Steady-state calculations of free surface flow around ship hulls and resistance predictions

A. PranzitelliUniversity of Leeds, Leeds, UK

C. de Nicola & S. MirandaUniversità di Napoli Federico II, Napoli, Italy

ABSTRACT: This study numerically simulated the free-surface flow around a semi-displacing motoryacht advancing steadily in calm water, recurring to two approaches implemented in two commercial software. The volume of fluid (VOF) method implemented in a Reynolds-averaged Navier-Stokes (RANS) equations solver and a panel method with free-surface capabilities were employed. Both methods were assessed on the Wigley parabolic hull and the Series 60, CB=0.6 hull. Particular importance was attached to the grid topology for the RANS simulation to minimize computational efforts without any lack of accuracy of the numerical solution. Indeed, all the computations presented here were carried out on a dual core single processor personal computer avoiding expensive hardware. Numerical results were finally compared with existing experimental data.The VOF method correctly predicted both the free-surface shape and the total resistance, while the panel method showed its shortcomings in simulating complex wave shapes typically generated around semi-displacing hulls. The validation results suggest also a strong dependency of the simulation on the trim of the hull.

1 INTRODUCTION

Experimental tests are one of the most important and reliable methods for performance predictions of a new ship. Unfortunately conducting towing tank test can be difficult and expensive, and sometimes it takes a long time. Moreover, because of the impossibility of simulating at the same time Froude and Reynolds numbers in model scale, it is necessary to recur to extrapolation methods to obtain powering performance of the ship from the model experimental data, with the subsequent addition of error.

An alternative to the experimental tests is the mathematical solution of the flow field. Mathematically speaking, analytical methods are beautiful and extremely efficient when the flow is simple and the non-linear effects are not significant, but they cannot be applied on complex geometries. Therefore, a numerical solution is needed. The first techniques of Computational Fluid Dynamics (CFD) are born together with

numerical computing, gaining quickly appreciation. In particular, panel methods based on potential flow were shown to be a valid analysis tool especially in pre-design. Free-surface wave flow is of direct interest to naval engineering and several CFD techniques were developed to simulate it under the hypotheses either of potential flow or turbulent viscous flow. Thanks to the advent of high-performance computing, its availability and its decreasing costs, CFD techniques solving Navier-Stokes equations are now abundant and largely employed in research and industrial engineering.

Generally speaking, free-surface flows are an especially difficult class of flows to simulate due to their moving boundaries. The position of the boundary is known only at the initial time and therefore its location at later times has to be determined as part of the solution. Two boundary conditions must be imposed on the free-surface: (i) the kinematic condition which requires that the

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free-surface is a sharp boundary separating the two fluids that allows no flow through it; (ii) the dynamic condition which requires that the forces acting on the fluid at the free-surface are in equilibrium, that is momentum conservation at the free-surface.

For Navier-Stokes equations there are two different types of free-surface wave solution methods: interface tracking and interface capturing. In interface tracking, the free-surface is a well-defined surface separating the two fluids considered. Boundary fitted grids are then used and the evolution of the interface is followed, satisfying boundary conditions on it and moving grid nodes at each time step or iteration. These methods accurately predict the free-surface shape in case of relatively small deformations of it, but cannot handle highly distorted or breaking waves. Moreover, they are unsuitable for complex geometries and the grid quality may degrade during the conforming process. In contrast, in interface capturing the interface is not defined as a sharp boundary and the computation of it is performed on fixed grids extending beyond the free-surface. The free-surface can be determined by following the motion of massless particles introduced at the free-surface at the initial time step, as the marker-and-cell (MAC) method (Harlow and Welsh 1981). Alternatively, the free-surface is determined by solving a transport equation for the fraction of the cell occupied by the liquid phase, as in the volume of fluid (VOF) method (Hirt and Nichols 1981). Interface capturing methods are suitable for complex geometries and can easily handle highly-distorted or breaking waves. Also, there is no need for grid conformation, thus the initial grid quality can be retained.

For their wide range of applicability interface capturing methods are most frequently used for free-surface flows, but they can be computationally expensive. For example, the MAC method in three-dimensional cases requires a large number of particles to be followed. In contrast, the VOF method requires that only one scalar hyperbolic equation (in two-phase flow) for the volume fraction has to be added to. This does not excessively increase computational efforts.

