Ocean Engineering 258 (2022) 111837
Available online 1 July 2022
0029-8018/© 2022 Elsevier Ltd. All rights reserved.
Experimental and numerical investigation on the resistance characteristics of a high-speed planing catamaran in calm water
Hui Wang , Renchuan Zhu
*, Le Zha, Mengxiao Gu
State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China
A R T I C L E I N F O Keywords:
Planing catamaran Model test CFD method Tunnel Resistance Lift force
A B S T R A C T
There has been an increasing interest in high-speed planing catamaran hulls for their exceptional resistance performance in military and maritime transportation fields. In this paper, experimental tests and numerical investigation on a planing catamaran under different displacements are performed to analyze the resistance characteristics and mechanism of the tunnel flow. Model tests are conducted in a towing tank for the Froude number ranging from 0.76 to 1.93. A computational fluid dynamics (CFD) code incorporated with dynamic overset mesh technique is employed to simulate the flow around the planing catamaran in calm water. The numerical results are validated with measured data and show good agreement. The mean error in the trim angle and the resistance took the whisker spray drag into account are 8.9% and 4.5%, respectively. Numerical results of pressure, comparisons of wave profile along transom stern centerline, lift force distribution on the tunnel, and components of tunnel lift are presented and compared for the analysis of tunnel flow under different displace- ments. The tunnel could contribute the maximum lift about 26% of ship weight in the case of M =202.9 kg. This study would provide a better understanding of hydrodynamics and the aerodynamics of the tunnel for the planing catamaran.
1. Introduction
The military unmanned surface vehicle (USV) is considered as an important direction for the development of unmanned equipment at sea in the future. USV in the military field generally adopts a planing hull or semi-planing hull as the carrier. Recently, planing crafts have been the research hotspot due to their high speed and excellent hydrodynamic performance (Ding and Jiang, 2021). The weight of the hull in the displacement and planing region is supported by the buoyancy and hydrodynamic lift, respectively (Yousefi et al., 2014). The high-speed planing catamaran is a newly developed tunnel-type planing craft, which is also named tunneled planing hull (Roshan et al., 2020). The air flow enters into the tunnel and generates the aerodynamic force over the hull during high-speed regions. The planing catamaran has similar resistance performance with the planing trimaran, the tunnel can create aerodynamic lift and achieve a significant resistance reduction (Jiang et al., 2017). The accurate prediction of total resistance and resistance components is the most significant stage in terms of hydrodynamics (Dogrul et al., 2020; Du et al., 2020). The hybrid hydrodynamic and aerodynamic characteristics of the planing catamaran hull are very
complex. However, there are only a few studies on the physics of tunneled hulls (Roshan et al., 2020).
The model tests and empirical methods are widely used for the investigation of the planing hulls. Savitsky (1964) carried out experi- ments on prismatic planing mono-hulls and proposed a semi-empirical formula for determining the resistance and attitudes of planing hulls.
There are some previous extensive tests on the resistance performance of the high-speed planing hull in calm water (Kim et al., 2013; Sukas et al., 2017; Dumortier et al., 2019; Wang et al., 2021; Ashkezari and Moradi, 2021). A series of model tests of high-speed planing trimaran with and without spray strips were conducted to measure the resistance, sinkage, and trim angle (Ma et al., 2013). Their resistance tests took several factors including step, step form, and types of planing surface.
The experimental method is challenging and expensive to capture the detailed flow field (Ding and Jiang, 2021). In recent years, the computational fluid dynamics (CFD) method has become a reliable and important approach for analyzing the hydrodynamic mechanism of the planing hull. Subramanian et al. (2007) carried out CFD simulation by Fluent software and experiments in a towing tank, using a model of the planing hull with and without the tunnel. They confirmed that the
* Corresponding author.
E-mail address: [email protected] (R. Zhu).
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Ocean Engineering
journal homepage: www.elsevier.com/locate/oceaneng
https://doi.org/10.1016/j.oceaneng.2022.111837
Received 30 January 2022; Received in revised form 21 June 2022; Accepted 22 June 2022
presence of the tunnel has improvement in the resistance by appreciable reduction. Panahi et al. (2009) proposed a fractional step method based on finite volume discretization to investigate the steady-state forward motion of a high-speed planing catamaran. Their numerical results had good concordance with available experimental data. Yousefi et al.
(2014) numerically simulated the flow around a Cougar high-speed planing hull and compared their results with experiments. They intro- duced two tunnels at the bottom section of the original hull and they found that the two tunnels reduced the total drag of the modified hull.
Ghassabzadeh and Ghassemi (2014) analyzed the hydrodynamic behavior of a multi-hull tunnel vessel in calm water by using the VOF model. The heave and pitch motions at each time step were simulated by a dynamic mesh restructuring method for grid generation. The resis- tance and trim angle of the CFD simulation was in good agreement with the measured data. Su et al. (2014) carried out a numerical simulation based on a RANS-VOF solver to study the hydrodynamic performance of a channel type planing trimaran. The numerical results indicated that the channel type planing trimaran has good hydrodynamic character- istics. Kazemi Moghadam et al. (2015) attempted to study the influence of the size of the tunnel. And they found that reducing the size of tunnel aperture can decrease the total resistance at a high Froude number.
Jiang et al. (2016) presented an analysis of tunnel hydrodynamic characteristics for a planing trimaran by model tests and numerical simulations. They also studied the influence of tunnel configuration on hull behaviors (Jiang et al., 2017).
Roshan et al. (2020) studied the details of the hydrodynamic behavior of a tunneled planing hull using Star-CCM +software. They introduced a new parameter as tunnel efficiency and tunnel efficiency shows appropriate speed in resistance reduction. Su et al. (2020) per- formed experimental tests and numerical simulations to find out the relationship between hydrodynamic performance and the main hull shape of planing trimaran. The comparison of numerical results indi- cated that shapes of the central body have a significant effect on the resistance performance. The inhibition and hydrodynamic analysis of twin side-hulls on the porpoise instability of high-speed craft were investigated and discussed (Wang et al., 2021). Zou et al. (2021) con- ducted an experimental study on the motion behavior and longitudinal stability assessment of a planing trimaran hull in calm water. And the numerical simulation showed that the presence of a tunnel could decrease the resistance of the main hull and improve the longitudinal motion stability at high speeds (Sun et al., 2020). The hull-waterjet interaction of a planing trimaran was studied by numerical and exper- imental methods (Jiang et al., 2017). Ding and Jiang (2021) adopted CFD technology to conduct a detailed study on the tunnel flow of planing trimaran and its effects on resistance at low & hump speed and planing speed separately. The tunnel can absorb the internal wave energy and provide additional lift force. They found that the tunnel could contribute the maximum lift about 24% of ship weight.
