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Calculation of hydrodynamic coefficients of ship sections in roll motion using Navier-Stokes equations
Ocean Engineering 133 (2017) 36–46
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Calculation of hydrodynamic coefficients of ship sections in roll motion using Navier-Stokes equations
MARK
⁎
A. Lavrov, J.M. Rodrigues, J.F.M. Gadelho, C. Guedes Soares
Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisbon, Portugal
A R T I C L E I N F O
A BS T RAC T
Keywords: Nonlinear roll motion Roll damping Hydrodynamic coefficients Navier Stokes equations
The flow in the vicinity of 2D ship sections carrying out forced roll motions is simulated using the solution of the Navier-Stokes equations implemented in OpenFOAM. A hybrid Lagrangian-Eulerian adaptive mesh scheme is applied to solve forced motions on ship sections oscillating in calm water as well as the free surface capturing problem. Three sections of a containership, with differing geometrical properties, are investigated by comparing viscous unsteady results with potential flow solutions. The method presented in this study has been validated by comparing numerical results with experiments performed on a rectangular section. The influence of viscosity, particularly on vortex shedding, on the roll motion is investigated for an array of oscillation frequencies.
1. Introduction Several 3D boundary element methods, incorporating a potential flow approach, have been developed to solve the radiation (and diffraction) problem. The frequency domain approaches proposed by Inglis and Price (1981), Newman and Sclavounos (1988) and Nakos and Sclavounos (1990), and the time domain methods developed by Lin and Yue (1990) and Nakos et al. (1993) represent some of the most significant milestones in applying Boundary Element Methods for the solution of ship motions. Nonetheless, 2D methodologies to compute hydrodynamic coefficients are still very much in use nowadays. This undoubtedly comes from the high speed in computation, due to the simplification of the domain, but also one should bear in mind that 3D methods also pose specific problems. An example is the inclusion of forward speed effects in 3D methods similar to that by Papanikolaou and Schellin (1992). In their method, usage is done of zero speed Green functions formulations with the advance speed being accounted in a somewhat similar manner to that of strip theory (Salvesen et al., 1970). The method is valid, in all frequencies, for slender bodies where lengthwise hydrodynamic pressure variation is small relative to its girthwise variation. Another factor to take into consideration is that evaluating the finite depth Green's function for the zero speed 3D radiation-diffraction problem is extremely complicated, as is the evaluation of the forward speed frequency domain Green's function for the 3D ship motion problem (Datta et al., 2011). The earliest of these two dimensional methods were based on
⁎
conformal mapping. From the seminal works of Lewis (1929), Tasai (1959,1961) and Ursell (1949), several conformal mapping techniques have been proposed and benchmarked throughout the years (Ramos and Guedes Soares, 1997); examples of recent studies using multiconformal mapping are the ones by Rajendran et al. (2015a,2015b). However, conformal mapping methods present limitations regarding the geometry of the ships’ sections to be assessed: Frank's close fit method (Frank, 1967) surpasses those limitations but suffers from the phenomenon of irregular frequencies (John, 1950). The method by Yeung (1973) is immune to this problem while fulfilling an arbitrary section geometry requirement in addition to implementing the possibility of shallow water consideration. The method was further developed by Sutulo and Guedes Soares (2004), and applied to the study of a set of sections of the S175 hull in different depths and bottom geometries in Sutulo et al. (2009, 2010). All the above mentioned methods are based on the idea that the fluid is ideal and implement the potential flow paradigm, failing to account for the viscous damping in roll, which typically becomes substantial relative to the (small) wave damping contribution by slender, ship-like, bodies (Lewandowski, 2004). So, to include the viscous damping component it is necessary to resort to forced roll or roll decay model tests, as discussed, for example, by Uzunoglu and Guedes Soares (2015), whenever the application of semi-empirical methods, e.g. Miller (1974) and Ikeda et al. (1978), are not deemed accurate. Experimental tests consistently acknowledge the nonlinear nature of such a coefficient, with 2nd or 3rd order polynomial regressions being usually proposed. However, Fernandes and
Corresponding author. E-mail address: [email protected] (C. Guedes Soares).
http://dx.doi.org/10.1016/j.oceaneng.2017.01.027 Received 6 February 2016; Received in revised form 9 December 2016; Accepted 27 January 2017 0029-8018/ © 2017 Elsevier Ltd. All rights reserved.
