Numerical investigation into the interaction of resistance components for a series 60 catamaran

Ocean Engineering 146 (2017) 151–169

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Numerical investigation into the interaction of resistance components for a series 60 catamaran Andrea Farkas, Nastia Degiuli *, Ivana Martic University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Croatia

A R T I C L E I N F O

A B S T R A C T

Keywords: Catamaran Interference resistance CFD RANS Separation

Numerical study of the flow around S60 catamaran and S60 monohull, as well as an investigation into interference phenomenon are performed within this paper. Numerical simulations of the free surface and double body viscous flow are carried out for six Froude numbers in the range 0.3–0.55, for four separation ratios in the range 0.226–0.4696 and for monohull. Verification and validation of the obtained results are provided. Validation is performed by comparison with available experimental data and satisfactory agreement is obtained. Firstly, catamaran interference resistance is studied through interference factor based on the total resistance of catamaran and monohull. Afterwards, the interference resistance is investigated through a procedure based on the resistance decomposition, in order to determine the viscous and wave interference effect on the total resistance of catamaran. The form factor for monohull is found to be different than the form factor for catamaran with identical demihulls. Also, form factor for catamaran is found to be in correlation with the separation. Analysing the flow around demihulls, it has been established that the strength of cross flow is correlated to interference resistance. Finally, benefits of CFD based on viscous flow as a tool in catamaran preliminary design are highlighted.

1. Introduction Catamaran and other multihull configurations have a better performance regarding speed, resistance, manoeuvrability and transverse stability compared to monohull configurations. Thus, for the last few decades significant growth of interest for multihull vessels in civil, recreational and military fields can be noticed. Consequently, many theoretical, numerical and experimental investigations concerning multihull vessels have recently been made (Zaghi et al., 2010). Despite that, catamaran resistance prediction still has a degree of uncertainty (Sahoo et al., 2007). Spacing between hulls (separation) represents one of the most important parameters in catamaran design because of the hydrodynamic interaction between hulls operating in a proximity to each other. Therefore, this parameter must be taken into account at the design stage (Bari and Matveev, 2016). The wave field generated by a multihull vessel is not a simple superposition of the wave fields generated by each hull, if the hulls are sufficiently close to each other (Faltinsen, 2005). These wave fields usually strongly interfere and therefore can cause either favourable or unfavourable effects (Souto-Iglesias et al., 2012). The resulting wave pattern in the inner region is very complex and strongly affects hydrodynamic characteristics of a catamaran. Wake angle of catamaran wave pattern is narrower than Kelvin cusp angle (He

et al., 2016). Also, wavelengths of the highest wave generated by catamaran and monohull vary considerably. The wavelength of the highest wave generated by a catamaran is approximately equal to the separation for values of Froude number (Fn) above 1, while the wavelength of the highest wave created by fast monohull ship is nearly equal to the beam of the ship for Fn values above 5 (Ma et al., 2016). The resistance of a catamaran is different than double resistance of the monohull because of the appearance of interference resistance (Jamaluddin et al., 2012). Interference resistance can be divided into two components, i.e. viscous and wave interference resistance. Asymmetric flow around demihulls is the main cause of the viscous interference, since it changes formation of the boundary layer and the development of vortices. Wave interference is caused by the interference between wave system of each demihull (Insel and Molland, 1992). Many authors have investigated the influence of the separation on catamaran resistance and the interference resistance. While most of the authors have investigated catamaran resistance for catamaran models, Haase et al. (2016) developed a novel full-scale resistance prediction method for large medium-speed catamarans based on Computational Fluid Dynamics (CFD). The method assumes that pressure drag is independent of Reynolds number (Rn). The estimation of the full-scale resistance is derived from simulations at full-scale Rn. These simulations are made for model

* Corresponding author. E-mail address: [email protected] (N. Degiuli). https://doi.org/10.1016/j.oceaneng.2017.09.043 Received 30 May 2017; Received in revised form 21 August 2017; Accepted 24 September 2017 0029-8018/© 2017 Elsevier Ltd. All rights reserved.

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Ocean Engineering 146 (2017) 151–169

non-parallel hulls. Authors indicated that for Fn below 0.8 catamarans with non-parallel hulls have larger total resistance, but for Fn above 0.8 these catamarans have smaller total resistance than catamarans with parallel hulls. Castiglione et al. (2014) showed that interference effects are more significant in shallow water than in deep water. He et al. (2015) demonstrated the applicability of Unsteady Reynolds Averaged Navier-Stokes (URANS) solver for the interference problems including the effects of sinkage and trim. They concluded that the main cause of the deviations in resistance, sinkage and trim using URANS solver is caused by the grid quality, but these numerical errors were found acceptable. This paper contributes to the previous studies with the main goal to evaluate capabilities of CFD in numerical assessment of the interference resistance. Even though Insel and Molland (1992) divided interference resistance into two components, most of the authors nowadays investigate interference resistance through interference factor. For determination of interference factor, they use total resistance or wave resistance. Within this paper, influence of Fn and separation (s) on both components of interference resistance is investigated utilizing CFD. Benefits of CFD for assessment of interference resistance are highlighted. Simulations of viscous flow around S60 monohull and four S60 catamaran configurations are performed for six Fn values in the range 0.3
Fn
0.55. Catamaran configurations differ in the separation and numerical simulations are performed for four separation ratios in the range 0.226
s/L
0.4696. In order to determine both components of the interference resistance, free surface simulations as well as double body simulations are performed. Firstly, verification of numerically obtained results is made for simulation of free surface flow around the monohull and the narrowest catamaran configuration. After the verification procedure, the obtained results utilizing fine mesh are validated against available experimental data from literature (Souto-Iglesias et al., 2012). After verification and validation procedure for the free surface simulations, the double body simulations are performed. These simulations are carried out utilizing grid that has comparable density to fine mesh used in free surface simulations. The obtained numerical results are used to gain better insight into interference phenomena and the flow between demihulls.