In this work, the VOF method implemented in a RANS solver was employed to simulate the free-surface wave flow around a semi-displacing motoryacht hull. The method was validated on two standard ship hull form, that is the Wigley parabolic hull and the Series 60, CB=0.6 hull. Together with the free-surface shape, the total resistance coefficient was determined, and the

numerical results were compared with experimental data. The effects of two classical turbulence model, the SST k - ω and Realizable k - ε on the free-surface and resistance were estimated for the Wigley hull. Moreover, a method to improve convergence was considered showing its effects on the solution. Then, a full-scale simulation of the motoryacht was carried out in order to avoid correlation formulas to transfer the results from the model to the ship. Finally, RANS solution of free-surface shape was compared with results from panel methods solver. All these simulations were carried out on a dual core single processor personal computer to show the possibility to obtain good estimations of the flow around ship hulls advancing steadily using cheap hardware.

2 COMPUTATIONAL METHOD

2.1 RANS solver and VOF method

Viscous flow computations were carried out using the general purpose CFD software FLUENT 6.3, which solves the Reynolds-Averaged Navier-Stokes (RANS) equations with a finite-volume approach on hybrid unstructured grids. Steady-state computations were performed to minimise the CPU time. The SIMPLE algorithm was used for pressure-velocity coupling. The QUICK scheme was used for discretising either the convection and diffusion terms. Two turbulence models were used, the SST k - ω and the Realizable k - ε, both with wall functions.

As mentioned in Section 1, the VOF method was employed to predict the free surface shape around ships. VOF formulation implemented in FLUENT (ANSYS 2006) is slightly different from the original formulation of Hirt and Nichols (1981) and more general. It relies on the hypothesis that two or more fluids (or phases) are not interpenetrating. For n phases included in the system, n-1 new variables that is the volume fraction of the corresponding phases are introduced, as well as their respective transport equations. In each cell, the volume fractions of all phases sum to unity. The fields for all variables and properties are shared by the phases and represent volume-averaged values, as long as the volume fraction of each phase is known at every location. Thus, the variables and properties in any given cell are either purely representative of one of the phases, or representative of a mixture of the phases, depending upon the volume fraction values. Based on the local value of αq, the appropriate properties and variables will be assigned to each control volume within the

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domain. In the hypothesis of incompressible flow and without mass transfer among phases, the transport equation for the qth fluid's volume fraction αq has the form:

0

qqq v

t

(1)

where v⃗ is the velocity vector in the Cartesian coordinate system. The volume fraction equation is not solved for the primary phase, but it is computed based on the constraint:

11

n

qq (2)

A single momentum equation is solved throughout the domain and the resulting velocity field is shared among the phases. The momentum equation is dependent on the volume fractions of all phases through the fluid properties, which are determined by the volume fraction of phases occupying each cell. For example, density ρ is given by:

n

qqq

1 (3)

where ρq is the density of the qth phase. In case of turbulence quantities, a single set of transport equations is solved and the turbulence variables are shared by the phases throughout the field.

In order to avoid excessive smearing of the interface due to numerical diffusion, is necessary a special high resolution interface capturing scheme for the convection term of the volume fraction transport equation, thus the modified HRIC scheme (Muzaferija et al. 1998) was used in this work. Standard interpolation schemes are used whenever a cell is completely filled with one phase or another. Only when the cell is near the interface between two phases, the HRIC is used.

2.2 Panel method solver

In this work, RANS solutions of the deformed free-surface shape were compared with solutions of the panel method solver included in the FLOWTECH SHIPFLOW software. SHIPFLOW software uses a zonal approach to solve the whole flow-field around a ship hull by recurring to the most opportune solver for each single region the flow-field is subdivided in order to optimize the numerical computing. In particular, free-surface shape and the flow far away the hull are determined by means of a panel method solver, the boundary layer on the forward surfaces of the hull is calculated using a boundary layer method while the flow aft the hull is calculated by recurring to a single-phase RANS solver operating below the

free-surface. Since we are mainly interested in the prediction of the free-surface shape, only the panel method solver of this software was used.

On the hypotheses of incompressible and irrotational flow the continuity equation becomes:

0 v

(4)Since v⃗=∇⃗Φ , where Φ is the velocity potential, the set of Navier-Stokes equations reduces to the Laplace equation:

02 (5)that is a linear equation. Thus, the potential Φ can be decomposed in an asymptotic potential ϕ∞

and a perturbation potential ϕ : (6)

Boundary condition on the impermeable surface of the hull is:

Sonnnv 0 (7)

that is:

Sonn

(8)

For a generic point P placed at a large distance r from the body, the (asymptotic) boundary condition is: rforP (9)

At the free-surface the kinematic and dynamic boundary conditions have to be imposed. From the kinematic boundary condition the normal velocity at the free-surface is zero:

0

fsnVn

(10)

Defining the free-surface as y=η(x,z) yields:

0

zyyxx

(11)

Dynamic boundary condition requires that the pressure is constant on the free-surface. Applying the Bernoulli theorem on the undisturbed free-surface far away from the body and at one point on the wavy part of the free-surface yields:

021 2

222

Vzyx

gh (12)

Furthermore, a radiation boundary condition has to be imposed to ensure that no waves upstream of the hull shall be created, and usually this is enforced numerically.

