Chunyu Guo, Qi Zhang and Yu Shen

1. College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China 2. Technical Center, COSCO Shipyard Group Co. Ltd., Dalian 116600, China

1 Introduction1

Over the last century, model testing has been widely used to study a ship’s performance. Currently, it also remains one of the main methods used to predict a ship’s performance.With the gradual accumulation of experience through actual ship building, large discrepancies have been identified between the wake values of model-scale and actual ships.This is called the scale effect (Sheng and Liu, 2004), which is mainly caused by differences between the Reynolds numbers of model-scale and actual ships despite both satisfying the same conditions in terms of geometric similarity and Froude number.

To convert the drag of a model into that of an actual vessel, a developed semi-empirical method can be applied with a self-modelling Reynolds number to account for macroscopic drag. However, predicting the wake field of an actual vessel is much more complex (Fu and Yang, 2009).The prediction and analysis of the wake field of a vessel’s hull are very important because these results help to reduce the propeller noise and vibration, and to improve propulsion efficiency.

Two methods are generally used to deter-mine the wake field of a ship: test measurements (at model- and full-scales)and numerical simulations using computational fluid dynamics (CFD). For experimental studies,the measurement of local flow fields has become a popular research and test topic in hydrodynamics. A five-bladed MAU propeller wake was measured by Paiket al. (2007).Di Felice (2004) and Felliet al. (2006) had gotten the main characteristics of propeller flow field measurement in a large cavitation tunnel by particle image velocimetry (PIV).Danget al. (2012) performed PIV measurements of the local flow fields of a ship’s hull and around its energy-saving appendages. There have been some efforts in measuring full scale wake, for example, in project EFFORT and DALIDA.However, wake data from these projects are not publically available. In China, Li (2010) performed detailed laser doppler velocimetry (LDV) measurements of a propeller’s wake field. Zhanget al. (2007) and Liet al. (2011) used PIV to measure under water fluid fields. In China, wake field measurements are mainly performed at the model-scale, but rarely at the full-scale.

For numerical simulations, the flow field around a ship’s hull is widely calculated using the CFD method, which is capable of accurately providing the Reynolds number of an actual vessel for theoretical calculations for use in predicting the hydrodynamic performance of a ship (Koronowiczet al.,2003). Shen and Su (2008) used block grids and multiple grids (comprising a mixture of structured and unstructured grids) for local grid refinement when performing mesh partitioning of the flow field around the hull of a full-formed ship. They were able to obtain wake fields that well matched the values obtained from model tests.

Huanget al. (2009) calculated the wake field of a particular container ship using models at three different reduced scales. By analyzing the wake field of the propeller disk in these models, they were able to study the scale effect of the wake field. Presently, the effect of different Reynolds numbers can be studied by CFD. In recent years, many numerical simulations have focused on the Reynolds number for the flow fields of actual vessels (Niet al., 2010).Kimet al. (2012) compared the model and full scale nominal wake differences for an Aframax product carrier equipped with pre-swirl stators. Alejandroet al. (2011) also studied flow fields at the full-scale. The assessment (Larssonet al. 2014) of results shows that a 3–4 million grid on a half hull with a 2nd order discretization scheme (Weiss and Smith, 1995) seems sufficient to make a good prediction at model scale.

The scale effect can be addressed by using various conversion methods. And the most common method is the semi-empirical wake conversion (or contraction) method(ITTC, 2011). The measured nominal wake is used as the starting value, after which the wake width is reduced along different paths. However, very few studies have verified the reliability of the results derived using different wake conversion methods. To date, a standard and reliable theoretical method to determine the wake field at the ship’s stern and a universal conversion method remain lacking.

Schuilinget al. (2011) proposed the concept of a smart dummy model (SDM) to simulate the scale effect of wakes during cavitation tests. An SDM is basically a ship model that is non-geometrically similar to an actual vessel but has a similar wake field. The construction of the virtual model is mainly reflected in the geometric distortion of the ship’s stern. Changes to the stern based on viscous flow calculations are then used to predict the wake fields of actual vessels. The SDM can generally simulate the wake field of ships accurately, with the exception of the lower quarter of the propeller disk. When the wake reaches a peak value at the top of the propeller disk, the wake fields of the SDM and actual vessel are well matched. However, this method is still in an exploratory stage.