Those previous studies mentioned above mostly focused on the planing trimaran and multi-hull tunnel vessel. However, the compre- hensive hydrodynamic analysis of planing catamaran hulls by experi- mental tests and numerical methods has been carried out rarely. To the best of the authors’ knowledge, a deep study on the hydrodynamic performance of the planing catamaran is needed besides the studies given above. Therefore, this paper investigates and discusses the resis- tance and lift characteristics of the tunnel flow for a planing catamaran under the influence of distinctive displacements at different forward speeds. Firstly, a description of planing catamaran and model tests are presented. And then, a brief introduction of numerical simulation is given. The CFD method incorporated with dynamic overset mesh tech- nique is conducted to model large motion of planing catamaran at different speed ranges. Subsequently, the convergence study and veri- fication are carried out. The CFD solver is validated using the benchmark experiment of the Fridsma hull (Fridsma, 1969). The sinkage, trim angle, and resistance obtained by the numerical method are compared with experimental results. Furthermore, pressure distribution on the tunnel
and tunnel-wave distance are analyzed. And then transom wake flow behind the stern, lift distribution on the tunnel, and components of tunnel lift are compared and discussed. Finally, the main conclusions are given in Section 6. And the detailed equations of the Savitsky method are shown in Appendix.
2. Model test
2.1. Description of planing catamaran
A 1:4 scale model of planing catamaran hull has been built and the geometry of the planing catamaran in this study is shown in Fig. 1. And the main parameters of this catamaran are presented in Table 1. The hull is divided into two demi-hulls arranged aside by a tunnel, so this kind of planing catamaran is also named tunneled planing hull (Roshan et al., 2020). The tunnel starts the bow like a horn and extends aft to the transom stern. When the planing catamaran hull is advancing at high speed, air flow enters from the mouth of the tunnel and forms an air-water mixture flow in the tunnel region under the action of ramming.
The higher the speed, the more air will enter, which is more conducive to the formation of an air layer. Ding and Jiang (2021) divided a planing trimaran into 4 parts to explore the influences of tunnel flow. To investigate behaviors of the tunnel flow, the roof and the side-wall of the tunnel are considered as all the parts of the tunnel in this paper.
2.2. Model tests
The model tests of the planing catamaran were conducted in the towing tank of China Special Vehicle Research Institute, Hubei, China.
The length, breadth, and depth of the towing tank are 510m, 6.5m, and 6.8m, respectively (Jiang et al., 2016; Ding and Jiang, 2021). The main dimensions of the basin are length 510m, width 6.5m, and depth 6.8m, respectively. The maximum steady speed of the carriage is 22 m/s, which satisfies the need of the planing catamaran. The temperature of the water in the towing tank is 16 ◦C.
The photograph of the model test is shown in Fig. 2. Moreover, the schematic view of the experimental setup is presented in Fig. 3. As shown in Fig. 3, the planing hull is attached to the carriage platform with only two degrees of freedom including sinkage and trim, and the towing point is aligned with the center of gravity longitudinally. The trim angle is measured by an electric angle sensor (fitted at the foredeck region).
The sinkage data is collected by a cable-extension displacement sensor, which is mounted at the gravity center of the hull (Jiang et al., 2017).
The resistance is measured by a dynamometer mounted on the carriage during the towing tests. Three different loads of M =127.4 kg, 159.5 kg, and 202.9 kg are considered in the model tests and the towing tests are conducted at the Froude numbers ranging from 0.761 to 1.925. A more detailed description of the experimental setup can be found in the ref- erences (Jiang et al., 2017; Wang et al., 2021).
2.3. Experimental results
The results of calm water resistance tests are summarized in Fig. 4.
The non-dimensional sinkage is defined as the ratio of sinkage to static draft and the non-dimensional resistance is defined as the ratio of resistance to the weight, respectively. And all the values are plotted against the length Froude number (Fn). In this paper, the length Froude number Fn is used and defined as:
Fn= U
̅̅̅̅̅̅
√gL (1)
where U is the speed of the ship, L denotes the length of the ship and g =9.81 m/s2 is the gravitational constant.
As shown in Fig. 4 (a), the non-dimensional sinkage increases with the increasing mass. At the lower speed (Fn =0.761), the hull operates
in the semi-planing region, and the hydrodynamic force acting on the bottom upraises the bow causing a trim angle. After the Froude number is over 1.267, the sinkage increases slightly with the increase of speeds.
The trim angle increases by increasing the mass in the range of 0.761
≤Fn≤1.773. With the reduction of the amount of mass, the hull comes up more from the water. And then the longitudinal center of the pressure moves to the astern of the vessel and it becomes closer to the center of the mass (Masumi and Nikseresht, 2017). In this situation, the trim angle is reduced due to the decrease in the trim moment. The trim angle goes up to a maximum and then decreases gradually with the speed increasing. At the higher speed (Fn =1.925), the trim angle becomes a Fig. 1. Geometry of the planing catamaran hull.
Table 1
Main parameters of planing catamaran.
Parameter Symbol Value
Length overall (m) L 2.625
Beam (m) B 1.0
Draft (m) D 0.24; 0.217; 0.192
Mass (kg) M 202.9; 159.5; 127.4
Longitudinal center of gravity (m) LCG 1.05
Vertical center of gravity (m) VCG 0.325
Fig. 2.The photograph of the experimental setup.
Fig. 3. Schematic view of the experimental setup.
relatively steady value.