Although the equation (5) is linear, the problem defined above is non-linear since boundary conditions (11) and (12) are non-linear and the potential ϕ and the wave height η are unknowns. Moreover these equations are to be applied on the free-surface, whose location is initially unknown. To solve the problem the free-surface boundary conditions are linearised about

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the previous solution in an iterative scheme, which may start either from the undisturbed flow or from a flow assuming the free-surface to be flat. In the first iteration the conditions are applied on the undisturbed surface, while in the later ones they are applied on the wavy surface from the previous iteration. If the process converges the difference between two subsequent solutions for ϕ and η tend to zero. The problem is solved by discretising the hull surface and the free-surface by quadrilateral panels. On each panel sources are continuously distributed. For further details see (Larsson and Janson 1999).

3 GRID GENERATION AND BOUNDARY

CONDITIONS

Since a numerical solution strongly depends on the grid quality, grid generation needs great accuracy and often requires a lot of hours of work. The time required for the grid generation depends on the type of grid to realise. Unstructured tetrahedral orprism-tetrahedral grids are fast to build and can yield good results especially for industrial applications, while structured hexahedral grids require more abilities and time but yield generally better numerical solutions. In the generation of grids for the case of a ship advancing on the free-surface, some particular tricks are necessary to obtain satisfactory results in free-surface shape as well as in resistance predictions.

To capture with accuracy the free-surface, a fine and good quality grid is necessary in the region where the free-surface expected, that is nearby the design water plane. Tetrahedral cells are not suitable for capturing the free-surface shape, especially in the regions far away the ship where the wave height is low. In fact in these regions the cells should be flat enough to capture the small variations of the wave elevation and sufficiently wide to avoid an excessive number of them. Tetrahedral-shaped cells cannot satisfy these requirements unless recurring to a high skewness. On the contrary hexahedral cells are more suitable to assume such a shape without compromising the quality of the solution.

A problem often encountered in presence of waves is their reflection on nonphysical boundaries of the flow field, that causes oscillations of the value of the coefficient of resistance. Since non-reflecting boundary conditions are not available in FLUENT for such simulations, the introduction of numerical diffusion is useful to avoid reflection at the boundaries of the domain. This can be done using

a strong growth rate of cell dimension from the model to the external boundaries.

To obtain a good calculation of the coefficient of resistance, it is necessary to have a fine grid close to the walls of the hull. In order to restrain the number of cells of the grid, wall functions were used; the height of the cells on the walls was determined to obtain a sufficiently low value of y+. As the resistance due to air is negligible, grid was not refined on the deck surface.

To satisfy these requirements structured hexahedral grids were generated for all the ship models considered here. Although this kind of grids required several hours of work, it allowed to obtain good results in the free-surface capturing as well as in the resistance predictions using a relatively small number of cells.

All the grids here realised are hexahedral boxes reproducing a towing tank. Naming L the length of the model (length between perpendiculars of the hull), bottom surface were at 1.0L below the design waterline, while inflow and outflow surfaces were posed respectively at 1L and 2L from the model. Since all the hulls in this work are symmetric and are advancing straight ahead, only one half of the hulls was considered, posing the side surface at 2L from the symmetry plane. Using the VOF model, the region occupied by air has to be modelled as well, thus the domain was extended for 1L above the design waterline with an additional block, sufficient to ensure that the air flow between the waves and the upper boundary had room enough to flow and does not artificially accelerate due to a blockage effect. This would increase the bow-wave height to some extent introducing additional errors in the solution.

Assumed boundary conditions were of pressure inlet and pressure outlet for the inflow and outflow surfaces respectively, under the hypothesis of open channel flow to ensure a correct pressure profile upstream and downstream the model, while top, bottom and side surface were considered as slip walls.

4 WIGLEY PARABOLIC HULL

4.1 Grid independence and CT calculation

Although Wigley parabolic hull is not employed to deaign actual ships, it is a good test case because of its simple geometrical form and the large amount of experimental and computational data available.