In this study, the findings of previous studies were summarized and used as the basis. The SDM was used to perform geometric distortions of the ship’s stern at the model-scale. The purpose is to achieve results that are consistent with the target (full-scale) wake field. The Korean Research Institute of Ship and Ocean Engineering (KRISO)container ship (KCS) was used as the sample to design a non-geometrically similar ship model. The shape of the model’s stern was modified substantially, but the rest of the hull was kept geometrically similar. Numerical calculations as well as a comparison between the results and the full-scale calculation data were performed.

2 Concept of smart dummy model (SDM)

Schuilinget al. (2011), Bosschers and van Wijngaarden(2012) and Johannsen and Wijngaarden (2012) reported the use of such SDMs, which is a ship model that is non-geometrically similar to an actual vessel but has a similar wake field (ITTC, 2014). It has already been known from the previous section that the SDM is a ship model that is non-geometrically similar to an actual vessel, but has a similar wake field. As with a conventional ship model, the portion of the SDM’s stern above the propeller is similar to the full-scale, as is the gap between the hull and the propeller shaft. Presently, in most cases it still uses the geometrically similar models for testing in towing tanks.However, to simulate a wake field that is closer to the full-scale one, the solution is to disregard the geometric similarity of the stern. The main difficulties lie in model design and achievement of effective wake fields similar to those of actual vessels. To satisfy the condition of movement similarity, the SDM should deviate from geometric similarity. The CFD tool must also be used to ensure the optimal state of the SDM as a wake field generator.

When establishing the SDM, the first step is to modify the width and length of the hull model, which results in simple intuitive changes. Four different ship profiles were selected according to the profile of the mother ship when establishing that of the smart dummy ship. The hull shape was maintained while its width was reduced. Separately, the shape of the stern was maintained while the hull length was reduced (by shortening the vessel’s body).

Thus, the following four shapes are generated, as shown in Fig. 1:

1) A ship model with original length and half breadth.

2) A ship model with half length and original breadth.

3) A ship model with half length and half breadth.

4) The original ship model (Geosim).

Fig. 1 Variations of width and length in hull form

Parnassos software, which is based on Wilcox (1994), has been used in literature to solve these models. The stack mould model was used for simulating all calculations. The calculation results of the axial wake field of the first three modified models were compared against the full-scale one.Next, the flow field distribution at four different planes–propeller disk and 95%, 85%, and 75%Lppcross sections–was selected for comparison. Comparative analyses based on the calculation results of all hull forms(including those not listed above) showed that the axial velocities at the top portion of the propeller disk were significantly lower than those at the full-scale. Thus, simply modifying the length and width was insufficient to generate a requisite wake field.

Modifications were also required to the stern. As reported in literature, such changes led to the SDM. Fig. 2 (left-hand side) shows the shape of the model’s rear. The final calculation results indicated that the wake field distribution at the fan-shaped area at the top portion of the propeller disk well matched that at the full-scale as shown in Fig. 3.

Fig. 2 View of ship’s stern

Fig. 3 Axial velocity distribution within fan-shaped area at top portion of propeller disk: SDM (left) and at full-scale (right)

3 The grid independence test

3.1 Numerical method

In this study, the shear stress transport (SST)k-ωmodel(Menter, 1994) was used to better simulate the strong adverse pressure gradient flow field taking into consideration the impact of shear stress on the model’s walls.This model combines the strengths of thek-ωandk-εmodels and it can be used to calculate the flow separation region. It is one of the most advanced two-equation turbulence models and it produces superior results when used to calculate viscous flow fields.

Currently, two types of methods are used to simulate the free surface when solving the viscous flow of a ship based on RANS: interface tracking and capturing. The volume of fluid (VOF) method (Hirt and Nichols, 1981) belongs to the latter type and is more widely applied to solve multiphase flows. The flow field around ships involves air-fluid two-phase flows in which the VOF method was used to deal with the free surface issue of flow fields (Fu and Ma, 2009).