At the lower speed (Fn ≤1.013), the resistance to lift ratio (R/Δ) of M =202.9 kg is larger than those of another two conditions. However, with the increment of the mass, the resistance to lift ratio (R/ Δ) de- creases obviously at high towing speed (Fn ≥1.267). This behavior of resistance is consistent with previous studies (Su et al., 2014), which implies that the planing catamaran hull has a superior resistance per- formance with a larger displacement.
3. Computational methodology 3.1. Numerical setup
In this study, the planing catamaran hull is assumed to be a rigid body, and the numerical simulation is carried out on the commercial software Star-CCM+. The unsteady Reynolds Averaged Navier Stokes (RANS) equations are solved using the finite volume method (FVM) based on an implicit and iterative solver (Guo et al., 2021). The shear stress transport (SST) k-ω turbulence model was used to close the RANS equations (Menter et al., 2003). This turbulent model is superior to the standard k-ω and k-ε models and exhibits desirable advantages in the numerical simulation of viscous flow, which is capable to predict ship hydrodynamics accurately (Song et al., 2020). The pressure-velocity coupling problem is solved by the semi-implicit method for the pressure-linked equations (SIMPLE). The convection terms are dis- cretized by the second-order upwind schemes. And the transient terms are discretized by the first-order temporal scheme due to less compu- tational time and show no oscillation of the numerical result (Farkas et al., 2018).
The Volume of Fluid (VOF) approach is applied to capture the free surface of the two-phase flow involving water and air (Hirt and Nichols,
1981). The High-Resolution Interface Capturing (HRIC) scheme is adopted to mimic the convective transport of immiscible fluid compo- nents, which is suited for tracking sharp interfaces.
In this study, the dynamic overset mesh method is used to handle the large amplitude motion of the planing catamaran. The Dynamic Fluid Body Interaction (DFBI) model was incorporated with the RANS solver to evaluate the force and moment on the hull (De Marco et al., 2017).
The DFBI model was activated to simulate and determine the running attitude of the planing catamaran, which includes the free motion of sinkage and trim angle (Guo et al., 2021). The number of inner iterations per time-step is considered as ten in the current numerical simulation.
3.2. Computational domain and mesh generation
The computational domain combining a static background domain and a dynamic overset domain and the applied boundary conditions are shown in Fig. 5. As can be seen, the computational domain of the planing catamaran can be approximated with dimensions of 6.0L ×2.0L×3.5L.
And the overset domain was set as 2.3L ×2.0B× 1.4L. The planing catamaran was defined as a wall with the no-slip condition. In this study, a linear interpolation scheme was applied to transform the solution between the background domain and the overset domain.
The trimmed cell mesher (Star-CCM+) was utilized to generate high- quality Cartesian cells above the computational domain (Guo et al., 2021). The orthogonal prismatic mesh with 8 layers was generated adjacent to the wall of planing catamaran, which is necessary to capture the exact flow behavior near the walls of the hull. The All y +wall treatment was used and the thickness of the first layer was considered to make the wall y+in the range of 30–100. The values of wall y +dis- tributions on the planing catamaran are presented in Fig. 6. It can be observed that the wall y +range on the hull at the maximum speed (Fn
=1.925, M =159.5 kg) is reasonable.
The mesh of the overall domain and refined mesh of the free surface region and hull region are illustrated in Fig. 7. There are three main refined mesh regions: the region around the planing catamaran, the free surface, and the wake flow region. The region near the free surface was refined to capture the elevation of waves and obtain a precise descrip- tion of the flow field. Local volume refinement was carried out using volumetric controls with a particular mesh size (Le et al., 2021). The surface mesh was used in all regions of the computational domain to obtain the smaller grid on the hull surface. In each region, the aniso- tropic mesh refinement was employed to capture flow features, which could achieve good accuracy compared with isometric refinement.
Moreover, the triangular wave region was also refined to make a distinct description of patterns of wake flow.
4. Verification and validation study 4.1. Convergence study and verification
The results of numerical simulation highly depend on the time step and mesh refinement (Lee et al., 2021). The mesh and time step should be compromised to obtain convergence results due to the limitation of computing equipment. In this study, the six cases including three sets of different grid numbers and three sets of different time steps are con- ducted at Fn =1.013 for the M =159.5 kg case (see Table 2).
The resistance, sinkage, and trim angle of CFD simulation under different meshes and time steps are summarized in Table 3 and Table 4.
It’s noted that EFD refers to experimental results of model tests. The maximum error in resistance is 8.50%, which can be acceptable for high- speed ships. The maximum error of sinkage reaches 18.97%, but the relative value is only 0.0076m corresponding to experimental data. The error in trim angle is 16.54% for the solution of the medium mesh with the medium time step.
It’s necessary to execute an extensive verification for mesh and time step (Le et al., 2021). The grid convergence index (GCI) is carried out by Fig. 4. Experimental results of three different masses: (a) sinkage, (b) trim
angle, and (c) resistance.
Fig. 5.The computational domain and boundary conditions.
Fig. 6.Wall y +visualization on the hull at Fn =1.925.
Fig. 7. The mesh of overall domain and refined mesh of free surface region and hull region.
Table 2
Comparison of different meshes and time steps.
Variables NO Case Number of meshes Time step(s)
Mesh A Coarse 1,360,000 0.005
B Medium 1,560,000 0.005
C Fine 1,700,000 0.005
Time step D Coarse 1,560,000 0.007
E Medium 1,560,000 0.005
F Fine 1,560,000 0.003
Table 3
Results of mesh convergence study.
Mesh Resistance
(N) Error
(%) Sinkage
(m) Error
(%) Trim
(deg) Error (%)
EFD 266.7339 – 0.0403 – 4.51 –
Coarse 244.0485 8.50 0.0472 17.07 3.25 28.02
Medium 249.7999 6.35 0.0450 11.73 3.76 16.54
Fine 245.0681 8.12 0.0443 9.99 3.81 15.57
Correction Factor (CF) (Stern et al., 2001) and Factors of Safety (FS) methods (Xing and Stern, 2010) based on Richardson Extrapolation (RE).
The refinement ratios are computed by considering the total numbers of mesh (Duman et al., 2018), as follows:
rk21= (N1
N2
)1/3 ,rk32=
(N2
N3
)1/3
(2)
where N1, N2, and N3 denote total numbers of mesh for fine, medium, and coarse mesh, respectively.