The model considered here has a length of L=4m and advancing at Fr=0.267, corresponding to Re=6.66x106 (both Reynolds and Froude numbers are based on L). For the RANS analysis,

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four H-type grids are generated: a coarse grid of 57'266 cells, two medium grids of 372'960 (medium 1) and 618'544 (medium 2) cells and a fine grid of 106 cells. The height of the first layer of cells on the hull walls are of 0.0005L for all the grids. This allowed to obtain y+<70 on the whole hull surface. The SST k – ω turbulence model was used. Figure 1 and Figure 2 show wave profile on the hull and wave pattern obtained by all the grids. Apart from the ones obtained by the coarse grid, all the results are similar, in particular the wave profile on the hull agrees very well with experimental data.

Analogous results were obtained for the coefficient of total resistance CT, (Table 1), where the difference ΔCT = (CT

CFD - CTexp)/CT

exp between numerical and experimental results is lower than 3% in all of the cases. As it can be noticed from results presented here, good free surface and CT

calculations can be obtained using one of the two medium grids reducing significantly in this manner CPU time without excessive loss of accuracy.Table 1: Wigley hull, total resistance coefficient CT

grid CT ΔCT CPU time [h]coarse 4.06x10-3 -2.40% 2.5 (1 core)medium 1 4.25x10-3 +2.16% 18 (1 core)medium 2 4.22x10-3 +1.44% 30 (1 core)fine 4.20x10-3 +0.96% 40 (1 core)exp. 4.16x10-3 -

4.2 Effect of turbulence model

The turbulent flow structure in the proximity of the free-surface can be much more complex than the turbulent flow structure of single phase flows. Existing turbulence models, which have been mainly proposed for single phase flows, may not adequately represent the turbulence structure at the free surface (Senocak and Iaccarino 2005). The creation of a turbulence model able to simulate

correctly turbulent free-surface flows and the interaction between free-surface and turbulent boundary layer represents a challenge in CFD. Moreover, surface tension effects are important for breaking waves at smaller scale. Among the turbulence models available in FLUENT, the SST k – ω and the Realizable k – ε models were compared to investigate possible differences in free-surface shape and resistance predictions. Since Weber and Reynolds numbers are both >>1 for all the cases in this work, surface tension effects were always neglected.

Figure 2: Wigley hull, wave pattern. Isocurves; Fr = 0.267, Re=6.66x106. From the top to the bottom: grid coarse, medium 1, medium 2, fine.

Figures 3 and 4 show the same solution of free-surface for both the turbulence models used. This absence of differences can be due to the absence of particular turbulent structure on the free surface for this case as breaking wave. On the contrary, significant difference appeared in the resistance calculation (Table 2), where the SST k – ω model overestimated CT value while Realizable k – ε underestimated it. However, both models gave a result close to the experimental value.

Figure 1: Wigley hull: wave profile on the hull; Fr=0.267, Re=6.66x106

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Figure 4: Wigley hull, wave pattern. Comparison between Realizable k − ε (top) and SST k − ω (bottom) turbulence models; Fr = 0.267, Re = 6.66x106, fine grid

Table 2: Wigley hull, CT. Comparison between the turbulence models Realizable k-ε and SST k − ω.

grid turb. model CTx10-3 ΔCT

medium 2 Real. k-ε 4.08 -1.92%medium 2 SST k-ω 4.22 +1.44%fine Real. k-ε 4.08 -1.92%fine SST k-ω 4.20 +0.96%exp. - 4.16 -

4.3 Improvement of the convergence

Using VOF model it is easy to encounter convergence problems in presence of a high difference between densities of fluids considered as equations become stiffer. This is the case of air and water where ρw/ρa ≈ 800. As shown in (Reichl and al. 2005), convergence can be improved by altering the ratios between densities and viscosities of fluids; a value of 100 for ρw/ρa and μw/ μa can be used without committing remarkable errors.

As one can see in Figure 5, residuals become more regular and reach lower values than in the real case. The effects on CT (Figure 6) are minimal and mainly due to the major forces acting on the surfaces exposed to air. However, this problem can be relevant only if superstructures are well defined and in the pre-design phase this error can be easily corrected or avoided by eliminating these surfaces from the CT calculation. The effects on the free-surface shape (Figg. 7 and 8) are also negligible since the reduction of the density ratio of the fluids are not so considerable to affect the wave-making.

5 SERIES 60, CB=0.6 HULL

5.1 Free-surface calculations at design trim

The Series 60 hull form is largely used to design ships and it was the subject of several numerical and experimental studies. Extensive towing tank tests were carried out at Iowa Institute of Hydraulic Research (IIHR) (Toda et al. 1992, Longo et al. 1993) and results are partially available on the web1 in digital form. IIHR Towing

1 http://iihr.uiowa.edu

Figure 5: residuals history, real fluids (top) and ρw/ρa = μw/μa

=100 (bottom); Fr=0.267, Re=6.66x106, medium grid.