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for very small roll amplitudes: time domain methods were deemed inevitable for proper levels of accuracy. Yu (2008) obtained a good correlation of his NSEQ code's results with the hydrodynamic coefficients computed with FLUENT, considering laminar and turbulent flow, for ship-shaped hull sections without bilge keels – turbulent effect was found small. Results were not so good for low frequency oscillations when bilge keels are present, though. Also worth of mention is the investigation of roll damping decay of an FPSO, fitted with bilge keels, by Avalos et al. (2014) by means of the incompressible two-dimensional Navier-Stokes equations. A strong dependence of the damping coefficient on the bilge radius was reported and, more interestingly, the occurrence of the so-called damping coefficient saturation phenomenon. This means that a fixed value of this quantity, irrespective of the roll motion amplitude, was observed, in agreement with the conclusions of Oliveira and Fernandes (2011). In what relates to the application of distributed applications, Quérard et al. (2008) carried out RANS based simulations with ANSYS-CFX 11.0, and concluded on the superiority of their predictions compared to potential flow analysis. Four sections’ geometries were investigated: rectangular, triangular, chine and bulbous bow; however, the bulbous bow section did not exhibit an actual protruding bulb. In Quérard et al. (2010), the same tool was used to simulate the roll motion of 2D FPSO sections. Another important study is that of Henning (2011), where predictions of large motions of simple shapes (triangle, square and demi-circle) were carried out resorting to the proprietary program FLUENT. The latter compared worse with the experimental results by Vugts (1968), probably due to the large amplitudes tested. Bonfiglio and Brizzolara (2013), have carried out comparisons between monohull, catamaran and SWATH midship sections, using OpenFOAM. However, only the heave motion is considered. More recently, when OpenFOAM was used by Kwon et al. (2014) to simulate the forced roll motion of a circular cylinder, a mesh with approx. 200 thousand cells resulted in moment amplitudes with errors ranging from 30% to 80%. In the following year, a study on forced motion of a circular cylinder, was carried out by Gadelho et al. (2015), where the potential flow damping coefficient was meant to be estimated by measuring the height of the radiated waves. OpenFOAM was used to calculate the flow for the heave and sway motions of the body, which provided good results for lower frequencies. But divergences with Sutulo et al. (2009) were witnessed for higher frequencies, probably due to the high wave slope of the corresponding radiated waves which violate the linear free surface assumption made. From the above, it is clear that research on forced roll motions of 2D sections, by using CFD techniques, has been mainly focused on, but not limited to, the effect of bilge keels on midship sections of FPSO hulls. In the present paper, three typical mono-hull sections, with clearly distinct geometrical topologies, are considered: bulbous bow, midship and skeg. OpenFOAM is used to determine the moment amplitude of forced roll motions of each, with comparisons between viscous unsteady results with potential flow solutions being carried out. The methodology applied is described and validated by comparing numerical results with experiments performed on a rectangular section. A laminar flow is considered while solving the Navier-Stokes equations using OpenFoam's interDyMFoam solver in an incompressible isothermal fluid; at each time step, the mesh is morphed, according to the motion lay imposed at the hull surface, following a hybrid Lagrangian-Eulerian approach. The ultimate objective of the present paper is the preliminary assessment of the need for CFD computations of roll motion hydrodynamic coefficients and its dependence on the sections topology. These are to be applied to methods resorting to 2D formulations of the coefficients, where such a formulation is regarded as sufficiently accurate for the specific problem being solved in the first place. Applying a full 3D viscous approach, with its much heavier computa-