The uncertainties of mesh and time step are presented in Table 5 when Fn =1.013. The mesh convergence ratio obtained from the so- lution of the resistance for different meshes is − 0.823, demonstrating the oscillatory convergence of the mesh. The convergence ratios of resistance, sinkage, and trim by the different time steps are in the range of 0–1, which implies that the monotonic convergence is satisfied. The maximum uncertainties for the mesh and time step are 3.39%D (where D is the corresponding experimental value) and 4.11%D respectively, which corresponds to satisfactory convergence behavior of the mesh and time step (Song et al., 2020). The medium grid and medium time steps are adopted in the current numerical simulations due to the limitations of the computational resources.
4.2. Validation study based on the benchmark Fridsma hull
The present numerical simulation is validated using the benchmark experiment of Fridsma (1969). The main parameters of the planing hull are CΔ =0.912, β =10 deg and L/B =5, respectively. And the center of gravity is located 0.4L from the stern longitudinally and 0.294B above the keel vertically.
The CFD results in comparison with experimental data and the Savitsky method are shown in Fig. 8. The present results of sinkage, trim angle, and mean wetted length match the experiments well at all speeds.
The resistance of the present study is slightly over-predicted compared with measured data at the speed of 0.9 ≤Fn≤1.2. But the resistance obtained by the CFD method is in good agreement with the experiments at Fn <0.9 and Fn >1.2. Although the results computed by the Savitsky method have a certain deviation from experimental results and CFD results, which has a similar tendency to other results. In general, the present numerical method is validated and acceptable for the simulation of the planing hull from the displacement region to the planing region.
4.3. Comparison of turbulent model
Another CFD model is established based on Star CCM +using the RANS equations with the Realizable k-ε turbulence model (Liu et al.,
2020). The CFD simulation with the Realizable k-ε model is employed to compare and validate CFD simulation with the SST k-ω turbulence model due to its high numerical stability and accuracy for the pressure gradient solution.
For the case of M =159.5 kg, the results of resistance, sinkage, and trim angle obtained by the CFD method with two turbulence models are compared with the available experiments. And the comparisons are presented in Fig. 9–11. It can be seen clearly that the resistance of both CFD results is in good agreement with the experimental data in the range of Fn ≤1.521. A certain deviation between the numerical results and experiments occurs when Fn ≥1.773. The resistance obtained by the Table 4
Results of time step convergence study.
Time
step Resistance
(N) Error
(%) Sinkage
(m) Error
(%) Trim
(deg) Error (%)
EFD 266.7339 – 0.0403 – 4.51 –
Coarse 262.1518 1.72 0.0479 18.97 3.97 12.02
Medium 249.7999 6.35 0.0450 11.73 3.76 16.54
Fine 244.3922 8.38 0.0438 8.78 3.65 18.96
Table 5
The uncertainties of mesh and time step.
Variables Parameter Rk pk Ck Uk-CF (%D) UkC-CF (%D) Uk-FS (%D) UkC-FS (%D)
Mesh Resistance (N) −0.823 NA NA 3.39 NA 3.39 NA
Sinkage (m) 0.318 25.039 1.113 2.12 0.23 2.17 0.43
Trim (deg) 0.098 50.781 1.010 1.22 0.11 1.39 0.28
Time step Resistance (N) 0.438 4.851 1.215 2.95 0.43 2.58 0.52
Sinkage (m) 0.409 5.257 1.189 4.11 0.56 3.73 0.75
Trim (deg) 0.536 3.662 1.260 4.10 0.07 3.37 0.67
Fig. 8.Comparison between present study, experimental data, and Savitsky method: (a) sinkage; (b) trim angle; (c) mean wetted length; (d) resistance.
SST k-ω turbulence model seems closer to measured data than that of the Realizable k-ε turbulence model, especially at high speeds.
According to Savitsky’s research (2007), the spray resistance can constitute over 15% of the total resistance in high-speed planing mode.
In this study, the whisker spray drag (WSD) is computed by Savitsky’s formula (Savitsky et al., 2007) and further added to the resistance ob- tained by the CFD simulation. The detailed formulas of the whisker spray drag are shown in Appendix. It can be seen that the CFD results incorporated with whisker spray drag show better agreement with ex- periments in comparison with only CFD results.
The sinkage and trim angle simulated by CFD results are in good accordance with the experimental results, which indicates that the viscous numerical simulation performed in the present study can obtain accurate values. The results of the two turbulence models seem similar, which implies that there is no significant difference between these two models for numerical simulation of high-speed planing hull (De Luca
et al., 2016; Roshan et al., 2020).
The contours of pressure distributions on the bottom of planing catamaran obtained by two turbulent models are plotted in Fig. 12. The pressure coefficient CP is the pressure coefficient relative to the atmo- spheric pressure p0, and the pressure coefficient is defined as follows:
Cp=p− p0
0.5ρU2 (3)
where p denotes the pressure and ρ is defined as the water density.
As the speed increases, the pressure distribution of the side-bottom becomes discontinuous and two major high-pressure regions occur, which is completely different from the conventional planing monohull (Wang et al., 2020). The pressure distribution obtained by two different turbulent models has a certain difference when Fn ≥1.521, the pressure distribution of the Realizable k-ε turbulence model becomes not smooth.
As can be seen, the peak of pressure on the tunnel moves astern with an increase of speed.
To deeply investigate the pressure distribution on the planing cata- maran, the pressure coefficients acting on the longitudinal sections are presented in Fig. 13. The pressure distributions on the Y/L =0.0 and Y/
L =0.038 in the tunnel-ventilating condition have a similar tendency.
The position of peak value on the Y/L =0.0 and Y/L =0.038 moves astern by increasing the speed. The peak of aerodynamic pressure on the tunnel has an obvious difference. The pressure coefficient on the side- bottom of the hull (Y/L =0.076) obtained by the two models shows good agreement. The oscillation occurs in the result (Y/L =0.076) of the Realizable k-ε turbulence model when Fn ≥1.521.