Figure 3: Wigley hull, wave profile on the hull. Comparison between Realizable k−ε and SST k−ω turbulence models; Fr=0.267, Re=6.66x106,fine grid

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Figure 6: Wigley hull, CT history in the actual case and with ρw/ρa = μw/μa =100; Fr=0.267, Re=6.66x106, medium 2 grid.

Figure 7: Wigley hull: wave profile on the hull, Fr=0.267, Re=6.66x106, fine grid, ρw/ρa = μw/μa =100.

Figure 8: Wigley hull, numerical wave pattern in the actual case (top) and with ρw/ρa = μw/μa =100 (bottom), isocurves; Fr = 0.267, Re = 6.66x106.

tank is 100m long and 3m wide, and tests were carried out on a model 3.048m long at Froude numbers 0.16 and 0.316. The wave pattern around the hull were measured by several longitudinal wave cuts with the model fixed at design trim, while the total resistance were measured with the

model free to vary both the longitudinal trim angle θ and the sinkage.

In this section, numerical simulations were carried out of the Series 60 hull model of L=3.048m for a direct comparison with the experimental tests cited. Froude number considered was 0.316, corresponding to Re=5.24x106. Three grids of 331'652 (coarse), 692'984 (medium) and 1'274'742 cells (fine) were created, with the height of the first layer of cells on the hull walls of 0.001L for all the grids. This allowed to obtain y+<50 on the whole hull surface. A first set of computation was performed with the model fixed at design trim. As in the case of the Wigley hull, from Figures 9 and 10 it appears that solution does not degrade significantly by using a less refined grid as the medium one. Moreover, Figures 9 and 11 show that the calculated free-surface match very well the experimental one both in the wave profile on the hull and in the wave pattern.

Figure 9: Series 60, CB=0.6 hull: wave profile on the hull, Fr=0.316, Re=5.24x106

5.2 Effects of trim and sinkage

As previously seen, the first set of numerical computations was performed with the model fixed at design trim, while experimental measurement of total resistance was executed with the model free to vary both the θ and the sinkage. Table 3 shows a difference between numerical and experimental values of almost 9%, greater than in the case of the Wigley hull. This can be due to the change in trim for this model during towing tank tests, not shown by the previous model considered. Thus, a second set of calculation was carried out considering the model fixed at the trim measured during tests, that is θ=-0.1° (corresponding to a bow down and stern up attitude) and sinkage of 1.19cm (positive for downward displacement).

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Figure 10: Series 60, CB=0.6 hull: wave pattern. Isocurves, Fr=0.316, Re=5.24x106. From the top to the bottom: grid coarse, medium, fine.

Figure 11: Series 60, CB=0.6 hull, wave pattern. Comparison between numerical solution (FLUENT, fine grid, top) and experimental data (bottom); Fr=0.316, Re=5.24x106

Table 3: Series 60, CB = 0.6 hull, CT. Design trim (ΔCT = (CT

CFD – Ctexp)/CT

exp).

grid trim CTx10-3 ΔCT CPU time [h]coarse design 5.41 -9.23% 18 (1 core)medium design 5.43 -8.89% 30 (1 core)fine design 5.45 -8.56% 57 (1 core)exp. free 5.96 - -

Although these values seems to be negligible, Table 4 shows a remarkable variation in the CT

calculation that now falls in the range of uncertainty of the experimental tests both for the medium and the fine grid.

Table 4: Series 60, CB=0.6 hull, CT. Trim correction by experimental data.

grid trim CTx10-3 ΔCT

coarse corrected 5.81 -2.52%medium corrected 5.94 -0.34%fine corrected 5.96 -0.00%exp. free 5.96 -

6 MOTORYACHT

6.1 Ship description and towing tank tests

In this section, the same approach used for the hull forms previously considered, was used for a motoryacht belonging to the category of luxury yacht, produced by ISA Ancona shipyard. Its geometry is quite different from the classical Wigley and Series 60 hull forms mainly because of the fin extended all the ship long and the transom stern. Moreover, the ship has other devices as bow-bulb, fin stabilizers and rudders that for the sake of simplicity are not considered here.

Towing tank tests are available for this ship, carried out in 2004 at the Dipartimento di Ingegneria Navale (DIN) of the University of Naples Federico II (DIN 2004) for several velocities and configurations. The towing tank is 145m long, 9m wide and 4.5m deep. The length of the model used is L=4.583m, corresponding to a scale factor λ=12. Characteristics of the ship in the full and model scale are resumed in Table 5.