speed, model dimensions and the initial model mesh with full-scale Rn obtained by changing the kinematic viscosity of the water. Souto-Iglesias et al. (2007) carried out experimental investigation in the towing tank for one commercial monohull and catamaran model to investigate the influence of the separation on the interference resistance. In order to ascertain the accuracy of the conducted measurements, an uncertainty analysis was included. Results showed that catamarans with greater separation have wider range of favourable interference effects. This investigation was revisited using the same catamaran model (Souto-Iglesias et al., 2012). The effect of fixed and free model condition on the interference resistance was investigated. Also, extensive towing tank tests were carried out to investigate interference resistance for fixed and free S60 monohull and S60 catamaran model for different separations. The obtained results were compared with results published in (Yeung et al., 2004), where the authors proposed numerical procedure that is capable of rapid and accurate evaluation of the wave resistance of any monohull or combination of demihulls. Results showed that free model provided more extreme cases than fixed model for both favourable and unfavourable interference regimes. Broglia et al. (2014) found that the interference effects are more significant for narrower catamaran configurations and at intermediate values of Fn. They claimed that for smaller Fn, the wave elevation is too small and it does not affect the catamaran total resistance significantly. At higher values of Fn, wave system of each hull diverges and therefore superposition between them is considerably reduced. Thus, a catamaran starts to behave like a combination of almost non-interacting vessels. Authors also found that interference effects are strongly connected with sinkage and trim of a catamaran model. As a result of advancement in computer science and numerical computation methods, improvement in accuracy and efficiency of CFD methods can be noticed. Consequently, an optimal choice to investigate hydrodynamic characteristics of catamaran becomes the combination of towing tank tests and CFD methods (Zha et al., 2015). Zhang et al. (2015) performed a validation of the Neuman-Michell theory. They highlighted the importance and robustness of this method for catamaran design. Reynolds averaged Navier-Stokes equations (RANSE) methods proved to be the most suitable for resistance prediction of medium-speed catamarans (Haase et al., 2013). Zha et al. (2015) carried out numerical simulations of viscous flow around catamaran model using the in-house RANSE solver naoe-FOAM-SJTU. Authors concluded that naoe-FOAM-SJTU is reliable software for solving general hydrodynamic problems with good efficiency and that is more flexible and extensible than commercial software. Broglia et al. (2011) carried out numerical simulations of viscous flow around catamaran and monohull models with the aim of studying the interference effects and their relationship with Rn. Catamaran and monohull models were fixed at the dynamic positions taken from the experiments. It was shown that dependence between interference effects and Rn is weak. In order to investigate dependence between interference effects and separation, Zaghi et al. (2011) performed extensive experimental and numerical investigations. Results of these investigations showed that interference, as well as maximum resistance coefficient, is higher for catamaran configurations with smaller separation. Also, maximum resistance coefficient for narrower catamaran configurations occurred at higher Fn values. Sarles et al. (2011) showed that section shape of demihull significantly affects the interference. Yengejeh et al. (2016) used RANSE solver to perform various viscous flow simulations around asymmetric planing hulls. Numerical simulations were performed for different separations, trim angles and Fn. Analysis of the obtained results showed that catamaran configuration has a significantly reduced wetted surface area than corresponding monohull having the same displacement. Utama et al. (2012) pointed out that interference effects due to separation and stagger are larger for symmetric catamaran hulls than for asymmetric ones. While most of the authors focused their research on catamaran configurations with the parallel hulls, Ebrahimi et al. (2014) studied a catamaran with

2. Governing equations In numerical simulations of incompressible viscous flow, RANSE along with averaged continuity equation are used as governing equations. RANSE and averaged continuity equation are obtained by time averaging of Navier-Stokes equations and continuity equation. RANSE and averaged continuity equation are given as follows (Ferziger and Peric, 2012):

 ∂ðρui Þ ∂
∂p ∂τij þ ρui uj þ ρu0 i u0 j ¼  þ ∂t ∂xj ∂xi ∂xj

(1)

∂ðρui Þ ¼0 ∂xi

(2)

where ρ is the fluid density, ui is the averaged Cartesian components of the velocity vector, ρu0 i u0 j is the Reynolds stress tensor and p is the mean pressure. The mean viscous stress tensor is defined with following equation:

 τij ¼ μ

∂ui ∂uj þ ∂xj ∂xi

 (3)

where μ is the dynamic viscosity. Equations (1) and (2) represent unclosed set of equations. In order to close this set, turbulence model is introduced. Eddy-viscosity model for the Reynolds stress tensor is based on a fact that effect of the turbulence

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  h   d μ ∂ε ∫ ρε dV þ ∫ ρε ui  ugi ⋅dai ¼ ∫ μ þ t ⋅dai þ ∫ fc Cε1 Sε dt V σ ε ∂xi A A V ε ε pffiffiffiffiffi Cε2 ρðε þ ðCε1 Cε3 Gb Þ  k k þ νε i  ε0 Þ þ Sε dV

Table 1 The main dimensions and particulars of the S60 monohull model. Particulars

Value

Length between perpendiculars (Lpp) Beam (B) Draft (T) Wetted surface (S) Displacement (Δ) Block coefficient (CB)

2.5 m 0.333 m 0.133 m 1.062 m2 65.7 kg 0.6

(8) where V is the cell volume, ui is the velocity vector, ugi is the grid velocity vector, ai is the face area vector, σ k and σ ε are turbulent Schmidt numbers (for this model: σ k ¼ 1 and σ ε ¼ 1.2), fc is the curvature correction factor, Gk is the turbulent production term as defined in (Launder and Spalding, 1974), Gb is the production term due to the buoyancy as defined in (Launder and Spalding, 1974), ε0 is the ambient turbulence value in the source terms that counteracts turbulence decay, ϒ M is the dilatation dissipation as defined in (Launder and Spalding, 1974), Sk and Sε are user specified source terms, Cε1 , Cε2 and Cε3 are model coefficients, S is the modulus of the mean strain rate tensor and ν is the kinematic viscosity. Within this form of k-ε turbulence model, Cμ is defined as a function of mean flow as follows (STAR-CCMþUser Guide, 2017):

Table 2 Catamaran configurations. Catamaran configuration

s, m

s/L

C1 C2 C3 C4

0.565 0.768 0.971 1.174

0.2260 0.3072 0.3884 0.4696

can be described as increased viscosity and it reads:

ρu0 i u0 j ¼ μt

  ∂ui ∂uj 2  ρδij k þ 3 ∂xj ∂xi

(4)

Cμ ¼

where k is the turbulent kinetic energy defined as follows:

1 k ¼ u0 i u0 i 2

(5)

k2 ε

(9)

where A0 ¼ 4, AS is given with:

AS ¼

k-ε turbulence model with wall function is used to describe turbulence effects. Within this model, eddy viscosity is described with equation:

μt ¼ ρCμ

1 A0 þ AS U ð*Þ kε

(6)

where Cμ is the critical coefficient of model and ε is the dissipation rate of turbulent kinetic energy. In this paper, Realizable k-ε Two-Layer (RKE2L) turbulence model was used. The transport equations for the Realizable k-ε model are defined as follows (STAR-CCMþUser Guide, 2017):

pffiffiffi 6cos ϕ

(10)


pffiffiffi  1 ϕ ¼ arccos 6 W 3

(11)

Sij Sjk Ski W ¼ pffiffiffiffiffiffiffiffiffi 3 Sij Sij

(12)

and U ð*Þ is defined according to equation:

U ð*Þ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sij ⋅Sij þ Wij ⋅Wij

(13)

The strain rate tensor is expressed as:

    d μ ∂k ∫ ρk dV þ ∫ ρk ui  ugi ⋅dai ¼ ∫ μ þ t ⋅dai þ ∫ ½fc Gk þ Gb dt V ∂x σ k i A A V

Sij ¼

 ρððε  ε0 Þ þ ϒ M Þ þ Sk  dV

  1 ∂ui ∂uTi þ 2 ∂xj ∂xj

The rotation rate tensor reads:

(7)

Fig. 1. The geometry of catamaran configuration C1. 153

(14)

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Fig. 2. Front view cross section of the volume mesh for monohull (left upper) and C1 (left lower) and top view cross section of the volume mesh for monohull (right upper) and C1 (right lower).