One of the most important factors determining the achievement of ideal results for a turbulence model is grid generation for the boundary layer. For greater calculation accuracy, the position of the first grid node must be estimated when dealing within ear-wall flows, particularly for flows with high and low Reynolds numbers. This will determine whethery+is within an appropriate range. The minimum grid size Dypfor the boundary layer during meshing is calculated using the following formula (Zhang,2007):

whereLis the characteristic length, generally treated asL=Lpp(Including the characteristic length of Reynolds number). For a relatively better first grid node, the value of the dimensionless distancey+is 30–500 (Fu and Ma, 2009).

3.2 Comparing the calculated results with different numbers of grids

This study focuses on the wake field at the stern of the KCS, a KRISO 3 600 twenty-foot equivalent unit (TEU)container ship. In this article,Lpprepresents length between perpendiculars,Bwlrepresents the breadth,Tmeans the draft,Vmeans the designed speed, andx,y,zrepresent the directions of length, width and depth of the ship,respectively. Table 1 shows the main parameters for the calculation model. Table 2 shows a comparison of the test value and drag with different grid density. All the calculation errors are below 0.3%.

Table 1 Main parameters for the KCS model

Table 2 Comparing the calculated results of drag with different numbers of grids

Fig. 4 Free-surface wave patterns with different numbers of grids

Free-surface wave patterns with different numbers of grids are compared in Fig. 4, for catching detail of free surface. The denser the grid is, the more meticulous the results are. However, all the accuracy values of free surface are fine with the different grid density.

Fig. 5 Comparison of axial wake values at z=?0.03Lppposition of propeller disk

A comparison of axial wake values is provided. Fig. 5 shows comparison of axial wake values with three different numbers of grids. Three curves almost overlap, in which the distinctions are very small, not only for the results of axial wake values but also for the calculation of drag and free-surface wave patterns. So the effect of grid density on the calculation results is little.

4 Establishment of SDMs and numerical simulations

4.1 Research subject and method

This section will carry out a series of numerical calculations for the wake field. Table 3 shows the main parameters for the various calculation models.

The calculation domain was rectangular. Lengthwise, the domain extended upstream from the ship’s bow by a ship’s length overall(LOA) and downstream from the stern by three ships’LOA. Breadthwise, the domain extended along the directions of the draft and the ship’s breadth by a ship’sLOA.Since a viscous flow field is symmetric, only the flow field of half the hull is considered in the calculation model, with the middle longitudinal section of the hull set as the plane of symmetry. The boundary conditions are provided in Table 4.

For this study, Eq. (1) was used during meshing to adjust the thickness of the first grid layer until the final value ofy+was between 30 and 500. Table 5 shows the number of grids and the maximum value ofy+for the various calculation models.

Table 3 Main parameters for calculation models

Table 4 Boundary conditions

Table 5 Number of grids used in calculation models

Fig. 6 shows the cross-sectional view of the original KCS hull. Modifications to the hull shape were based on the KCS model shown in Table 1, and the involved portions of the stern before Station 2 and below the design waterline. The next section focuses on the three typical models developed during the stern modification process to examine the resultant impact patterns on the flow field. The established model was termed the KCS SDM. The other parameters of the ship remained unchanged. Modifications to the shape of the stern were mainly based on the profile of the mother ship.The results after modification were imported into computer aided tri-dimensional interface application (CATIA) for model building. Next, the ICEM CFD software was used for meshing. As stated earlier, the flow field of only half the hull was considered in the calculation model, given the symmetry in viscous flow fields. The results after meshing were imported into STAR–CCM+ for the purpose of calculation.

Fig. 6 Cross-sectional view of KCS hull

4.2 Calculation results and analysis

Flow field data around the propeller disk of the various models were obtained after the calculations had reached convergence. Wakes can be categorized as either nominal or effective. This study focuses upon the former type. The velocity field of wakes can generally be expressed using three components: axial, circumferential (or tangential), and radial velocities relative to the propeller. The measurement results indicate that the axial wake velocity is the most important component. The other two components, being of second order and relatively small, can often be ignored when designing propellers. Thus, only axial wake was considered in this study.