Oliveira (2009) and Oliveira and Fernandes (2011) question the ability of the application of polynomials to properly account for the viscous damping in large roll amplitudes, even though the traditional quadratic approach is seen to remain accurate for V shaped hulls. A (nonpolynomial) bi-linear approach, which acknowledges the existence of two vortex shedding regimes separated by a roll amplitude threshold, has been developed and tested by the same authors (Oliveira and Fernandes, 2013). Still, the time and cost necessary to carry out such tests may be significant: Computational Fluid Dynamics (CFD) based calculations become attractive - e.g. Wanderley et al. (2007). CFD methods are, thus, being increasingly used nowadays for roll damping estimation in an effort to improve, or substitute entirely, the application of traditional semi-empirical approaches to it. Few CFD based studies on the roll radiation problem consider a domain incorporating the whole vessel. The complexity of the domain discretization, problem setup and the cumbersome computational burden result in a strip-wise approach being the standard. Chen et al. (2001) carried out time domain simulations of the entire ship using Reynolds-Averaged-Navier-Stokes equations (RANS) and applying a Chimera domain decomposition approach with approx. 65,000 cells. In their study, forced large amplitude roll motions of a ship fitted with bilge keels with a hull presenting a skeg were simulated. The authors realized the high viscosity and low wave damping generated by the bilge keels, while both damping effects were seen significant as a result of the skeg. Also using RANS, the roll motion of a surface combatant ship was numerically predicted by Wilson et al. (2006), but a much larger number of cells was part of the domain: approx. 2.3 million. In both studies, the strong viscous effect of bilge keels was evident; furthermore, it was shown that such methods are accurate for practical geometries. Studies dealing with 2D roll motion using CFD typically carry out simulations concerning the effect of bilge keels fitted amidships of vessels with a long midbody having a close to rectangular sectional shape. In Sarkar and Vassalos (2000) a RANS-based technique for simulation of the flow near a 2D rolling rectangular cylinder was applied; only approx. 8000 cells were required for successful comparisons with Vugts (1968) experiments and Yeung et al. (1998) inviscid calculations. A relatively accurate RANS based numerical prediction of the time history of the roll motion of a rectangular cylinder, fitted with bilge keels, was achieved by Yeung et al. (2001). However, a total of 40,000 cells were necessary for a good comparison with experimental data, despite the slight over estimation of the motion amplitude. Kakar (2002) solved the Euler equations regarding the inviscid flow around a 2D rectangular section, with and without bilge keels, under forced roll motion. Grids with 37, 74 and 118 thousand elements were tested and the results revealed a convergence to the damping coefficients obtained by Yeung et al. (1998), though much higher values of added mass were observed, especially in low frequency range. On the other hand, Vugts (1968) reports the less accurate experimental results of the hydrodynamic coefficients in roll, due to the mechanical difficulties of the tests, which adds to the smaller order of these values relative to translational modes. Later, Kacham (2004) resorted to Navier-Stokes equations (NSEQ) to solve the same problem, using three grids with 17, 21 and 25 thousand cells. Comparisons with Kakar (2002) and Vugts (1968) confirmed the over prediction of added mass coefficients experienced by Kakar (2002) for large periods and revealed some inaccuracies in the damping coefficient as well. Finally, Kinnas (2004) further developed the code used by Kacham (2004) and Kakar (2002) and managed to achieve good results for all frequency values. Vinayan et al. (2005) studied the combined effect of the free surface and flow separation at the bilge keels and Kinnas et al. (2007) found that in the case of roll motion nonlinear effects are considerable, even 37
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→ ⎯ ∇(U ) = 0
tional burden, would thus be counterproductive, albeit the more realistic results. A hybrid approach to the radiation problem can thus be formulated, where CFD based solutions are only applied to zones of the hull where these are deemed necessary, with the potential formulation still being applicable to the remaining sections. Although past investigations have provided general remarks on this issue, the methodology of the present study and the specific consideration of ship sections in forced roll motion, is, to the best of the authors’ knowledge, a novelty. In Section 2, the problem formulation for the forced roll motion, the fluid's governing equations and their numerical implementation are presented. The process to obtain the hydrodynamic coefficients is described in Section 3. The validation of the scheme is done in Section 4, by comparing its results to those by Vugts (1968) and Kacham (2004), regarding a rectangular section; results for the three ship sections are also presented and compared with those by Sutulo et al. (2009). Finally, some conclusions are drawn in Section 5.