Fig. 3. Mesh refinement around the bow (left) and the stern (right).

Fig. 4. Computed y þ distribution in the first cell next to the wall for monohull.

Wij ¼

  1 ∂ui ∂uTi  2 ∂xj ∂xj

and it is determined according to averaged continuity equation. Physical properties depend on the presence of the fluid in a particular cell and are calculated according to the following equation:

(15)

RKE2L turbulence model can work with both low-Rn type of mesh and with wall function type of mesh. Wall function type of mesh represents a mesh in boundary layer which is set in order to obtain value of y þ parameter in the first cell above 30, while for low-Rn type of mesh, a mesh in boundary layer is set in order to obtain value of yþ parameter in the first cell around 1. The height of the first cell is set according to this condition (STAR-CCMþUser Guide, 2017). In order to track sharp interfaces and locate a free surface in free surface simulations, Volume Of Fluid (VOF) method combined with High Resolution Interface Capturing scheme (HRIC) is used. VOF method is based on the volume fraction of i-th fluid (αi ). This parameter represents volume fraction occupied by i-th fluid inside an arbitrary closed volume

X¼

X

Xi αi

(16)

i

Finite Volume Method (FVM) is used to represent and evaluate partial differential equations in the form of algebraic equations. The solution domain is subdivided into finite number of control volumes by grid. FVM allows using either structured or unstructured grid. Since unstructured grid greatly simplifies grid generation for complex geometries, it was used within this research. More details about domain discretization are presented in subsection 3.2 of this paper.

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Fig. 5. Computational domain for monohull.

3. Computational model

catamaran configurations, which differ in separation, are listed in Table 2. The geometry of catamaran configuration C1 is shown in Fig. 1. The origin of the coordinate system for monohull is located at the intersection of the baseline and aft perpendicular. The origin of the coordinate system for catamaran is located at the intersection of the baseline and aft perpendicular of left demihull. x axis is directed towards the bow, y axis is directed to portside, and z axis is directed vertically upward, Fig. 1.

3.1. Catamaran and monohull models Numerical simulations were performed for S60 monohull and catamaran models. In order to validate numerically obtained results, comparison with experimental data available in the literature (Souto-Iglesias et al., 2012) is performed. Experiments were conducted for slightly modified S60 model, compared to the one defined as benchmark model for Tokyo 1994 CFD Workshop. Therefore, numerical simulations are performed for slightly modified S60 model as well. According to Souto-Iglesias et al. (2012), modifications were made because the benchmark geometry had too many surface patches with not enough quality matching. Furthermore, in order to cope with waves generated at higher Fn, vertical extension of the hull was made (Souto-Iglesias et al., 2012). The main dimensions and particulars of model are given in Table 1. The

3.2. Computational domain and numerical setup The computational domain is discretized with unstructured hexahedral mesh, which is created using STAR-CCMþ meshing tools: surface remesher, automatic surface repair, prism layer mesher and trimmed cell mesher. Firstly, S60 geometry composed of NURBS surfaces is triangulated. Only half of the computational domain is modelled, since

Fig. 6. Applied boundary conditions for free surface simulations (upper) and double body simulations (lower). 155

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monohull and catamaran models are symmetric with respect to the centreline plane. Therefore, only portside of S60 model is considered in the numerical simulations of viscous flow around monohull. For simulations of viscous flow around catamaran configurations only demihull, whose centreline is placed at half of the separation from the symmetry plane, is considered. After the triangulated surface is imported into STAR-CCMþ, surface remesher is used to improve the overall quality and to optimize triangulated surface for volume mesh generation. Surface remesher retriangulates the imported surface based on a target edge length and feature refinement that is based on curvature and surface proximity. Automatic surface repair is used to automatically correct geometric type problems that can appear after surface is remeshed. Hexahedral mesh is obtained using trimmed cell mesher. This meshing tool utilizes a template mesh, constructed from hexahedral cells, from which it cuts or trims the core mesh utilizing the input surface. Before the core mesh is created, subsurface is generated at specified prism layer thickness value. Prism layers are orthogonal prismatic cells next to wall surfaces or boundaries. A prism layer is defined in terms of its thickness, the number of cell layers within it, the size distribution of the layers and the function that is used to generate the distribution. The core mesh is created using this subsurface. In order to refine the mesh in the bow and stern region, close to the hull and in the free surface region, additional geometry parts are created and custom mesh controls are set within those parts. After the core mesh with refinements has been created, prism layer mesh is generated by extruding the cell faces from the core mesh to the initial surface. Refinement near free surface and refinement for capturing Kelvin wake can be seen in Fig. 2. In this figure, grids containing around 2 million cells, for both monohull and C1 are shown. Mesh refinements around the bow and stern for the double body simulation are shown in Fig. 3. To simplify remeshing within mesh sensitivity studies, mesh parameters are defined as relative values of the cell base size. Only prism layer thickness is set as absolute value, to keep the same value of yþ parameter in the first cell next to the wall for all meshes. yþ parameter is defined according to following equation:

yþ ¼

ρuτ y μ

(17)

where y is the distance from the wall and uτ is the shear velocity defined as follows:

uτ ¼

rffiffiffiffiffi τw ρ

(18)

where τw is the wall shear stress. To ensure that the first cell next to the wall lies within the logarithmic region of the boundary layer, the yþ value is kept above 30, i.e. prism layer thickness is set in order to keep yþ value above 30. Therefore, wall functions can be used and the nondimensional velocity profile can be calculated according to the following equation:

uþ ¼

u 1 ¼ lnðyþÞ þ B uτ κ

(19)

where u is the mean velocity parallel to the wall, κ is the Von Karman constant and B is the constant of integration. It is important to notice that wall functions are based on twodimensional flow, typically at zero pressure gradient. Their validity becomes less with increasing adverse pressure gradients, which are present near a ship stern. Thus, wall functions should be considered as a trade-off between accuracy and computational effort (ITTC, 2011). Demirel et al. (2017) have shown the applicability of the wall function approach to account for the roughness effects of AF coatings and biofouling on full-scale ship hulls. Khor and Xiao (2011) showed the influence of yþ values on the change of relation between lift and drag coefficient for NACA 4424 air foil and for DREA hull. Authors highlighted the

Fig. 7. Wave patterns obtained using time steps: T/100 (upper), T/200 (middle), T/ 400 (lower).

importance of the near-wall grid density for capturing the turbulence for DREA hull. Since large number of simulations was performed, wall