Due to a general lack of full scale data all over the world,the accuracy of calculation results is difficult to be verified.The current practice is to perform some form of validation study at model scale. In this article, calculation results from the model-scale hull were compared with the test values.First, the drag value of the KCS bare hull was calculated to be 40.39 N. In comparison, the test value was 40.50 N. The results satisfied the 5% accuracy requirement. Next, the calculated and test values of the isolines of the dimensionless axial velocities at the propeller disk were compared as shown in Fig. 7. The radii of the inner and outer circles shown in Fig. 7 are 30% and 110% of the propeller radiusRp, respectively. The calculated and test values show a relatively better match.

Fig. 7 Isolines of dimensionless axial velocity components for propeller disk: test value (left) and calculated value (right)

Fig. 8 Dimensionless axial velocity components for back of propeller disk (at 0.25DP, z=?0.03Lpp)

This is also the case when comparing the calculated and test values of the dimensionless axial velocity componentuat the planex=0.4825Lpp, which is 0.25DP(propeller diameter) behind the propeller disk and under the design waterlinez=?0.03Lppas shown in Fig. 8.

For this study, the wake fields at the stern of the full-scale KCS and the hull of the original geometrically similar model were calculated as shown in Fig. 9. The calculation results at full-scale were then used to establish the virtual model. It was started with SDM 1, for which the stern profile was offset toward the centre. Using the profile at Stations 0.75, 1,and 1.5 as examples (along the direction of the captain, from after perpendicular to forward perpendicular, the ship can be divided into 20 frames and every frame is a station), it was found that there was a relatively large profile offset at all three stations toward the ship’s central axis of symmetry.The hull shape at the central portion above the propeller shaft became more slender, whereas the bottom of the ship remained unchanged as shown in Fig. 10. Figs. 11 and 12 show the calculation results. The former shows a comparison of the wake distribution at the propeller disk between SDM 1 and the full-scale, whereas the latter shows a similar comparison between SDM 1 and the original geometrically similar model.

Figs. 11 and 12 show that modifications to the stern had a relatively great impact on the wake field. The wake field of SDM 1 did not match the full-scale one. However, compared with the original geometrically similar model, the isolines of the wakes for SDM 1 had begun to contract toward the centre,with the 0.5 isoline of the axial wake being broken.Contractions of the axial wake isolines were mainly concentrated at the top of the propeller disk. This indicates that modifications to the hull above the centre of the propeller shaft have a definite amount of impact on improving the wake field. It can also be observed from the figures that there are significant changes to the wake field near the centre of the propeller shaft. Compared to the original model, the 0.2–0.5 isolines of the axial wake are even offset toward the external portion of the propeller shaft.

Fig. 9 Comparison of wake fields between original geometrically similar model (left) and the full-scale one (right)

Fig. 10 Line chart for KCS SDM 1

Fig. 11 Comparison of wake fields between KCS SDM 1 (left)and the full-scale one (right)

Fig. 12 Comparison of wake fields between KCS SDM 1(left) and original model (right)

The calculation results from KCS SDM 1 were used for further modifications of the stern. The entire profiles at Stations 0.75, 1, and 1.5 were offset toward the centre of the propeller shaft. And the portion of the hull above the centre of the propeller shaft was again modified and made slender. Modifications to the profile as shown in Fig. 13 led to the establishment of SDM 2. Fig. 14 shows a comparison of the wake distribution at the propeller disk between SDM 2 and the full-scale, whereas Fig. 15 shows a similar comparison between SDM 2 and the model prior to modifications.

Figs. 14 and 15 show that the wake isolines for SDM 2 had contracted further toward the centre of the propeller disk compared to SDM 1. The calculated and full-scale values corresponded better within the fan-shaped area at the top portion of the propeller disk. However, the changes arising from modifications to the central region of the propeller shaft and the central region below the propeller disk remained unsatisfactory.

Fig. 13 Line chart for KCS SDM 2

Fig. 14 Comparison of wake fields between KCS SDM 2(left) and the full-scale one (right)

Fig. 15 Comparison of wake fields between KCS SDM 2(left) and original model (right)

SDM 3 was created after combining the analysis of the calculation results for SDMs 1 and 2. The hull shape above the centre of the propeller shaft was made even more slender(as shown in Fig. 16). Figs. 17 and 18 show the obtained calculation results. The former shows a comparison of the wake distribution at the propeller disk between SDM 3 and the full-scale, whereas the latter shows a similar comparison between SDM 3 and the model prior to modifications. For the axial wake values calculated at the 0.7Rposition, Fig. 18 shows a comparison between SDM 3, the original geometrically similar model, and the full-scale (horizontal position was set to 0°, with one point selected at intervals of 10°, and counter-clockwise was treated as the positive direction).