4.4. Comparison with the delft catamaran model
It is important and necessary to compare the proposed planing cat- amaran model with a conventional catamaran hull form for evaluating the resistance characteristic in similar working conditions (Ding and Jiang, 2021). In this paper, the widely used benchmark Delft catamaran model 372 is chosen for a comparison with the proposed planing cata- maran. The Delft catamaran is a typical high-speed catamaran form with transom sterns. The Delft catamaran model 372 was designed at Delft University of Technology and the experimental work in cam water was conducted in the Delft Ship Hydrodynamic Laboratory (Van’t Veer, 1998). The experiments in calm water include sinkage, trim angle, and resistance at constant speeds ranging from Fn =0.18–0.75. The main particulars of the Delft catamaran model 372 are summarized in Table 6.
The mesh of the Delft catamaran model 372 simulated by the CFD method is depicted in Fig. 14.
The resistance characteristics of the Delft catamaran obtained by the present study are validated by comparison with the available experi- ments under the same experimental conditions. The trim angle is posi- tive when the bow moves up, while the sinkage is negative when the ship moves down the calm water at rest (Castiglione et al., 2011). The comparison of present numerical results and experimental results (Van’t Veer, 1998) of the Delft catamaran model, as well as other numerical results (Castiglione et al., 2011; Dogrul et al., 2021), are shown in Fig. 15. Castiglione et al. (2011) used an unsteady RANS code CFDSHIP-Iowa with the level-set method to simulate the flow around the Delft catamaran. Dogrul et al. (2021) investigated the interference factor in motions and added resistance for Delft catamaran model 372 by the commercial CFD code Star-CCM+. It can be found that the sinkage, trim angle, and resistance of the present study are in good agreement with the experimental data and other numerical results in a wide range of Froude numbers. It indicates that the present study has acceptable accuracy in predicting the resistance of the Delft catamaran model in calm water.
It should be noted that the Delft catamaran model 372 is introduced only for the comparison of resistance performance between the pro- posed planing catamaran model and the conventional catamaran hull form. Therefore, the hydrodynamic analysis of the Delft catamaran in Fig. 9. The comparison of numerical and experimental resistance.
Fig. 10.The results of sinkage with Froude number.
Fig. 11.The results of trim angle with Froude number.
detail is not carried out in this paper. The comparison of resistance of the planing catamaran and the Delft catamaran model in a wide speed range (0.25 ≤Fn≤1.925) is shown in Fig. 16. Firstly, the resistance of the planing catamaran is larger than that of the Delft catamaran at low speeds (Fn ≤1.15). As the speed increases (Fn ≥1.2), the growth rate of resistance for the planing catamaran is relatively mild at high speeds.
However, the growth rate of resistance for the Delft catamaran is much steeper and the resistance of the Delft catamaran becomes larger than that of the planing catamaran at the high speeds (Fn ≥1.2). Those be- haviors of the comparison indicate that the proposed planing catamaran model has a better resistance performance than the Delft catamaran model 372 at planing speeds (Fn ≥1.2).
Fig. 12.The contours of pressure distributions by two turbulent models.
Fig. 13.The pressure distributions longitudinally by two turbulent models.
Table 6
Main particulars of the Delft catamaran model 372.
Parameter Symbol Value
Length overall (m) LOA 3.11
Length between perpendiculars (m) L 3.00
Beam overal (m) B 0.94
Hull separation (m) H 0.70
Draft (m) D 0.15
Displacement (kg) M 87.07
Longitudinal center of gravity (m) LCG 1.41
Vertical center of gravity (m) VCG 0.34
5. Results and discussion
5.1. Comparison between numerical and experimental results
The two-phase flow around the planing catamaran was simulated at speeds ranging from Fn =0.250 to Fn =1.925. The modes of motion can be divided into three regions: displacement, semi-planing (transition), and planing (Yousefi et al., 2013). In the planing mode, the weight is almost supported by the hydrodynamic lift generated by water and the aerodynamic lift generated by air, which is due to the presence of the tunnel. Comparisons of the numerical solution and experimental results in sinkage, trim angle, and resistance for two displacement cases are presented in Fig. 17. It can be seen that the numerical results are in good agreement with experimental data.
For the attitude, the trend of the curve in sinkage and trim angle is consistent with that measured in the model test. The trough value of sinkage occurs for all three cases at the speed of Fn =0.500. In this situation, the transom stern of planing catamaran sinks, and the sinkage becomes a negative value. And the numerical results seem larger than the sinkage obtained by experiments at high speed. The average nu- merical error in sinkage is 10.3%, which is acceptable in the CFD simulation. The speed of the peak trim is predicted accurately by the present numerical method, but the numerical method underestimates the peak value compared with experiments. The average deviation in trim angle is 8.9%.
For the total resistance, the numerical results are in good agreement with experimental data at low speed (Fn ≤ 1.0). With the speed increasing, a small deviation occurs. And the deviation increases with the speed. The average numerical error in resistance with correction of the whisker spray drag is 4.5%. It can be seen from Fig. 17 that the numerical results incorporated with the whisker spray drag (WSD) remarkably improve the accuracy of numerical resistance.
The attitudes and resistance simulated by numerical results are in good accordance with the experimental results, which indicates that the current numerical solution can be implemented for a comprehensive investigation of the characteristics of the planing catamaran.
5.2. Pressure distribution on the tunnel
The pressure distribution on the tunnel surface of three cases at different Froude numbers is shown in Fig. 18. There are some positive and negative pressure values acting on the tunnel. As shown in Fig. 18, the position of the peak value moves to astern with the speed increase.
At the same speed, the peak value of pressure for M =202.9 kg is the largest. This behavior has a relationship with the internal flow of the
Fig. 14.The mesh of the Delft catamaran model 372.
Fig. 15.The comparison of numerical results and experimental results of the Delft catamaran model: (a) sinkage; (b) trim angle; (c) resistance.
Fig. 16. The resistance of the planing catamaran and the Delft cata- maran model.
tunnel. The comparisons of volume fraction of air contours on the tunnel for three cases are presented in Fig. 19. It can be seen from Fig. 19 that the wetted surface of the tunnel decreases by increasing the speed. The tunnel is ventilated at all speeds, which implies that the lift force acting on the tunnel is mainly generated by aerodynamics.