→ ⎯ → ⎯ → ⎯ ∂ρU + ∇(ρU ⊗ U ) = − ∇τ , ∂t
(4) ⎯ → where ρ represents the fluid density, U stands for the velocity vector, with components u1, u 2 , u3 in the x1, x 2 , x3 directions, respectively, and the tensor τij is the stress tensor in a Newtonian fluid, with general form:
⎛ ∂u ∂uj 2 ∂u ⎞ τij = −δijp + μ⎜⎜ i + − δij k ⎟⎟ , i , j , k = 1, 2, 3 ∂xi 3 ∂xk ⎠ ⎝ ∂xj ∂u
2.1. Problem formulation
∂ωq ∂t
The case of a bi-dimensional semi-submerged body, forced to perform small angular oscillatory motion with prescribed amplitude and period, is to be investigated and the sectional hydrodynamic coefficients determined. The motion, φ , may be described by the following kinematic equation:
+ ui
∂ωq ∂xi
= 0 , q = 1, 2
(6)
The governing equations are evaluated on the domain bounded by the bottom, a wall on each side, the oscillating body contour and a top limit ceiling representing the atmosphere. The boundary condition for the first four boundaries sets the normal pressure gradient to be null:
(1)
∂p =0 ∂n
where φ0 is the amplitude of the motion, ω its circular frequency of oscillation and t represents time. The radiation problem is formulated by not considering body mass inertial effects or restoring forces, while imposing a single mode oscillation to the body. By doing so, only the forced motion forces and the fluid hydrodynamic reaction forces are present. The fluid forces can in turn be decomposed between forces in phase with the acceleration and those in phase with the velocity. These forces correspond to added masses and damping coefficients multiplied by the acceleration and velocity, respectively. The single mode of oscillation in roll will have fluid reactions in more than one direction; considering only the nonzero components, the following equations of motion hold:
(7)
The atmosphere boundary has the following boundary conditions:
∂U = 0, ∂n
∂p = 0, ∂n
ωI = 0
(8)
2.3. Numerical implementation A schematic view of the geometry of the computational domain is presented in Fig. 1. The inertial coordinate system is located at the intersection of the section symmetry plane and the undisturbed water surface. The domain must have sufficient horizontal extent to prevent too early reflection of the water waves on the lateral walls. In addition, the depth must be such so as to prevent any significant shallow water effects on the body and on the free surface, i.e. it must ensure the validity of the dispersion relation: k =ω2 / g , where k is the wave number and g the standard gravity acceleration. Both these issues are more significant to lower frequencies. On the other hand, too big a domain will force the presence of a large number of cells. The level of mesh refinement in way of the free surface does not allow for sufficient mesh coarsening in areas away from the body, in order to lessen the increase number of cells as would otherwise be possible. This concern becomes more serious in higher oscillation frequencies as the wave slope becomes significant. The minimum domain half-width, L1/2 , may be determined by the total number of oscillations, N , without interference from reflected waves, at some point located at a distance ∆x from the centre of the domain:
μ24 φ + η24φ = Fx sin(ωt + ϕ24 ) μ34 φ + η34φ = Fysin(ωt + ϕ34 ) μ44 φ + η44φ = Mzsin(ωt + ϕ44 )
(5) ∂uj
In the abovementioned equation, p is the pressure, ∂xi + ∂x the j i u the normal stress and μ denotes the shear stress, ∂uk /∂xk = ∇⋅→ dynamic viscosity coefficient. The Volume of Fluid method (VoF) (Hirt and Nichols, 1981) is used for capturing the water-air interface. If the concentrations of water and air in each cell are defined by ω1 and ω2 , respectively, these sum to one at all times and are determined through the continuity equation:
2. Numerical method
φ = φ0sin(ωt ),
(3)
(2)
where μi4 and ηi4 are the sectional added mass and damping coefficients, respectively, in the ith direction, due to the roll motion, Fx , Fy and Mz represent the sway force, heave force and roll moment amplitudes, respectively, with their corresponding phase angles: ϕ24 , ϕ34 , and ϕ44 ; phases are relative to the motion. The subscripts x , y and z denote alignment with the base components x1, x 2 and x3 of a direct coordinate system with origin at the intersection of the section vertical mid line and the undisturbed waterline. Evidently, the force Fy is zero for a body symmetric about the x 2 axis. From the knowledge of the right hand sides of Eq. (2), and the prescribed motion characteristics included in Eq. (1), the hydrodynamic coefficients may be determined. These forces represent the action of the fluid on the body and are calculated from the equations which govern the flow. 2.2. Flow governing equations A laminar flow regime is assumed, as in Inok et al. (2014). The Navier-Stokes equations for the time-depended unsteady incompressible flow in Cartesian coordinates are:
Fig. 1. Sketch of the computational domain (not to scale).