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stopped when the oscillation of the total resistance force vanished. Elimination of wave reflections at the domain boundaries can be achieved using several approaches. Nowadays, the most common are grid damping and damping layer approaches. Grid damping approach is based on progressively increasing the cell size towards the corresponding boundary where reflection is expected. Damping layer approach is based on a damping zone, which is set up next to the corresponding boundaries, where reflection is expected. Within this zone, momentum sinks are included in the governing equations, in order to damp the waves that propagate through the zone. Even though both approaches are used successfully, damping layer approach is more predictable regarding the damping quality. For sufficiently fine discretization, damping layer approach is independent on grid, time step, temporal and spatial discretization schemes, which is not the case for grid damping approach (Peric and Abdel-Maksoud, 2016). In this paper, reflection of VOF waves is prevented using damping layer approach. Within STAR-CCMþ, Choi and Yoon method for VOF waves damping is implemented (STAR-CCMþUser Guide, 2017). This method adds the source term in the z-momentum equation. Added source term introduces vertical resistance to the vertical velocity component uz and it is represented with following equations:

functions are used to enhance computational efficiency because computational resources were limited, as done in (Yengejeh et al., 2016; Choi et al., 2010). When wall functions are not used, prismatic cells near the wall can possess a high aspect ratio of order of 103 in order to assure yþ<1 (Yengejeh et al., 2016). Therefore, mesh around the wall should be refined, what results in greater cell number. Prismatic cells consist of six layers. Calculated yþ values in the first cell next to the wall, utilizing fine grid for monohull at Fn ¼ 0.45 are shown in Fig. 4. The location of the domain boundaries is important and it must be ensured that boundaries do not influence the flow solution, i.e. that no reflection is present (Date and Turnock, 1999). From the literature it can be seen that domain size varies (Tezdogan et al., 2015). In this paper, for free surface simulations, inlet, outlet and side boundaries are placed 2Lpp away from the model, while top and bottom boundaries are placed Lpp away from the model, Fig. 5. Since wave damping is applied at the inlet, side and outlet boundaries, as explained in detail later, no reflection is present and therefore boundaries do not influence the flow solution. Selected boundary conditions at domain boundaries can be seen in Fig. 6. These boundary conditions are used according to the (STAR-CCMþUser Guide, 2017). Due to symmetry, for double body simulations, only hull surface below waterline is considered. The double body model assumes that the immersed part of ship model is doubled with respect to the undisturbed flat free surface and thus the flow field is symmetrical. Therefore, streamlines lie on the symmetry plane, i.e. there is no wave formation and no wave-making resistance (Lin and Kouh, 2015.). Inlet, outlet and side boundary are placed 2Lpp away from the model, while bottom boundary is placed Lpp away from the keel line. For double body simulations, selected boundary conditions are shown in Fig. 6. A cell based FVM is used for discretization of the governing equations. Double body simulations are performed as steady state calculations. Therefore, unsteady term in RANSE is zero and there is no need for temporal discretization. Free surface simulations are performed as unsteady calculations, using implicit unsteady solver. Both free surface and double body simulations are modelled using segregated solver, which solves momentum equations in turn, one for each dimension. Rhie-Chow type of pressure velocity coupling combined with SIMPLE algorithm is used to obtain pressure and velocity fields. Convection terms in RANSE are discretized with a second order upwind scheme, while temporal discretization is carried out utilizing first order temporal scheme. Algebraic Multigrid (AMG) solver is used to solve the discrete linear system iteratively. In order to improve the convergence, the obtained solution in each time step was under relaxed by Gauss-Seidel relaxation scheme. Simulations are performed for fixed models and fluid velocity is imposed at inlet boundary. Free surface simulations are initialized with the initial velocity and pressure field, while double body simulations are initialized with imposed velocity. The simulations are stopped when the total resistance force became steady, i.e. when the oscillation amplitude near the converged value fell below 0.5% of the total resistance value. Since double body simulations converge faster, those simulations are

qdz ¼ ρðf1 þ f2 juz jÞ  κ¼

x  xsd xed  xsd

eκ  1 uz e1  1

(20)

n (21)

where f1 is the linear damping constant, f2 is the quadratic damping constant, n is the damping exponent, x is the wave propagation direction, xsd is the start and xed is the end coordinate of the damping layer. Thickness of the damping layer (damping zone) is represented as:

xd ¼ jxed  xsd j

(22)

VOF wave damping is applied to inlet, outlet and side boundary. Damping constants are set as follows: f1 ¼ 10, f2 ¼ 10, n ¼ 2, while xd is set using dvar function:

dvar  Lpp þ Lpp cos2


π t  2 10T

(23)

where T is the period defined as ratio between Lpp and velocity and t is the physical time. Since domain boundaries are placed 2Lpp away from the model, damping zone at the beginning of the simulation is 2Lpp. As physical time passes, xd decreases up to 10T. From then on, xd is set to Lpp until the end of simulation. Using the larger damping zone in the beginning of the simulation is found to ensure faster convergence of the results.

Table 3 Verification of time step for monohull and C1 for Fn ¼ 0.4.

MH C1

b S i;3 , N

b S i;2 , N

b S i;1 , N

εi;21

εi;32

Ri

pi

δRE

U, %

17.975 36.052

17.991 35.990

18.000 35.985

0.009 0.005

0.016 0.062

0.563 0.081

0.830 3.632

0.0189 0.0004

0.1311 0.0015

Table 4 Verification of mesh size for monohull and C1.

MH

C1

Fn

b S i;3 , N

b S i;2 , N

b S i;1 , N

εi;21

εi;32

Ri

pi

δRE

U, %

0.35 0.45 0.55 0.35 0.45 0.55

9.217 29.219 44.261 19.084 77.930 114.710

9.186 29.157 44.315 19.193 77.732 115.076

9.167 29.129 44.342 19.296 76.904 115.380

0.019 0.028 0.027 0.103 0.828 0.304

0.031 0.062 0.054 0.109 0.198 0.366

0.613 0.452 0.500 0.945 4.177 0.831

2.118 3.440 2.999 0.245 / 0.803

0.030 0.023 0.027 1.768 / 1.491

0.410 0.099 0.076 11.454 / 1.615

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Table 5 Relative deviations of total resistance values for MH, C1, C2, C3 and C4. Fn

REXP, T N

MH

0.3

5.744

0.35

8.643

0.4

17.089

0.45

27.282

0.5

36.375

0.55

43.890

,

MH RCFD, , N RD, % T

REXP, T N

C1

6.360 þ 10.724 9.167 þ6.063 17.967 þ5.138 29.129 þ6.770 37.870 þ4.110 44.342 þ1.030

14.429

,

18.899 35.223 74.621 98.814 115.012

C1 RCFD, , N RD, % T

REXP, T N

C2

14.961 þ 3.687 19.296 þ2.101 36.601 þ3.912 76.904 þ3.059 102.243 þ3.470 115.380 þ0.320

12.859

,

17.442 40.300 73.438 89.624 100.386

Verification has been carried out for two input parameters, time step and grid spacing. Convergence study is made according to ITTC procedure (ITTC, 2011) for three solutions. Changes between solutions are defined as follows:

(24)

εi;32 ¼ b S i;3  b S i;2

(25)

REXP, T N

C3

13.553 þ 5.397 17.605 þ0.937 43.108 þ6.744 77.584 þ5.645 92.932 þ3.691 102.800 þ2.404

12.061 17.217 43.320 67.002 81.635 93.234

,

C3 RCFD, , N RD, % T

REXP, T N

C4

12.721 þ5.472 17.675 þ2.660 45.213 þ4.370 70.942 þ5.880 85.113 þ4.260 95.593 þ2.530