According to the clear ventilation in the tunnel above, the relation- ship between the internal flow of the tunnel and pressure on the tunnel surfaces is should be investigated deeply. The comparison of tunnel profile and wave cut along the centerline (Y/L =0) for M =159.5 kg is illustrated in Fig. 20. As the speed increases, the position of the tunnel
changes with the attitudes of the hull (Jiang et al., 2017). As Fig. 20 shows, the distance between the wave profile and tunnel centerline gradually decreases and then becomes a stable value in the direction from the bow to the transom stern. And the position of the start of the stable value moves astern as the speed increases, which indicates that the ventilated tunnel region becomes wider. It’s noted that the tunnel roof has a certain distance with wave profile at all speeds for the case of M =159.5 kg. In other words, the tunnel of planing catamaran can be considered to advance on an air cushion and the aerodynamic forces become dominant in this situation.
Fig. 17.Comparisons of numerical and experimental results for two cases.
Fig. 18.Comparison of pressure distribution on the tunnel for three cases.
The pressure distribution on the tunnel has a direct relationship with the internal flow of the tunnel (Jiang et al., 2017). Comparisons of pressure distribution and tunnel-wave distance along the centerline (Y/L =0) for three cases are shown in Fig. 21. ZH denotes the position of the tunnel roof and ZW represents the wave profile, respectively. The value of (ZH-ZW)/B is larger than zero in all cases, which implies that the tunnel is ventilated completely for the case of M =202.9 kg, 159.5 kg, and 127.4 kg. The tunnel-wave distance decreases and then becomes a stable value from the bow to the transom stern. The start position of the stable value is similar to the stagnation point of water flow, which is corresponding to the position of the peak value of pressure. The position of the peak value moves astern and the value decreases with the speed increasing under the same mass. Besides, the position of maximum value shifts to stern and the value decreases with the mass decreases under the same speed.
5.3. Streamline and transom wake flow
The streamline of the air and water phases could be used to track the flow pattern around the planing catamaran. The volume fraction of air of
streamline around the hull at Fn =1.013 for M =159.5 kg is displayed in Fig. 22. In this study, the streamlines are computed using forward integration mode. As can be seen from Fig. 22, the complex flow occurs near the spray region and separation of water. A mixed vortex flow of the air and water is generated near the entrance of the tunnel. The air streamlines above the deck detach from the hull continuously and are twisted toward astern (Khazaee et al., 2019). Khazaee et al. (2019) divided the local flow pattern around the prismatic planing hull into three portions. In this study, the behaviors of local fluid flow near the planing catamaran hull (see Fig. 22) can be divided into four regions due to the presence of the tunnel.
1) The air streamlines are located above the deck of a planing cata- maran hull, which detaches from the deck.
2) The water flow under the side-bottom and behind the transom stern.
3) The air flow above the free surface, which can be defined into three parts according to the position. The first part separates from the spray zone and breaks away from the sides. The second part sepa- rates from the bow and continues to the transom with a twisting form. The last part flows along the side of the hull and leaves the hull at the transom stern.
4) The group of air streams under the tunnel of the hull, can support the aerodynamic lift force and reduce the frictional resistance.
Transom stern is widely used for high-speed hulls, which can reduce the resistance (Haase et al., 2016). It’s necessary and useful to study the characteristics of stern waves for stern shape optimization (Duy et al., 2017). To investigate the transom wake flow of the planing catamaran, the sketch of the wave profile along the centerline and definition of parameters are shown in Fig. 23. The dry transom causes a hollow behind the transom stern centerline at high speed. And the flow reaches a minimum value and then rises to a rooster tail (Lugni et al., 2004). As shown in Fig. 23, the presence of the tunnel causes three rooster tails behind the transom stern, in correspondence to the two demi-hull transom sterns and the centerline, respectively. The behavior of wake flow behind the planing catamaran is quite different from that of the Fig. 19.Comparison of VOF contours on the tunnel for three cases.
Fig. 20.The comparison of tunnel profile and wave cut for M =159.5 kg.
planing monohull. The wave profile along the centerline (Y/L =0) is only chosen to compare and discuss since the positions of rooster tails behind demi-hull transom sterns change with speed. The length of the hollow ΔL is defined as the longitudinal distance between the stern position and the location where the flow reaches zero firstly (Song et al., 2020). The height of rooster tail HRT could be defined as the vertical distance between the maximum value of stern wave and calm water
(seen Fig. 23).
The comparisons of wave profiles along the transom stern centerline at different speeds are displayed in Fig. 24. The wave profiles of three displacements have obvious differences near the transom stern (-X/
L =0.0). And wave profiles become similar when the position -X/L is larger than the location of the rooster tail, especially at high speed.
Furthermore, the length of the hollow and the height of the rooster tail at Fig. 21.Comparison of pressure distribution and tunnel-wave distance longitudinally.
Fig. 22.The volume fraction of streamline at Fn =1.013 for M =159.5 kg.
Fig. 23.The sketch of wave profile along the centerline (left) and definition of hollow length and height of rooster tail (right).
different speeds are shown in Fig. 25. It can be found that the length of the hollow increases with the speed. And the hollow length is not particularly affected by the displacement except for high speed (Fn ≥ 1.521). The height of the rooster tail shows an increment with the Froude number in the range of Fn ≤1.013. And then the value of the roster tail height has little change with speed increasing. The behaviors of the planing catamaran are quite different from those of the monohull with a transom stern (Wang et al., 2022). The rooster tail height of three displacements shows a similar tendency, which implies that the height of the rooster tail is slightly affected by the displacement.
5.4. Lift distribution on the tunnel
The running attitude of the planing catamaran is directly affected by the lift distribution. To investigate the lift distribution on the tunnel, the tunnel is divided into 20 sections along the longitudinal direction (Sun et al., 2020). The tunnel lift can be obtained by pressure integration and lift distributions on the tunnel at various speeds for three cases are plotted in Fig. 26- Fig. 28. The hump of the lift curve moves astern with the increase of speed (see Fig. 27). The comparison for different displacement at the same speed shows that the maximum lift on the tunnel decrease with the increase of the displacement, and the position of peak value moves to the stern in the meanwhile. It’s noted that the variation of tunnel lift distribution is closely related to the relative
position between the wave profile and tunnel roof, which is consistent with the above analysis (see Fig. 21). As the tunnel is fully ventilated at all speeds, the aerodynamic lift has been dominant for the tunnel lift.
