axial wake fraction.

Comparisons of characteristic quantity of nominal wake at r/R¼0.5, r/R¼ 0.7, r/R¼0.9 between extrapolated full scale results adopting the above procedure from λ ¼ 63, λ ¼32 and numerical results at full scale ship are summarized in Tables 12–14 respectively. It can be seen from Table 12–14 that the characteristic quantities of extrapolated full scale results agree quite well with the numerical results at full scale ship, and the differences of extrapolated results from different model scale are relatively small. To evaluate the reliability of the procedure above, several similar ships should be collected and verified. Unfortunately the validating studies of other hulls have not carried out because of the lack of similar ship with extensive experimental data at model scale.

4. Conclusions

Fig. 16. Axial wake pattern at a characteristic radius of a single screw ship.

A 4000TEU container ship was studied without considering free surface effect, and the viscous flow fields of ship in different

Table 7 Amplitude of wake peak 1 at each radius in different scales λ

log10(Re)

r/R ¼0.3

r/R ¼0.4

r/R ¼0.5

r/R ¼0.6

r/R ¼0.7

r/R ¼0.8

r/R ¼0.9

r/R ¼1.0

63 32 16 8 4 2 1

6.830 7.272 7.723 8.175 8.626 9.078 9.529

0.63312 0.62309 0.61267 0.60317 0.59785 0.58776 0.57648

0.60617 0.59258 0.57796 0.56448 0.55300 0.54025 0.52738

0.58864 0.57268 0.55558 0.53990 0.52602 0.51347 0.49898

0.57596 0.55769 0.53899 0.52240 0.50739 0.49623 0.48116

0.56653 0.54661 0.52679 0.50956 0.49384 0.48299 0.46918

0.55985 0.53867 0.51783 0.49987 0.48386 0.47235 0.46012

0.55633 0.53371 0.51205 0.49320 0.47655 0.46529 0.45312

0.55616 0.53228 0.50974 0.48996 0.47174 0.46127 0.44824

448

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Table 8 Amplitude of wake peak 2 at each radius in different scales λ

log10(Re)

r/R ¼0.3

r/R ¼0.4

r/R¼0.5

r/R¼ 0.6

r/R¼ 0.7

r/R ¼0.8

r/R¼ 0.9

r/R¼1.0

63 32 16 8 4 2 1

6.830 7.272 7.723 8.175 8.626 9.078 9.529

0.71067 0.62187 0.56252 0.51250 0.45972 0.43279 0.41269

0.62069 0.53527 0.48650 0.44217 0.39815 0.37133 0.35586

0.52733 0.45879 0.41884 0.37921 0.34402 0.31704 0.30253

0.43461 0.38318 0.34913 0.31417 0.28689 0.26323 0.24805

0.33446 0.29186 0.25639 0.22473 0.20535 0.18651 0.17531

0.22597 0.18552 0.14726 0.12229 0.11269 0.09617 0.08829

0.13130 0.09969 0.07734 0.06638 0.06491 0.06262 0.06336

0.07429 0.05924 0.05536 0.05617 0.05725 0.05891 0.06018

Table 9 Mean axial wake fraction at each radius in different scales λ

log10(Re)

r/R¼0.3

r/R¼ 0.4

r/R¼ 0.5

r/R¼0.6

r/R¼ 0.7

r/R¼ 0.8

r/R¼ 0.9

r/R ¼1.0

63 32 16 8 4 2 1

6.830 7.272 7.723 8.175 8.626 9.078 9.529

0.56423 0.50540 0.44971 0.40070 0.36077 0.32763 0.29974

0.46191 0.38950 0.33742 0.29903 0.26940 0.24548 0.22502

0.36396 0.30621 0.26800 0.23908 0.21549 0.19665 0.18148

0.29571 0.25352 0.22304 0.19933 0.18068 0.16700 0.15627

0.24936 0.21582 0.19023 0.17173 0.15836 0.14846 0.14042

0.21596 0.18806 0.16744 0.15367 0.14328 0.13514 0.12821

0.19157 0.16903 0.15322 0.14229 0.13342 0.12640 0.12036

0.17488 0.15726 0.14440 0.13493 0.12689 0.12052 0.11496

Fig. 17. Variation of the amplitude of wake peak 1with Reynolds number.

Fig. 18. Variation of the amplitude of wake peak 2 with Reynolds number.

scales including full scale were solved numerically in RANS method and SST k  ω turbulence model, scale effect of nominal wake was analyzed in detailed. The research shows that: (1) For the single screw ship without bilge vortex, the “w” pattern appears in circumferential distribution of axial nominal wake

Fig. 19. Variation of the mean axial wake fraction at each radius with Reynolds number.

Fig. 20. Variation of reciprocal of the amplitude of wake peak 1 at each radius with Reynolds number.

fraction, two wake peaks are found, one is right above propeller disc corresponding to the angular position of 12 o’clock and the other is right down propeller disc corresponding to the angular position of 6 o’clock.