12.007 19.329 41.840 62.774 78.902 90.350

,

C4 RCFD, , N RD, % T

12.120 þ0.938 19.421 þ0.476 43.403 þ3.736 66.317 þ5.644 81.383 þ3.144 92.821 þ2.735

parameters. Convergence ratio Ri , which is defined as the ratio between εi;21 and εi;32 , is used to estimate convergence conditions. Monotonic convergence is achieved when 0 < Ri < 1, oscillatory convergence when Ri < 0 and divergence is achieved when Ri > 1. For monotonic convergence condition, generalized Richardson extrapolation (RE) is used to estimate uncertainties and errors. The order of accuracy is calculated with following equation:

3.3. Verification procedures

εi;21 ¼ b S i;2  b S i;1

C2 RCFD, , N RD, % T

ln ðεi;32 εi;21 Þ pi ¼ ln ðri Þ

where εi;21 stands for changes between medium-fine solutions, εi;32 stands for changes between coarse-medium solutions, b S i;1 , b S i;2 and b S i;3 correspond to solutions obtained utilizing fine, medium and coarse input

(26)

where ri stands for uniform refinement ratio. RE error is computed according to:

Fig. 8. The curves of IF as a function of Fn for C1 (upper left), C2 (upper right), C3 (lower left), C4 (lower right). 158

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δRE ¼

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ε21 r pi  1

(27)

IF ¼

The uncertainty for monotonic convergence is calculated according to safety factor method as follows:

U ¼ FS δRE

(28)

U ⋅100% b S i;1

4.2. Decomposition method for the total resistance of catamaran The total resistance coefficient of catamaran CT;C , can be decomposed according to the decomposition procedure described in (Insel and Molland, 1992):

CT;C ¼ ð1 þ ϕkÞσCF; MH þ τCW; MH

(29)

Interference resistance represents a difference between the total resistance of catamaran and the double total resistance of monohull. Commonly, the interference resistance is investigated through the interference factor IF. IF is defined as ratio between the interference resistance and the double total resistance of monohull. IF can be determined on the basis of the wave resistance as well. Additionally, interference resistance can be studied through the decomposition method for total resistance of catamaran. Insel and Molland (1992) proposed the decomposition method for the total resistance coefficient of catamaran on the basis of viscous and wave resistance interference factors. Within this paper the interference is investigated through IF and decomposition method for total resistance of catamaran proposed by the authors. This method is modification of the one defined in Insel and Molland (1992).

CT;C ¼ ð1 þ βkÞCF; MH þ τCW; MH

CT;C ¼ ð1 þ kÞβCF; MH þ τ CW; MH

CT ¼ 1 2

ϕ

σ

β

0.08956 0.09004 0.08966 0.08997 0.09043 0.09047 0.09002

1.07184 1.07492 1.07214 1.08401 1.06802 1.07494 1.07431

1.00482 1.00499 1.00489 1.00515 1.00546 1.00492 1.00504

1.01002 1.01042 1.01011 1.01122 1.01036 1.01038 1.01042

RT ρv2 S

(34)

where RT is the total resistance obtained utilizing free surface simulation, v is the monohull or catamaran speed and S is the wetted surface for v ¼ 0 m/s. The frictional resistance coefficient is calculated analogously, using the frictional resistance RF obtained utilizing double body simulations for monohull. Form factor for monohull is obtained utilizing double body

Table 6 Averaging process of k, ϕ, σ and β for C4. k

Table 7 Averaged values of k, ϕ, σ and β. MH

C1

C2

C3

C4

0.08422 1 1 1

0.10822 1.30725 1.01732 1.03984

0.09729 1.16662 1.00985 1.02202

0.09243 1.10470 1.00648 1.01410

0.09002 1.07431 1.00504 1.01042

(33)

This procedure is proposed in order to assess the viscous and wave interference effect on the total resistance of catamaran. In (Insel and Molland, 1992) CF is calculated according to the ITTC 1957 correlation line. To obtain viscous resistance, Insel and Molland carried out a wake transverse analysis. This analysis is based on a very complex measuring process which requires extensive time. In this paper, total resistance coefficient is calculated according to the following equation:

Interference factor can be calculated based on the total or wave resistance, as already mentioned. If interference factor is calculated on the basis of the wave resistance, total resistance must be decomposed. In order to obtain the wave resistance, form factor should be determined for the monohull and for catamaran configurations. Souto-Iglesias et al. (2012) calculated IF taking into account wave resistance and assumed the form factor to be identical for both monohull and catamaran configurations. To avoid the error due to this assumption, IF is calculated based on the total resistance as follows:

0.3 0.35 0.4 0.45 0.5 0.55 average

(32)

In this paper, ϕ and σ are combined into the viscous resistance interference factor β according to the following equation:

4.1. Interference factor

Fn

(31)

where CF; MH is the frictional resistance coefficient of monohull, CW; MH is the wave resistance coefficient of monohull, k is the form factor of monohull, ϕ is the factor that takes into account pressure field change around the demihull, σ is the factor that takes into account velocity augmentation between demihulls and τ is the wave resistance interference factor. In order to simplify equation (31), Insel and Molland (1992) suggested to combine ϕ and σ into one parameter, viscous resistance interference factor β. Consequently equation (31) can be expressed as:

4. Interference resistance

k ϕ σ β

(30)

where RT; C is the total resistance of the catamaran and RT;MH is the total resistance of monohull.

where FS ¼ 1:25 is the safety factor used in all cases in this study as recommended in (Yengejeh et al., 2016). The uncertainty is neither calculated for oscillatory convergence case, nor for divergence case. The normalized uncertainties, U, are given as follows:

U¼

RT; C  2RT;MH 2RT;MH

Fig. 9. The dependence of ϕ and σ on the separation ratio. 159

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with double body simulations. β is defined as a ratio between the viscous resistance of catamaran and the double viscous resistance of monohull both obtained with double body simulations. Form factor for catamaran configurations is calculated as a ratio between the pressure resistance and the frictional resistance of catamaran obtained utilizing double body simulations. Wave interference factor can be determined according to the equation:

simulations according to the following equation:

k¼

RP; MH RF; MH

(35)

where RP; MH is the pressure resistance and RF; MH is the frictional resistance of monohull obtained with double body simulations. The pressure resistance obtained in double body simulations is equal to viscous pressure resistance, since there are no free surface effects and therefore wave resistance is equal to zero. Since ϕ and σ are equal to 1 for monohull, CW; MH can be calculated as follows:

CW; MH ¼ CT; MH  ð1 þ kÞCF; MH

τ¼

CW; C CW; MH

(37)

where CW; C is calculated as follows:

(36)

CW; C ¼ CT; C  ð1 þ kÞβCF; MH

After CW; MH is obtained, double body simulations for catamaran configurations have to be carried out to determine ϕ, σ and β. ϕ is defined as a ratio between the pressure resistance of catamaran and the double pressure resistance of monohull both obtained with double body simulations. σ is defined as a ratio between the frictional resistance of catamaran and the double frictional resistance of monohull both obtained