Fig. 29 shows the proportions of lift generated by the tunnel for three cases. It can be found that the tunnel contributes the maximum lift about 26% of ship weight at the speed of Fn = 1.93 (M = 202.9 kg). The proportion of maximum lift generated by the tunnel is 16.8% for M =159.5 kg and 9.3% for M =127.4 kg, respectively. In other words, the majority of lift to balance the gravity is provided by the other parts except for the tunnel in the whole speed range. The tunnel lift increases Fig. 24.Comparison of wave profile along the transom stern centerline.
Fig. 25.Comparison of hollow length (left) and height of rooster tail (right).
Fig. 26.Lift distribution on the tunnel for M =202.9 kg.
with the speed increase for M =202.9 kg and M =159.5 kg. For the same speed, the tunnel lift decreases as the mass decreases. The pro- portions of resistance generated by the tunnel for three cases are illus- trated in Fig. 30. As shown in Fig. 30, the maximum proportion of resistance generated by the tunnel is 17%, 14.25%, and 12% for M =202.9 kg, 159.5 kg, 127.4 kg, respectively.
Tunnel efficiency was introduced, which is defined that the lift to resistance ratio of the tunnel is considered as a fraction of the total lift to resistance ratio (Roshan et al., 2020). The expression is shown as follows:
η=LTS/RTS
LT/R (4)
where LTS and RTS denote the lift and resistance generated by the tunnel, respectively. And LT and R denote the lift and resistance generated by the planing catamaran, respectively.
The comparisons of tunnel efficiency at different speeds for three cases are presented in Fig. 31. The tunnel efficiency for M =202.9 kg increases with the speed. For the cases of M = 159.5 kg and M =127.4 kg, when the speed increases, the tunnel efficiency increases gradually and then decreases. Those behaviors are different from the case of M =202.9 kg, which indicates that this planing catamaran has more advantages in terms of larger displacement.
5.5. Components of tunnel lift
The tunnel of planing catamaran is a special part to improve the resistance performance compared with planing monohull. In this study, the rest parts include the main hull, deck, and transom stern only except for the tunnel. Fig. 32 shows the proportions of resistance generated by the tunnel and rest parts at three displacements. It can be found that resistance generated by rest parts for M =202.9 kg is a bit larger than that for M =127.4 kg when Fn ≤1.013, which is the opposite when Fn ≥ 1.267. The resistance generated by the tunnel increases with the displacement. But the value of RTS for three different displacements becomes similar at high speed.
To explore the components of lift and resistance generated by the tunnel, the lift and resistance are separately obtained for the hydrody- namic and aerodynamic parts. The resistance and lift generated by hull parts S can be written in the form as follows:
RS= −
∫
S
pcos(p,x)dS−
∫
S
τcos(τ,x)dS (5)
LS=
∫
S
pcos(p,z)dS+
∫
S
τcos(τ,z)dS (6)
where p denotes the pressure and τ is the shear stress. And x represents the direction of horizontal motion of the planing catamaran and z rep- resents the upward direction, respectively. S denotes the wetted area of the hull parts, which is divided into the water and air condition in this study.
The Volume of Fluid (VOF) method is adopted to trace the free surface, which assumes that all phases in each control volume share velocity, pressure, and other field functions. The volume fraction of air has a value 0≤α≤1 that represents the volumetric ratio of fluids (seen Fig. 33). In the threshold value 0≤α<0.5, it’s assumed that the hull part is occupied with the water phase. And in the threshold value 0.5<α≤1.0, it’s assumed that the hull part is occupied with the air phase. For these two parts, the hydrodynamic and aerodynamic lift can be computed by area integral separately.
Fig. 27.Lift distribution on the tunnel for M =159.5 kg.
Fig. 28.Lift distribution on the tunnel for M =127.4 kg.
Fig. 29.The proportions of lift generated by the tunnel for three cases.
Fig. 30. The proportions of resistance generated by the tunnel for three cases. Fig. 31.The comparisons of tunnel efficiency for three cases.
The tunnel lift generated by the air for three cases at various speeds is shown in Fig. 34. The results indicate that the aerodynamic lift of the tunnel is dominant at all speeds. As the comparison shown in Fig. 34, the aerodynamic lift generated by the tunnel accounts for almost 100% of the lift of the tunnel. In other words, the tunnel lift of this planing cat- amaran for three cases is almost generated by the air when the Froude number Fn ≥0.76. Due to complete ventilation in the tunnel region, the air flows through the tunnel and then the aerodynamic force is gener- ated to act on the wall of the tunnel.
To find out the resistance components of the tunnel, Fig. 35 shows the resistance components of the tunnel for three cases. The tunnel resistance generated by air for M =202.9 kg is over 50% of the total tunnel resistance when Fn ≥0.50. This implies that tunnel resistance generated by air becomes dominant at high speeds when M =202.9 kg.
The proportion of maximum tunnel resistance generated by air is 77.5%, 69.3%, and 74.9% for the M = 202.9 kg, M = 159.5 kg, and M =127.4 kg, respectively. The resistance to lift ratio of the tunnel generated by the air is also depicted in Fig. 36. Compared with the resistance to lift ratio of the hull (as shown in Fig. 17), the RTS-air/LTS-air
is smaller than the value of R/ Δ at the same speed. It’s to say that the resistance to lift ratio of the tunnel generated by the air is smaller than that of the hull. The tunnel produces relatively smaller air resistance while contributing the same lift of the hull. Those behaviors indicate
that the aerodynamic lift of the tunnel plays an important role in the excellent resistance characteristics of the planing catamaran.
6. Conclusion
In this paper, a combined experimental and numerical investigation of the planing catamaran hull for three displacements in calm water is carried out. The results of resistance, sinkage, and trim angle have been measured for three different displacements in the towing tank. The three-dimensional incompressible viscous unsteady RANS method incorporated dynamic overset mesh technology has been utilized to simulate the motion of the planing catamaran in calm water. According to numerical analyses, the following conclusions can be drawn.