Z.-Z. Wang et al. / Ocean Engineering 104 (2015) 437–451

449

(2) The reciprocal of mean axial wake fraction of propeller disc exhibits a near linear dependence on Reynolds number in logarithmic scale. (3) Linear relationship is found between the mean axial wake fraction at each radius, reciprocal of amplitude of wake peak right above propeller disc and Reynolds number in logarithmic scales. (4) In inner radius area of propeller disc, the reciprocal of amplitude of wake valley and wake peak right down propeller disc reveal nearly linear dependence on Reynolds number in logarithmic scales. (5) In medium and outer radius area of propeller disc, the amplitude of wake valley and wake peak decline rapidly to the potential wake fraction. The wake width reveals negative exponent power function dependence on Reynolds number in logarithmic scales. Fig. 23. Variation of the amplitude of wake valley at each radius with Reynolds number.

Fig. 21. Variation of reciprocal of the amplitude of wake peak 2 at each radius with Reynolds number.

Fig. 24. Variation of the reciprocal of amplitude of wake valley with Reynolds number.

Table 11 Width at some typical radius in different scales λ

log10(Re)

r/R ¼0.5

r/R¼ 0.7

r/R¼ 0.9

63 32 16 8 4 2 1

6.830 7.272 7.723 8.175 8.626 9.078 9.529

2.01398 1.55193 1.25738 1.04324 0.85501 0.70875 0.58498

2.14293 1.79536 1.53212 1.32783 1.15017 1.00505 0.88286

2.39573 2.10642 1.85901 1.65881 1.47997 1.32580 1.18754

Fig. 22. Variation of the mean axial wake at each radius with Reynolds number.

Table 10 Amplitude of wake valley at each radius in different scales λ

log10(Re)

r/R¼ 0.3

r/R ¼0.4

r/R¼ 0.5

r/R¼ 0.6

r/R¼ 0.7

r/R¼ 0.8

r/R¼ 0.9

r/R¼ 1.0

63 32 16 8 4 2 1

6.830 7.272 7.723 8.175 8.626 9.078 9.529

0.50208 0.42569 0.34966 0.28757 0.24416 0.21508 0.19334

0.36722 0.27470 0.22023 0.18680 0.16364 0.14607 0.13108

0.24369 0.18780 0.15580 0.13243 0.11363 0.10002 0.09197

0.17497 0.14017 0.11448 0.09527 0.08332 0.07883 0.07829

0.12753 0.09831 0.07856 0.07091 0.07031 0.07095 0.07162

0.09388 0.07232 0.06403 0.06401 0.06508 0.06633 0.06726

0.06961 0.05887 0.05880 0.06024 0.06142 0.06245 0.06313

0.05525 0.05413 0.05536 0.05617 0.05725 0.05873 0.05931

450

Z.-Z. Wang et al. / Ocean Engineering 104 (2015) 437–451

Table 14 Comparison of characteristic quantity of nominal wake at r/R ¼0.9 between extrapolated full scale results and numerical result Characteristic quantity

Numerical results

Extrapolated from λ¼ 63

Extrapolated from λ ¼32

ωh1 ωh2 ωl 2m

0.4531 0.0634 0.0631 1.1875

0.4501 0.0658 0.0592 1.2137

0.4509 0.0658 0.0601 1.1945

Acknowledgments

Fig. 25. Variation of width at typical radius with Reynolds number.

The hull and experimental data are provided by Marine Design and Research Institute of China, and the present work is supported by the National Natural Science Foundation of China (Grant no. 51479207, 51179198) and the High Technology Marine Scientific Research Project of Ministry of Industry and Information Technology of China (Grant No.[2012]534). We would like to express our deep appreciation to Marine Design and Research Institute of China and Ministry of Industry and Information Technology of China.

References

Fig. 26. Comparison of width at typical radius between original numerical results and regressive data.

Table 12 Comparison of characteristic quantity of nominal wake at r/R¼ 0.5 between extrapolated full scale results and numerical result Characteristic quantity

Numerical results

Extrapolated from λ ¼63

Extrapolated from λ ¼32

ωh1 ωh2 ωl 2m

0.4990 0.3025 0.0920 0.5850

0.4994 0.2979 0.0908 0.5969

0.4995 0.2987 0.0909 0.5931

Table 13 Comparison of characteristic quantity of nominal wake at r/R¼ 0.7 between extrapolated full scale results and numerical result Characteristic quantity

Numerical results

Extrapolated from λ ¼63

Extrapolated from λ ¼32

ωh1 ωh2 ωl 2m

0.4692 0.1753 0.0716 0.8829

0.4677 0.1731 0.0693 0.8925

0.4683 0.1732 0.0699 0.8849

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