(38)

where CT; C is the total resistance coefficient of catamaran, k is the monohull form factor, β is the viscous interference factor and CF; MH is the frictional resistance coefficient of monohull. 5. Results In this section, the obtained results in terms of total resistance value, wave pattern, hydrodynamic pressure, cross flow and interference factor are presented. 5.1. Convergence studies Convergence studies are performed for two input parameters: time step and mesh size for monohull and the narrowest catamaran configuration C1. The narrowest catamaran configuration is chosen for the verification procedure since the flow around its demihulls is the most complex one. Firstly, verification for time step is made using coarse mesh for both monohull and catamaran configuration C1 for Fn ¼ 0.4. Chosen time steps are as follows: T/100, T/200 and T/400 and therefore ri is equal to 2. The obtained wave patterns using three different time steps are shown in Fig. 7. For monohull and catamaran configuration C1, monotonic convergence is achieved. The obtained results are shown in Table 3 and it can be seen that the obtained normalized uncertainties, U, are low. It was found that the time step T/200 is good compromise between numerical accuracy and computational time, since the obtained results for the total resistance value, as well as the obtained wave pattern using coarse, medium and fine input parameter (time step) do not differ significantly. Therefore, convergence study for grid size is performed for medium time step T/200. In order to perform convergence study for mesh size, coarse, medium and fine meshes are generated. Coarse mesh for both monohull and catamaran configurations has around half of million cells, medium mesh has around 0.95 million cells, while fine mesh has around 2 million cells. Uniform refinement ratio parameter ri for mesh size is determined according to following equation:

Fig. 10. The obtained form factor values.

ri ¼

rffiffiffiffiffiffiffiffiffi Ni 3 Ni1

(39)

Fig. 11. Dependence of β on the separation ratio.

where Ni is the number of cells of the i-th mesh. Therefore, ri for mesh size is equal to 1.26. Convergence study for mesh size is performed for three Fn values: 0.35, 0.45 and 0.55. The obtained results are shown in Table 4. In the case of monohull for all Fn values monotonic convergence is achieved. For catamaran configuration C1 for Fn values of 0.35 and 0.55 the monotonic convergence is achieved, and for Fn value of 0.45 the divergence is obtained. For monohull, normalized uncertainties are sufficiently low and the largest U is obtained for Fn ¼ 0.35 and it equals to 0.41%. For catamaran configuration C1 larger normalized uncertainties

Table 8 The wave interference factors. τ Fn 0.3 0.35 0.4 0.45 0.5 0.55

C1

C2

C3

C4

1.53772 1.07782 1.00221 1.45965 1.48839 1.42106

1.18074 0.83630 1.33173 1.48601 1.31851 1.22220

0.96289 0.86361 1.44586 1.31916 1.17272 1.10723

0.80000 1.15736 1.35961 1.20205 1.10312 1.06330

compared to monohull values are obtained. The largest U for C1 is 160

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obtained for Fn ¼ 0.35 and it amounts –11.454%. Larger U are expected for C1 than for monohull, since the flow around catamaran is more complex than the flow around monohull because of the interference effects. The obtained results for all catamaran configurations showed that interference effects are most significant for intermediate Fn values, as was found in (Broglia et al., 2011). Therefore, the largest normalized uncertainties were expected at intermediate Fn values. 5.2. Validation of results Since medium time step T/200 was found to be a good compromise between accuracy and computational time, it was used in validation procedure. As U values are significantly larger for mesh size than for time step, fine mesh is used for validation of numerical results. Relative deviations of the total resistance values obtained utilizing fine mesh for monohull (MH) and all catamaran configurations, RCFD T , from the experimental values (Souto-Iglesias et al., 2012), REXP , are given in T Table 5. Relative deviations are calculated as follows:

RD ¼

RCFD  REXP T T ⋅100% REXP T

(40)

As it can be seen from Table 5, numerical results overestimate experimental results for monohull and all catamaran configurations for all Fn values. The overestimations are more significant for monohull than for catamaran configurations. The largest relative deviation for monohull equals to 10.724% for Fn ¼ 0.3. It should be noted that the total resisMH tance values are measured in kilograms. This means that REXP, value T for Fn ¼ 0.3 amounts only 0.5855 kg. For such a small value, even small mistake or uncertainty in experimental measurement can lead to relatively large overestimation (Farkas et al., 2017). Since experimental uncertainty is unknown, it can be considered that the obtained numerical results show satisfactory agreement with experimental results. The largest relative deviation for C1 using fine mesh equals 3.912%, for C2 6.744%, for C3 5.880% and for C4 5.644%. The total resistance values obtained with free surface simulations utilizing fine mesh were used to calculate IF. IF values are calculated for four catamaran configurations for six Fn values using both numerical and experimental total resistance values. Curves of IF, obtained numerically and experimentally, as a function of Fn for four catamaran configurations are given in Fig. 8. Although deviations between experimentally and numerically obtained IF values are present, both curves follow the same trend. The maximum value of IF is obtained for C1, the narrowest catamaran configuration. In general, IF values for C1 are higher than IF values for other three catamaran configurations. This is in accordance with results obtained in (Broglia et al., 2014; Zaghi et al., 2011). The value of Fn where the minimum of IF curve occurs is very important for catamaran

Fig. 13. Velocity field for monohull (upper) and C1 (lower) at x ¼ 0 m for Fn ¼ 0.4 obtained utilizing double body simulations.

design, because for this Fn interference resistance is the smallest for particular catamaran configuration. This Fn can be considered as optimal Fn regarding interference resistance. Since the trend of experimentally and numerically obtained curve is the same, optimal Fn is also the same. This confirms that viscous CFD methods can be used in catamaran preliminary design for the prediction of the interference resistance. It can be seen from Fig. 8 that as separation decreases, the optimal Fn moves towards higher values, what will be explained in subsection 5.4 in detail.

Fig. 14. Velocity field for monohull (upper) and for C1 (lower) at x ¼ 0 m for Fn ¼ 0.4 obtained utilizing free surface simulations.

Fig. 12. Wave resistance interference factor as a function of Fn. 161

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Fig. 15. The wave profile along the C1 (upper), C3 (middle) and MH (lower) for Fn ¼ 0.5.

separation ratios. It should be noticed that parameter σ has larger influence on the total resistance of catamaran, since frictional resistance is significantly larger than viscous pressure resistance. The form factor values obtained for monohull and catamaran configurations are shown in Fig. 10. Form factor for the monohull and catamaran configurations cannot be considered the same. As separation ratio increases, form factor value for catamaran converges to the form factor value for monohull as was expected since monohull can be considered as catamaran with infinite separation ratio. The dependence of the viscous interference factor on the separation ratio is shown in Fig. 11. Since β is proven to be constant across the whole Fn range it can be concluded that viscous interference is also constant across the Fn range, as obtained in (Insel and Molland, 1992). It is important to notice that β is dependent on separation ratio and it cannot be assumed as constant for different catamaran configurations. Viscous interference is the highest for the narrowest catamaran configuration and it decreases as the