The same test conditions are reproduced with the numerical simu- lation by using two different turbulent models, i.e. SST k-ω and Realiz- able k-ε turbulence models. The sinkage and trim angle obtained by numerical CFD simulation compared with experiments show good agreement, which implies that there is no significant difference between SST k-ω and Realizable k-ε turbulence models for numerical simulation of high-speed planing hull. The comparison of resistance implies that the SST k-ω turbulence model can give slightly closer results to the experi- ments in comparison with the Realizable k-ε turbulence model, espe- cially at high speeds.
The average errors of computed sinkage and trim angle compared with measured results are 10.3% and 8.9%, respectively. The sinkage and trim angle of CFD simulation are in good agreement with experi- mental data at different displacements and speeds, which shows the capability of the CFD method in attitudes prediction of planing cata- maran. The average error of resistance including the whisker spray drag is 4.5%, which implies that the CFD method incorporated with Savit- sky’s formula can improve the accuracy of numerical resistance of planing catamaran, especially at high speeds.
The pressure distribution on the tunnel has a relationship with the internal flow of the tunnel. For the fixed mass, the position of peak value moves to astern with the speed increase. For the fixed speed, the peak value of pressure for M =202.9 kg is the largest. The tunnel-wave dis- tance (ZH-ZW) was introduced and the value decreases and then becomes a stable value from the bow to the transom stern. The start position of Fig. 32.The proportions of resistance generated by tunnel and rest parts.
Fig. 33.The sketch of tunnel parts in the water and air phase.
Fig. 34.Lift components of tunnel for three cases.
Fig. 35.Resistance components of tunnel for three cases.
Fig. 36.The resistance to lift ratio of tunnel for three cases.
the stable value is corresponding to the position of the peak value of pressure on the tunnel. The local flow around the planing catamaran hull can be divided into four regions. And the streamline of fluid flow has another portion, air streams under the tunnel of the hull due to the presence of tunnel, compared with conventional planing monohull. The height of the rooster tail behind the transom stern is slightly affected by the displacement.
The hump of the lift force moves astern with the increase of speed, which is consistent with pressure distribution. The tunnel region of this planing catamaran is ventilated at all speeds, which illustrates that the source of tunnel lift mainly stems from the aerodynamic force. The numerical results evidenced that the tunnel lift plays an important role in total lift force since the tunnel contributes the maximum lift about 26% of ship weight at the speed of Fn =1.93 (M =202.9 kg). For the cases of M =202.9 kg, the tunnel efficiency increases with the speed.
Since complete ventilation occurs in the tunnel region, the aerodynamic lift generated by the tunnel accounts for almost 100% of the lift of the tunnel of this planing catamaran when the Froude number Fn ≥ 0.76.
This behavior implies that the aerodynamic lift force plays the dominant role in the total lift of the tunnel.
An investigation of the motion and added resistance of planing cat- amaran in waves will be carried out in future works.
CRediT authorship contribution statement
Hui Wang: Writing – original draft, Investigation, Methodology, Software. Renchuan Zhu: Conceptualization, Writing – review & edit- ing. Le Zha: Data processing. Mengxiao Gu: Validation.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Data availability
The authors do not have permission to share data.
Acknowledgments
This work was supported by the YEQISUN Joint Funds of the Na- tional Natural Science Foundation of China (U2141228).
Appendix. Savitsky method
Savitsky (1964) carried out a comprehensive contribution to understanding and modeling of planing crafts and proposed a semi-empirical regression formula to evaluate the hydrodynamic performance of the planing hull, which is named Savitsky’s method. It’s noted that the whisker spray effect on the total resistance has been taken into consideration in the present study. The symbols used in Savitsky’s method are shown in Fig. A.
Fig. A. Symbols used in a planing hull analysis by Savitsky method.
The center of pressure and lift coefficient is defined as follows:
lp
λwB=0.75− 1 5.21Fn2B/
λ2w+2.39 (A.1)
CLβ=CL0− 0.0065βCL00.6 (A.2)
CL0=τ1.1deg(0.0120λ0.5w +0.0055λ2.5w / F2nB)
(A.3)
where lp represents the center of hydrodynamic pressure, λw is the aspect ratio and B is the breadth of the hull, FnB is the Froude number based on breadth. β is the deadrise angle, CLβ is the lift coefficient when the deadrise angle is equal to β, CL0 represents the lift coefficient when β=0, τdeg is the dynamic trim angle. An iterative solution to solve the above equations is carried out to obtain the value of λw, CLβ and τdeg.
The vertical position of gravity center above the mean water surface when the ship has a constant trim angle can be written as:
ZWL=VCG ⋅ cosτ− (LK− LCG)sinτ (A.4)
Then the value of sinkage can be obtained by
sinkage=ZWL− (VCG− D) (A.5)
where VCG and LCG denote the center of gravity along the vertical and longitudinal directions, respectively. Lk represents the wetted length of the keel and D is the draft of the hull.
The total resistance of the planing hull is given as follows:
Rt=RP+Rf+RS (A.6)
where RP is pressure-induced resistance, Rf is friction resistance which is accorded to the 1957-ITTC formula, RS represents the whisker spray resistance.
RP=Mgtanθ (A.7)
Rf=0.5( Cf+ΔCf
)ρU2S (A.8)
A high-speed planing catamaran has obvious water spray at semi-planing and planing regions, which causes the whisker spray drag. Whisker spray drag of high-speed planing hull can be expressed as a function of deadrise angle, trim angle, and speed (Savitsky et al., 2007). In this study, the result of whisker spray drag (WSD) is incorporated into the CFD results to improve the numerical simulation when Fn ≥0.761. It’s noted that the trim angle in this function is replaced by the CFD results. The function of the whisker spray drag can be defined as follows:
RS=0.5ρU2 B2
4 sin 2αcosβCf (A.9)
tanα=πtanτ
2 tanβ (A.10)
where β represents the deadrise angle of stern, α denotes the angle between keel and stagnation line of spray in the horizontal plane.
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