Figs. 17 and 18 can be used to compare SDMs 1 and 2.The wake distribution at the propeller disk in SDM 1 is highly consistent with that at the full-scale, especially for the fan-shaped area at the top portion of the propeller disk,where the wake value peaks. There was a certain amount of contraction toward the centre for the 0.1 and 0.2 isolines of the axial wake, where the radius of the propeller disk is relatively large. However, there remains a certain amount of discrepancy compared to that at the full-scale.

Fig. 16 Line chart for KCS SDM 3

Fig. 17 Comparison of wake fields between KCS SDM 3(left) and the full-scale one (right)

Fig. 18 Comparison of wake fields between KCS SDM 3(left) and original model (right)

Fig. 19 Comparison of axial wake values at 0.7R

Fig. 19 shows a general downward offset of the axial wake distribution curve at the 0.7Rposition of the SDM.This curve is actually closer to the calculated values at full-scale. The amount of offsets reduces with the increase of the angle. The wake in the area above the centre of the propeller disk shows greater improvements. However, there is no significant change in the wake distribution near the centre of the propeller shaft. This indicates that modifications to the hull above the centre of the propeller disk have a significant impact on the wake. Making this portion more slender can also generate a wake field that is closer to that at full-scale.

5 Conclusions

The wake field at the stern has a great impact on the thrust, torque, and efficiency of a ship’s propeller. Making the wake distribution at the propeller plane as close as possible to the conditions of actual ships is conducive for research on the hydrodynamic performance of propellers,and it provides a new way of thinking when predicting ship wakes. Through modifications made at the model-scale, this study attempts to generate wake fields that are more consistent with that at the full-scale.

Using the results of previous studies, a non-geometrically similar model of the KCS was designed. Numerical simulations were performed using the model, and the obtained results were compared with the full-scale calculation results. The main modification to the hull shape was made at the stern. Specifically, the stern portion above the centre of the axial shaft was modified and made more slender. Finally, it was managed to create a wake field that was more ideal. A special significance of SDM is to improve the accuracy and reliability of experimental prediction of cavitation extent and pressure fluctuation in a cavitation test. Although the model did not achieve results consistent with the target wake field, it nevertheless exhibited a trend that was much closer to the calculated values at the full-scale.

This is only a preliminary study, using a reverse design method for finding a way to get wake field of full-scale ships by model-scale ships. Follow-up experimental studies are needed to carry out model tests for further analyses of the wake fields at ships’ sterns. By comparing the software simulation results with the wake fields of actual ships,further and gradual model modifications can be made to refine the numerical simulation method for establishing SDMs. In doing so, numerical simulations can be used to generate wake field distributions that are more realistic,thereby saving time and cost for future experimental studies.

Alejandro MC, Pablo MC, Frederick S (2011). Full scale self-propulsion computation using discretized propeller for the KRISO container ship KCS.Computers >amp; Fluids, 51(1), 35-47.DOI: 10.1016/j.compfluid.2011.07.005

Bosschers J, van Wijngaarden E (2012). Scale effects on hull pressure fluctuations due to cavitating propellers.10th International Conference on Hydrodynamics, Petersburg,Russia, 26-30.

Dang J, Chen H, Dong G, van der Ploeg A, Hallmann R, Mauro F(2012). An exploratory study on the working principles of energy saving devices (ESDs).31st International Conferenceon Ocean, Offshore and Arctic Engineering, Rio de Janeiro,Brazil, 24-35.

Di Felice F, Di Florio D, Felli M, Romano GP (2004).Experimental investigation of the propeller wake at different loading conditions by particle image velocimetry.Journal of Ship Research, 48(2), 168-190.