5.3. Results of total resistance decomposition Double body simulations are performed in order to determine k, ϕ, σ and β utilizing mesh that has similar density to fine mesh used for free surface simulations. Comparable mesh density near the hull was used to obtain reliable estimations of wave resistance in (Raven et al., 2004). The authors used 0.6 million cells for double body simulations and 2.3 million cells for free surface simulations. Therefore, the mesh used for double body simulations in this paper has around 0.6 million cells. For each configuration, k, ϕ, σ and β values are averaged, since these values are proven not to be dependent on Fn. The averaging process for C4 is shown in Table 6. Obtained results for k, ϕ, σ and β for monohull and four catamaran configurations are shown in Table 7. The dependence of ϕ and σ on the separation ratio is shown in Fig. 9. It can be seen that parameter ϕ decreases significantly as separation ratio increases. On the contrary, parameter σ remains almost constant for all

Fig. 16. Hydrodynamic pressure distribution on the starboard side of left demihull for C1 for Fn ¼ 0.3 (upper), Fn ¼ 0.4 (middle) and Fn ¼ 0.5 (lower). 162

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Fig. 17. Hydrodynamic pressure distribution on the starboard side of left demihull for C3 for Fn ¼ 0.3 (upper), Fn ¼ 0.4 (middle) and Fn ¼ 0.5 (lower).

In Fig. 12 it can be seen that τ decreases more rapidly for catamaran configurations with the greater separation ratio for higher Fn. The minimum of τ curve shifts towards higher values of Fn for lower separation ratios.

separation ratio increases. Insel and Molland (1992) obtained similar trend for catamaran with Wigley demihulls. Since interference is the most significant for intermediate Fn values, where viscous resistance represents very important part in the total resistance of catamaran, interference phenomena cannot be investigated using inviscid numerical simulations because they can give misleading results. Therefore, investigation of the interference phenomena has to include viscous interference and consequently the analysis of viscous flow. After k, ϕ, σ and β are determined, wave interference factors in dependence of Fn and separation ratio are obtained according to the procedure described in 4.2., Table 8. The curves of the wave interference factor as a function of Fn for four catamaran configurations are shown in Fig. 12. When τ is smaller than 1, favourable wave interference is present. This means that wave resistance of catamaran is smaller than the wave resistance of the double monohull.

5.4. Flow analysis around demihulls The flow around demihulls is very complex since it includes viscous and wave interference. The presence of demihulls has minor influence on the flow in the outer region, but significantly affects the flow in the inner region. Insel and Molland (1992) suggested performing the viscous wake analysis, along with the routine model testing to experimentally investigate the viscous and the wave interference. The authors stated that viscous wake analysis was very time consuming and that it was very difficult to achieve the satisfactory standard of accuracy. Therefore, this

Fig. 18. Hydrodynamic pressure distribution on the starboard side of monohull for Fn ¼ 0.3 (upper), Fn ¼ 0.4 (middle) and Fn ¼ 0.5 (lower). 163

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Fig. 19. Hydrodynamic pressure distribution on the portside of left demihull for C1 for Fn ¼ 0.3 (upper), Fn ¼ 0.4 (middle) and Fn ¼ 0.5 (lower).

especially in inner region. The wave profiles along the starboard side of left demihull for C1 and C3 and along the starboard side of monohull for Fn ¼ 0.5 are shown in Fig. 15. The wave profile is significantly influenced by the interference, especially for the narrowest catamaran configuration C1. With the increase of the separation, the wave profile along the starboard side of left demihull converges to the wave profile along the starboard side of monohull. Hydrodynamic pressure distributions for catamaran configurations are significantly influenced by interference phenomena. These distributions are asymmetric for starboard side and portside. Pressure differences are significantly larger on the starboard side of the left demihull. The obtained hydrodynamic pressure distributions for Fn values 0.3, 0.4 and 0.5 on the starboard side of left demihull for C1 and C3 and monohull are shown in Figs. 16–18., respectively. The obtained hydrodynamic pressure distributions for portside of left demihull for C1 and C3 are shown in Figs. 19 and 20 respectively. Differences between pressure distribution

analysis is not recommended as a routine for commercial testing, unless the routine is fully automated. Numerical simulations based on viscous flow take into account both viscous and wave interference effects. In addition, these simulations give detailed representation of the flow around the demihulls. In this subsection, an interference phenomenon is investigated through pressure and velocity fields and the wave patterns. Velocity fields obtained utilizing double body simulations for Fn ¼ 0.4, for monohull and C1 at the plane located at aft perpendicular are shown in Fig. 13. Velocity field for C1 is asymmetric with respect to symmetry plane of the demihull. Velocity decrease around left demihull at the starboard side of C1 is caused by viscous interference, Fig. 13. The obtained velocity fields utilizing free surface simulations for Fn ¼ 0.4, for monohull and C1 at plane located at aft perpendicular are shown in Fig. 14. From Fig. 14 it can be seen that velocity field of C1 is distorted compared to the velocity field of monohull. Also, velocity decrease, caused by interference, can be noticed around the demihull,

Fig. 20. Hydrodynamic pressure distribution on the portside of left demihull for C3 for Fn ¼ 0.3 (upper), Fn ¼ 0.4 (middle) and Fn ¼ 0.5 (lower). 164

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velocity is present. In the areas of positive velocity, the fluid moves towards outer region. The strength of cross flow and location at demihull where it occurs depends on Fn and separation. For higher Fn, cross flow is significantly stronger and it occurs around midship section, Fig. 21. This relatively strong cross flow for C1 at higher Fn values, causes significant change in the wave profile along starboard side of left demihull for C1, Fig. 15. In the region of low hydrodynamic pressure, where negative transverse velocities occur, the amplitude of the wave profile along left demihull for C1 is significantly reduced. The location where the minimum hydrodynamic pressure occurs, and consequently the location of the maximum cross flow, is moving downstream for larger separation. The same result was obtained in (Broglia et al., 2011; Zaghi et al., 2011). The strength of the cross flow was found to be directly connected with the interference phenomena, i.e. IF. For Fn ¼ 0.3 and 0.5, the cross flow around C1 demihull is stronger than the cross flow around C3 demihull, Figs. 21 and 22. Therefore, IF for these Fn values are higher as can be seen form Fig. 8. On the other hand, for Fn ¼ 0.4, the cross flow around C3 demihull is stronger and consequently IF for C3 is higher. The occurrence of cross flow is one of the main causes of the viscous interference (Broglia et al., 2011). The main reason for this is the

on the starboard side and portside of demihull are largest for C1 and they are caused by the proximity of the demihulls. From Figs. 16–20 both longitudinal and transverse pressure gradients can be noticed. Longitudinal pressure gradient can also be noticed in Fig. 16., for the case of monohull. For Fn ¼ 0.3 there are three high pressure and two low pressure regions. Therefore, wave profile along the monohull has smaller wave length and three wave crests and two wave troughs can be noticed. As Fn increases, wave length of wave profile along the monohull also increases, and only two wave crests and one wave trough can be noticed. With an increase of Fn, low pressure region moves towards the stern and pressure decreases, Figs. 16–20. Transverse pressure gradient is caused by asymmetric distribution of the hydrodynamic pressure on the starboard side and portside. It can be noticed that first high pressure region is almost symmetrical. This was expected, since waves of each demihull do not interfere at the bow region where first high pressure region occurs. The largest asymmetry, and consequently transverse pressure gradient, is present in the low pressure region. Transverse pressure gradient causes cross flow, from starboard side to the portside of demihull and vice versa, around the keel of the demihulls. Cross flow is one of the main causes of the total resistance increase. In Figs. 21 and 22., cross flow for Fn ¼ 0.3, 0.4, 0.5 for C1 and C3 are shown. The fluid moves towards inner region in the areas where negative

Fig. 21. Cross flow for C1 for Fn ¼ 0.3 (upper), Fn¼0.4 (middle) and Fn¼0.5 (lower). 165

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Fig. 22. Cross flow for C3 for Fn ¼ 0.3 (upper), Fn¼0.4 (middle) and Fn¼0.5 (lower).