Table 5: Motoryacht, geometric characteristics and simulated conditions.

symbol model shipLength between perpendiculars [m]

LPP 4.583 55.000

Water line length [m] LWL 4.183 50.200Length overall [m] LOA 4.583 55.000Hull beam [m] B 0.831 9.970Water line beam [m] BWL 0.804 9.650Fore draft [m] TF 0.250 3.000Aft draft [m] TA 0.250 3.000Average draft [m] TM 0.250 3.000Displacement [t] Δ 0.426 755.000Wetted surface [m2] S 3.989 574.410

Since during these tests only the total resistance and longitudinal trim angle were measured, additional runs were carried out to take longitudinal wave cuts, sinkage and some photos of the wavy free-surface close to the model. For the longitudinal wave cut, eight microprocessed capacitive probes (Figure 12) were used, the first of them placed 1.15m from the middle of the tank and the remaining probes 23cm the one from the other. The arm mounting the probes were placed at half the lenght of the tank. During the experimental tests the model was free to vary trim and sinkage. Because of the impossibility to remove some devices from the model without damaging it, these further tests were carried out with fin stabilizers and shaft brackets.

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6.2 Model-scale simulation

RANS simulations in model-scale were carried out at velocities from 1.485 to 2.524m/s corresponding to Froude number from 0.221 to 0.376. The velocity of 2.524m/s corresponds to the ship design speed of 17kn. For these simulations a grid of 1.9x106 cells was created (Figure 13). The height of the first layer of cells was 0.002m, that allowed to obtain y+ < 100 for the highest velocity simulated.

The model was considered fixed at design trim, that is θ=0 and no sinkage. Figure 14 shows the CT

calculated using the SST k - ω turbulence model. It appears a relevant underestimation of experimental data up to 16.7% for the velocity of 2.524m/s. Then, for this velocity the trim of model was corrected using only the value of longitudinal

trim angle exhibited during towing tank test (θ=0.4°), and subsequently with the sinkage of 2.37cm as well. Moreover, a simulation using the Realizable k - ε turbulence model was also performed for the ship at the correct trim. Table 6 resumes that, showing a substantial variation of calculated CT. As seen for the Series 60 hull, the trim has a strong influence on the calculation of the CT and the Realizable k - ε model gives a smaller value than the SST k - ω.

Figure 14: Motoryacht, coefficient of total resistance versus the Froude number; FLUENT: design trim, experimental: set-free

Table 6: Motoryacht, coefficient of total resistance calculated on model-scale; Fr=0.376.

turb. model

trim correction

CTx10-3 ΔCT

SST k - ω no 7.33 -16.7%SST k - ω θ 8.28 -5.9%SST k - ω θ, sinkage 9.75 +10.7%Real. k - ε θ, sinkage 9.60 +9.1%exp. - 8.80 -

Figure 15 and Figure 16 show respectively the numerical wave pattern and longitudinal wave cut from the first probe compared with the measured data at Fr=0.376. The experimental wave pattern was obtained by interpolation from the wave cuts measured. Data sampled by the probes were filtered in order to eliminate noise due to small perturbations of the water surface.

Figures 17 and 18 show the computed tree-dimensional wavy free-surface directly compared with photographs taken during the towing tank tests for the same Froude number. Both the bow and the stern waves were reproduced correctly as well as the roll-down of the former and the transom stern.

Figure 13: Motoryacht, computational grid

Figure 12: Probes for the longitudinal wave cut

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Figure 15: Motoryacht, wave pattern. Comparison between experimental data (top) and numerical solution (FLUENT, bottom). Fr=0.376, Re=1.15x107.

Figure 16: Motoryacht, comparison between numerical (FLUENT) and measured longitudinal wave cut; y=1.15m, Fr=0.376, Re=1.15x107

6.3 Full-scale simulation

A full-scale simulation of the ship was attempted to evaluate the feasibility and the accuracy of such a calculation on low-cost hardware. The grid employed was similar to the one for the model scale simulation, but with more cells lengthwise and nearby the design water plane, for a total of 2.8x106 cells. The height of the first layer of cells on the hull walls was of 0.007m to avoid an excessive aspect ratio of them. Since the calculation showed 400 < y+ < 700, the grid was refined only on the hull walls by splitting each cell twice. Doing this, the final grid was made of 3.6x106 cells. The simulation was performed for Fr=0.378 and considering ρw/ρa = μw/μw = 100, as seen in Section 6. The ship was placed at trim and sinkage measured from the model tests. In absence of experimental data for the actual ship, calculated coefficient of resistance was compared with the results from the ITTC '57 procedure (Table 7). Also, the calculated CT was decomposed in viscous and pressure coefficients of resistance, and then compared respectively with the viscous and

the residuary coefficients of resistance derived from ITTC.