Felli M, Di Felice F, Gui G, Camusi R (2006). Analysis of the propeller wake evolution by pressure and velocity phase measurements.Experiments in Fluids, 41(3), 441-451.DOI: 10.1007/s00348-006-0171-4

Fu Huiping, Ma Ning (2009). Demands of free surface and wake computation on mesh.Journal of Shanghai Jiao Tong University, 43(10), 1573-1576. (in Chinese)

Fu Huiping, Yang Chenjun (2009). The effects of Reynolds number on resistance and wake of ship.Journal of Shanghai Jiao Tong University, 43(10), 1555-1558. (in Chinese)

Hirt CW, Nichols BD (1981). Volume of fluid (VOF) method for the dynamics of free boundary.Journal of Computational Physics, 39(1), 201-225.

Huang Jiabin, Chen Xiaping, Zhu Renchuan, Chen Changyun(2009). Study on the scale effect on nominal wake field using CFD method.Proceedings of the 22nd National Conference on Hydrodynamics, Chengdu, 685-692. (in Chinese)

ITTC (2011).Specialist committee on scaling of wake field. Final report and recommendations to the 26thITTC, Rio de Janeiro,Brazil, 408-412.

ITTC (2014).Propulsioncommittee. Final report and recommendations to the 27thITTC, Copenhagen, Denmark,62-65.

Johannsen C, Wijngaarden E (2012). Investigation of hull pressure pulses, making use of two large scale cavitation test facilities.8th International Symposium on Cavitation (CAV), Singapore,196-202.DOI: 10.3850/978-981-07-2826-7_196 Kim K, Leer-Andersen M, Werner S, Orych M, Choi Y (2012).Hydrodynamic optimization of pre-swirl stator by CFD and model testing.29th Symposium on Naval Hydrodynamics,Gothenburg, Sweden.DOI: 10.3233/ISP-130092

Koronowicz T, Tuszkowska T, Wac?awczyk T (2003). A computer method for prediction of the velocity field behind a full-scale ship hull.Polish Maritime Research, 10(1), 3-9.

Larsson L, Stern F, Visonneau M (2014).Numerical ship hydrodynamics: an assessment of the Gothenburg 2010Workshop. Springer Netherlands, Berlin.

Li Guangnian (2010). LDV measurements of propeller trailing vortex.Journal of Experiments in Fluid Mechanics, 24(4),75-79. (in Chinese)

Li Guangnian, Zhang Jun, Chen Zhengshou, Xie Yonghe (2011).Propeller trailing vortex analysis based on PIV experimental data.Journal of Ship Mechanics, 15(10), 1110-1115. (in Chinese)

Menter FR (1994). Two-equation eddy-viscosity turbulence models for engineering applications.AIAA Journal,32(8), 1598-1605.DOI: 10.2514/3.12149

Ni Chongben , Zhu Renchuan, Miao Guoping, Fan Sheming (2010).A method for ship resistance prediction based on CFD computation.Chinese Journal of Hydrodynamics, 25(5), 75-79.(in Chinese)

Paik BG, Kim J, Park YH, Kim KS, Yu KK (2007). Analysis of wake behind a rotating propeller using PIV technique in a cavitation tunnel.Ocean Engineering, 34(3), 594-604.DOI: 10.1016/j.oceaneng.2005.11.022 Schuiling B, Lafeber FH, van der Ploeg A, van Wijngaarden HCJ(2011). Influence of the wake scale effect on the prediction of hull pressures due to cavitating propellers.Second International Symposium on Marine Propulsors(SMP’11), Hamburg,Germany.

Shen Hailong, Su Yumin (2008). Mesh partitioning methods for numerical simulation of wake fields of full-formed ships.Journal of Harbin Engineering University, 29(11), 1190-1198.(in Chinese)

Sheng Zhenbang, Liu Yingzhong (2004).Principle of naval architecture. Shanghai Jiao Tong University Press, Shanghai,China, 41-63. (in Chinese)Weiss JM, Smith WA (1995). Preconditioning applied to variable and constant density flows.AIAA Journal, 33(11), 2050-2057.DOI: 10.2514/3.12946

Wilcox DC (1994).Turbulence modeling for CFD. DCW Industries,Inc. La Canada, USA, 11-30.

Zhang Jun, Zhang Zhirong, Zhu Jianliang, Xu Feng, Lu Linzhang,Dai Qin (2007). Investigation of internal flow field of ducted propellerusing particle image velocimetry.Journal of Experiments in Fluid Mechanics, 21(2), 82-88. (in Chinese)