Fig. 23. Wave patterns for C1 for Fn ¼ 0.3, 0.5 (upper left, lower left), for C3 for Fn ¼ 0.3, 0.5 (upper right, lower right). 166

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Fig. 24. Longitudinal wave cuts for C1 and MH (upper) and for C3 and MH (lower) for Fn ¼ 0.3 and Fn ¼ 0.5.

Fn ¼ 0.3 and Fn ¼ 0.5 are shown in Fig. 24. It should be noticed that wave cuts of monohull are doubled. At lower Fn values in region between demihulls, two wave crests and two wave troughs can be noticed. The first wave trough is caused by an interference of the transverse bow wave system. This wave trough is greater than the second wave trough. As Fn increases, wave crests and troughs move downstream. The location of the first wave trough is found to be connected with the total resistance coefficient of catamaran. The maximum of the total resistance coefficient is reached when the first wave trough is located around the stern shoulder. Deviations between wave cuts for catamaran and monohull are significantly smaller for C3. Consequently, IF values are also significantly smaller at these Fn values for C3. In order to achieve favourable wave

existence of the vorticity around the demihull keel and the change of the boundary layer formation. Beside the occurrence of cross flow, the wetted surface also changes considerably, Fig. 15. Consequently, the viscous interference occurs and it represents an important part of the total interference for these catamaran configurations. Cross flow causes also high wave crests and troughs between two demihulls. Location of these wave crests and troughs also moves downstream for higher separation, Figs. 23 and 24. In Fig. 23., the obtained wave patterns for Fn ¼ 0.3 and 0.5 for C1 and C3 are shown. Longitudinal wave cuts along the centreline of the catamaran configurations (solid lines) as well as for the monohull (dash and dot lines) at the same distance from the centreline of the monohull (y ¼ s/2) for

Fig. 25. Wave patterns for C2 (left) and C3 (right) for Fn ¼ 0.35.

Fig. 26. Longitudinal wave cuts for C2 and MH (upper) and for C3 and MH (lower) for Fn ¼ 0.35. 167

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hydrodynamic pressure distribution between starboard side and portside, cause the cross flow. The cross flow causes the increase in total resistance of catamaran and its strength was found to be directly connected with the interference, i.e. IF. Since interference depends on the separation and Fn, the strength of the cross flow is not always stronger for narrower catamaran configurations. The analysis of the wave patterns and longitudinal wave cuts showed that wave crests and wave troughs move downstream as Fn and separation increase. The location of the first wave trough caused by the interaction of the transverse bow wave systems, was found to be related to the total resistance coefficient of catamaran. The total resistance coefficient for S60 catamaran configuration has maximum value when the first wave trough is located near the stern shoulder. By comparison of the longitudinal wave cuts for catamaran and monohull, it was concluded that longitudinal wave cuts could be very useful in the wave interference investigation. In this paper, the total resistance coefficient of catamaran is decomposed into the viscous resistance coefficient and the wave resistance coefficient. These two coefficients were related to the viscous and wave resistance coefficients of monohull by introducing the viscous and wave interference factor. In order to determine total resistance of catamaran at the full-scale, influence of the scale effect on the viscous and wave interference factor should be investigated. This will form a part of the future investigations.

interference, energy of the waves generated by the catamaran should be smaller than the double energy of the waves generated by the monohull. Since wave energy is proportional to the squared wave amplitude, longitudinal wave cuts are useful in estimation of the wave interference. It is important to notice that wave beam is also important parameter in the determination of wave energy, so this longitudinal wave cut should not be used as the only parameter in wave interference determination. In Fig. 25., obtained wave patterns for C2 and C3 for Fn ¼ 0.35 are shown. In Fig. 26., the obtained longitudinal wave cuts, for the same Fn, compared to the double wave cuts of monohull are shown. For both catamaran configurations for this particular Fn wave interference is favourable. 6. Conclusion An extensive numerical study of the flow around S60 catamaran and S60 monohull, as well as investigation into interference phenomenon was performed within this paper. The emphasis was put on the influence of the separation and Froude number on the interference, as well as on the flow itself. Therefore, numerical simulations of the free surface viscous flow, as well as numerical simulations of the double body viscous flow were carried out utilizing the commercial software package STARCCMþ for six Fn in the range from 0.3 to 0.55 and for four separation ratios in the range from 0.226 to 0.4696. Numerical simulations of free surface flow were verified based on the recommended ITTC procedure. On the basis of convergence studies, optimal mesh, as well as the optimal time step, were established. The results of the free surface simulations were validated against experimental data (Souto-Iglesias et al., 2012) and satisfactory agreement was achieved. An interference phenomenon was investigated trough interference factor, which was based on the total resistance value for catamaran and monohull. The comparison between experimentally and numerically obtained IF curves as a function of Fn showed that these curves have the same trends. Therefore, location of the minimum IF occurs at the same Fn for both curves. Interference is decomposed into viscous interference and wave interference. This decomposition method provided better understanding of the total resistance components of catamaran. Viscous interference was investigated through the double body simulations which were performed utilizing the mesh that had comparable mesh density to the optimal one used in the free surface simulations. The results showed that form factor for catamaran configuration is influenced by the viscous interference and consequently is not the same as for monohull. It was found that form factor for catamaran is not dependent on the Fn, but it is dependent on the separation. As separation increases, form factor of the catamaran decreases and converges to the form factor of the monohull. Viscous interference was investigated through viscous interference factor. This factor was found to be dependent on the separation and independent on Fn. Viscous resistance represents an important part in the total resistance of catamaran for intermediate values of Fn, where interference effects are the greatest. In order to predict the total resistance of catamaran more realistically, it is important to take viscous interference into consideration. Therefore, viscous CFD methods should be used for interference investigations, rather than potential CFD methods. Since experimental determination of the viscous interference is very time consuming and difficult to perform, the method for determination of viscous interference, described within this paper, can be used in the catamaran design, especially in the preliminary design phase. The detailed investigation of viscous flow around catamaran configurations utilizing free surface simulations was performed to investigate and to clarify the wave interference phenomena. The obtained hydrodynamic pressure distributions indicated the existence of the longitudinal and transverse pressure gradients at the demihull. With the increase of Fn and separation, high and low pressure regions move downstream. Transverse pressure gradients, which occur due to asymmetric

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