Figure 17: Motoryacht, images of the bow wave formation. Comparison between the actual wave and the numerical solution (FLUENT); Fr=0.376, Re=1.15x107.

Figure 18: Motoryacht, images of the stern wave formation. Comparison between the actual wave and the numerical solution (FLUENT); Fr=0.376, Re=1.15x107.

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Table 7: Motoryacht, full-scale simulation. Comparison among calculated coefficients of resistance and results from ITTC ’57 procedure; Fr = 0.376, Re = 4.79x108.

CV CR, CP CT

ITTC ’57 1.92x10-3 5.87x10-3 7.79x10-3

FLUENT 1.86x10-3 6.66x10-3 8.53x10-3

7 COMPARISON WITH THE FREE

SURFACE POTENTIAL FLOW SOLUTION

The free-surface solutions obtained by RANS/VOF simulations were compared with the solutions of the potential flow obtained by means of the panel method solver included in the SHIPFLOW software.

7.1 Wigley and Series 60 hulls

Wigley and Series 60, CB=0.6 are standard hull forms that can be easily treated by a panel method. Hull surface and free-surface are both discretized with a total of 4128 panels for the Wigley and 2400 panels for the Series 60 hull. The number of panels was determined checking the convergence of the coefficient of wave resistance Cw as it increases (see Figure 19 for example) and considering a minimum of 25-30 panels/wavelength. The hulls were considered fixed at the design trim. Discretization of the free-surface for the two hulls is shown in Figures 20 and 21.

The wave profile on the hull, computed from the displacement of the free-surface panels an compared whit the RANS/VOF solution and the experimental data (Figures 22 and 23) shows an incorrect prediction of crests and hollows. In particular the amplitude of the bow wave of the Wigley hull is strongly underestimated.

Furthermore, even if the wave frequency is correctly predicted, it appears a phase displacement for both hulls. This phase displacement does not appear in the VOF solution.Figures 24 and 25 show respectively the potential solution compared with the VOF solution for the Wigley hull and with experimental data for the Series 60 hull.

Figure 20: Wigley hull, SHIPFLOW, discretization of the free-surface; Fr=0.267.

Figure 21: Series 60, CB=0.6 hull: SHIPFLOW, discretization of the free-surface; Fr=0.316

Figure 22: Wigley hull, wave profile on the hull. Comparison between the solution RANS (FLUENT) and the potential solution (SHIPFLOW); Fr=0.267.

Figure 19: Wigley hull: SHIPFLOW, convergence of Cw

versus the number of panels; Fr=0.267.

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Figure 23: Series 60, CB=0.6 hull, SHIPFLOW, wave profile on the hull; Fr=0.316.

Figure 24: Wigley hull: wave pattern. Comparison of the RANS solution (FLUENT, top) and the potential solution (SHIPFLOW,bottom), isocurves; Fr=0.267.

Figure 25: Series 60, CB=0.6 hull, wave pattern. Comparison between the potential flow solution (SHIPFLOW, top) and experimental data (bottom); Fr=0.316.

7.2 Motoryacht

Because of the complexity of its geometry, the motoryacht described in Section 6.1 could not be treated as a standard hull form using the SHIPFLOW panel method. The geometry required a more accurate discretization because of the

presence of the fin and the shape of the stern, as well as the free-surface. A problem encountered during the solution of the potential flowfield was due to the stern that becomes perpendicular to the symmetry plane and causes highly distorted panels on the free-surface in that region. Considering this case as transom stern and recurring to the specific option available in SHIPFLOW (FLOWTECH 2004) did not produce any results. The only way to avoid this problem was spacing out free-surface panels from the stern (see Figure 26), although not properly correct.

Figure 26: Motoryacht, SHIPFLOW, discretization of the free-surface; Fr=0.376.

Figure 27 shows the computed wave patten compared with experimental results. While in the previous cases the wave sequence were correctly predicted, apart from a phase displacement, for this hull the potential-flow wavy surface differs significantly from the actual one. This is due to the free-surface phenomenology shown in Figures 17 and 18 that cannot be simulated under the hypothesis of potential flow. Figure 28 shows that the characteristic wave shape near the hull was totally unpredicted because of the impossibility of the free-surface panels to roll-down and intersect themselves.

Figure 27: Motoryacht, wave pattern. Comparison between the potential flow solution (SHIPFLOW, bottom) and experimental data (top); Fr=0.376.

The wrong prediction of the free-surface affected the calculation of the coefficient of wave resistance, which exhibited a nonphysical trend at low Froude number, as shown in Figure 29. In this

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Figure the coefficient of wave resistance is compared to the coefficient of residuary resistance obtained by the methodology ITTC '57.

Figure 28: Motoryacht, three-dimensional view of the free-surface obtained by the code SHIPFLOW; Fr=0.376.

Figure 29: Motoryacht, comparison between the wave coefficient of resistance calculated by SHIPFLOW and the residuary coefficient of resistance obtained experimentally (methodology ITTC'57).

8 CONCLUDING REMARKS

In this work the Volume of Fluid (VOF) method implemented in the RANS software FLUENT was employed to determine the free-surface wave flow around three different ship hulls advancing steadily in calm water. Particular care was given to the grid generation to avoid problems of reflection of the waves and to minimize the computational efforts. It was shown that the convergence can be improved by increasing the density ratio between air and water without any relevant lack of accuracy in both free-surface and resistance predictions.

Both the SST k - ω and the Realizable k - ε turbulence models gave similar results concerning the free-surface shape for the Series 60 standard hull form. Differences appeared in the coefficient of total resistance calculation, for which the former produced a slightly higher value that the latter.

Proposed results showed that it is possible to obtain a good estimation of the free-surface shape and CT using the RANS/VOF approach without particular hardware resources for trim-fixed ship hulls advancing steadily in calm water. Furthermore, a numerical solution can be obtained for ship hull of moderate dimensions directly on the full-scale model and so avoiding extrapolation methods. For that purpose an optimized, good quality grid is necessary. Also, CT depends strongly on the trim, so a preliminary estimation of it is necessary. Alternatively a dynamic numerical simulation is required. The VOF method showed to be reliable in predicting complex wave shapes as the ones generated by a semi-displacing ship hull form.

The free-surface shape obtained by means of the VOF method was compared with the one obtained using the panel method of the SHIPFLOW software. The potential flow solution appeared to be sufficiently accurate for standard hull forms, although showing some shortcomings compared to the RANS/VOF solution. On the contrary, panel methods fails for those geometries that generate complex wave forms because of the impossibility of the panels to intersect each other and of the hypotheses themselves this approach is based upon. Furthermore, the presence of the free-surface makes the discretisation more complicated than for classical panel methods, giving rise to several numerical problems for complex geometries, especially for hulls with transom stern.

9 REFERENCES ANSYS (2006) FLUENT® 6.3 User’s Guide. Fluent Inc. Ferziger, J. and Perić, M. (2001) Computational Methods for Fluid Dynamics. Springer.

FLOWTECH (2004) Shipflow® 3.1 user’s manual. Flowtech International AB, Goteborg, Sweden.

Harlow, F. and Welsh, J. E. (1965) Numerical calculation of time dependent viscous incompressible flow with free surface. The Physics of Fluids: 8, 2182-2189.

Hirt, C. W. and Nichols, B. D. (1981) Volume of fluid /VOF/ method for the dynamics of free boundaries. Journal of Computational Physics: 39, 201-225.

Larsson, E. and Janson, C. E. (1999) Potential Flow Calculations for Sailing Yachts. Proceedings of 31st

WEGEMT School on CFD for Ship and Offshore Design.

Longo, J., Stern, F. and Toda, Y. (1993) Mean-Flow Measurements in the Boundary Layer and Wake and Wave Field of a Series 60 CB = 0.6 Ship Model - Part 2: Scale

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Effects on Near-Field Wave Patterns and Comparisons with Inviscid Theory. Journal of Ship Research: 37, 1, 16–24.

Muzaferija, S., Perić, M., Sames, P. and Schellin, T. (1998) A Two-Fluid Navier-Stokes Solver to Simulate Water Entry. Proc. 22nd Symposium on Naval Hydrodynamics: 277–289.

Pranzitelli, A. (2008) Previsione numerica del campo fluidodinamico e della resistenza al moto di carene. Ph.D. thesis, Università degli Studi di Napoli Federico II.

Reichl, P., Hourigan, K. and Thompson, M. C. (2005) Flow past a cylinder close to a free surface. Journal of Fluid Mechanics: 533, 269–296.

Senocak, I. and Iaccarino, G. (2005) Progress towards RANS simulation of free-surface flow around modern ships. Annual Research Briefs. Center for TurbulenceResearch, Stanford University.

Toda, Y., Stern, F. and Longo, J. (1992) Mean-Flow Measurements in the Boundary Layer and Wake and Wave Field of a Series 60 CB = 0.6 Ship Model - Part 1: Froude Numbers 0.16 and 0.316. Journal of Ship Research,36, 4, 360–377.

DIN (2004) Carena C 0403 - 560 Motoryacht ISA Ancona, Italy. Relazione 71. Dipartimento di Ingegneria Navale, Università degli Studi di Napoli Federico II.

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