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L1/2 =
λN +2Δx 4
the values listed in Table 2 were set for the ship sections simulations. Eq. (9) gives a close to eight cycles of pure radiation oscillations of the ship sections when measurements are taken at the body surface, for a period of 13.83 s (the longest considered in the present study). For the same period, the minimum depth would be about 150 m. But the d=95 m value implemented was shown to result in the bottom presence not influencing the flow about the sections tested significantly. In addition, the results for depths equal to or higher than 95 m are virtually the same in Sutulo et al. (2009). Four mesh levels of increasing cell density were used and denominated M1 through M4. The total number of cells (hexahedra and prisms), for each section, is listed in Table 3 - only hexahedral cells are used for the rectangular section case. Meshes for the ship sections mostly include hexahedral cells; a small number of prismatic cells is situated near the section's contour. Only Section S22 required using the finest mesh, for it has a complicated shape and generates a lot of vortices. For the M3 mesh, the dimensions of the cells in the x1, and x2 directions, εx and εy, near the contour of the rectangular and S02 sections are εx/B≈εy/B≈0.003 and εx/B≈εy/B≈0.008, respectively; other ship sections have mesh elements’ dimensions of the same order of magnitude. All calculations are conducted using Courant number equal to 0.5, while the time step varied between 8*10−4 s and 10−2 s, depending on the section and the mesh. The number of cycles, for which measurements at the body surface are free of reflection effects, drops to two for the case of the semisubmerged rectangular section oscillating at the maximum period considered, T=3.59 s. The practical procedure adopted for obtaining the hydrodynamic coefficients was to calculate the value of μ and ν from the evaluation of Eqs. (12) and (13) in the second and remaining periods of oscillation, which are not affected by reflected waves. For the particular case of the rectangular section with T=3.59 s, this means that only one period, the second, was analysed, as the first is the most subjected to transient effects as depicted in Fig. 3 for T=0.9 s. Still regarding the rectangular section simulations’ setup, the minimum depth, as per the deep waters approximation, would be about 10.1 m. From Table 1 it is seen that these limits are not respected in the simulations carried out, but one should recall that this minimum depth decreases for all the other (higher) frequencies simulated. An Intel Core i7-4770K CPU @3.5 GHz×8 processor was used for calculations. The computational time varied between 10 and 100 h, depending on the mesh applied and the period considered.
(9)
where λ=2πg / ω2 is the wave length. If the evaluation of the quantities of interest is to be done on the section contour then Δx ≅B /2 , with B denoting the sectional breadth. Domain discretization for the case of rectangular sections may be carried out by resorting to OpenFoam's utility BlockMesh, while more complicated geometries require OpenFoam's SnappyHexMesh tool. The latter manipulates an initial grid created with BlockMesh to adapt it to the actual shape of a section in its contour's vicinity. The SnappyhexMesh tool is more suitable to work with full 3D meshes, so OpenFoam's tools ExtractMesh and CreatePatch are used to create meshes with a single cell in the direction perpendicular to the sections. Smooth element size transition was applied to the mesh: a finer mesh is present in way of the section contour, getting progressively coarser at farther locations. Finally, the interDyMFoam solver of OpenFOAM is used for solving Navier-Stokes equations, which are evaluated on the aforementioned adaptive mesh. It implements the PIMPLE algorithm, which is based on the integration of the PISO algorithm of Issa (1986) and the SIMPLE algorithm of Patankar and Spalding (1972). 3. Determination of the hydrodynamic coefficients A precise way of determining the hydrodynamic coefficients is by substituting Eq. (1) into Eq. (2) and solving it for both coefficients, e.g. for roll into sway, of a symmetric contour, one has:
Fx
μ24 =
φ0ω2
cosϕ24
(10)
Fx sinϕ24 φ0ω
υ24 =
(11)
The force amplitudes may be obtained directly from the force time record, taken when the force oscillation has reached a steady phase, whereas the phase angles may be found from the lagging of the force up-zero crossings relative to the motion up-zero crossings. Nevertheless, a more general and robust method is to take the cosine and sine Fourier transforms of the force or moment, e.g. for the rollinto-roll case one has:
μ44 =−
ν44 =
1 πφ0ω
1 πφ0
∫0
∫0
T
Mz(t )sin(ωt )dt
(12)
4.1. Square section
T
Mz(t )cos(ωt )dt
The square section corresponds to the one experimentally investigated by Vugts (1968) and whose hydrodynamic coefficients were numerically calculated by Kacham (2004). The time history of Mz , with different mesh density levels is shown in Fig. 3, relative to a period T=0.9 s: a successful convergence is evident. The comparison between the results of the present study, calculated from Eqs. (12) and (13), with those from the aforementioned two authors is shown in Fig. 4, where the following non-dimensionalizations are applied:
(13)
where T is the period of oscillation. 4. Results The forced roll motion of 2D floating bodies is investigated, where the hydrodynamic coefficients are calculated for a set of oscillations with given amplitude and frequency. Four semi-submerged sections are considered as case studies: one semi-submerged rectangular section, with bilge radius 2.5 mm, and three semi-submerged sections – S02, S10 and S22 – of the container ship S175. Each section's breadth (B) and draft is listed in Table 1 and their geometries are shown in Fig. 2. From previous experience acquired in Gadelho et al. (2014,2015),
μˆ = μ44 /4ρQb 2
νˆ = ν44
ωˆ = ω44
Table 1 Main parameters of the investigated sections. Section:
Rectangular
S02
S10
S22
Breadth (m) Draft (m)
0.4 0.2
3.06 9.5
24.84 9.5
9.0 9.5
b /4ρQb 2 g b g
(14)
(15)
,(16)
where Q is the submerged area of the rectangular section and b represents the draft. The amplitude of the forced roll motions is 0.05 rad and the period of the oscillations ranges between 0.6 s and 3.59 s. A good agreement with the experimental values, regarding the damping coefficient, for both the present study and the one by Kacham (2004) is observed, while a divergent trend is seen for the lower 39
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Fig. 2. Sectional shapes and mesh views: a) rectangular section; b) S02; c) S10; d) S22; e) mesh for rectangular section; f) mesh for S02 (not to the same scale). Table 2 Domain semi-width, water and air height. Section
Semi-width L1/2 (m)
Water height d (m)
Air height a (m)
Rectangular S02, S10, S22
12 600
2 95
0.6 10
Table 3 Number of cells (hexahedra and prisms) in the simulations. Meshes Section
M1
M2
M3
M4
Rectangular S02 S10 S22
58,584 251,932 251,476 251,906
118,262 327,726 319,448 327,320
234,604 454,876 419,818 430,332
589,902
Fig. 4. Dependence of added mass and damping on frequency for roll of rectangular section.
frequencies. This divergence is much more pronounced in the results of Kacham (2004), though. The same behaviour is seen regarding the added mass, but with not so good correlations with the experimental values. The larger deviation of the computed values from the experi-
Fig. 3. Time history of moment Mz applied on the rectangular section, with three different meshes; Period of oscillation is T=0.9 s.
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Table 4 Influence of the domain semi-width on added masses and damping coefficients regarding rolling of rectangular section for T=3.59 s and λ=14.2 m. L1/2=12 m M3
μˆ44 0.0523
L1/2=71 m 926,738 cells
νˆ44 0.00318
μˆ44 0.0532
νˆ44 0.00311
Fig. 6. VoF snapshots for section S02 during a roll half-cycle with T=3.91 s, using mesh M3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
mental ones, for lower frequencies, is probably a consequence of the shallow water effect in those longer wavelengths. To confirm that the chosen value of L1/2 allows getting results that are not exposed to the influence of the reflected wave, an additional calculation for the rectangular section with period 3.59 s was carried
Fig. 5. Time history of moment Mz applied on sections S02, S10 and S22, with three different meshes; Period of oscillation is T=13.83 s.
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roll with an amplitude φ0=0.056rad , for eight different oscillation periods between 3.91 s and 13.83 s, during 60 s. The time series of the roll moment, for an oscillation period T=13.83 s, is shown in Fig. 5. Convergence is manifest for all three sections and it is seen that the moment amplitude is significantly higher for S02, with values in excess of 1.75 times that of S10 and S22. VoF snapshots for section S02 during a roll half-cycle, for the smallest period and using mesh M3, are shown in Fig. 6. In each cell, red corresponds to ω1=1 and blue to ω2=1;
out with a wider domain; corresponding parameters and results are listed in Table 4. It is seen that hydrodynamic coefficients resulting from this variation differ approx. 2%, which allows for the L1/2 domain width to be seen as suitable. 4.2. Ship sections In the simulations carried out, each ship section has been forced to
Fig. 7. Evolution in time of pressure fields for section S02; T=3.91 s and t0=7.82 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Evolution in time of pressure fields and vortices dynamics for section S10; T=3.91 s and t0=7.82 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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where L ref is the maximum of the draught and the half breadth at the waterline, for each contour. Due to symmetrical geometry of the sections, the roll-into-heave coefficients are null and are not presented. A good agreement is manifest for section S02 roll-into-roll and rollinto-sway results, for both coefficients, strengthening the conviction that potential flow solvers suffice for accurately predicting the flow about similar bulbous bow sections, due to their streamlined geometric nature. It is but for the coarsest mesh that the results diverge from the potential ones, mainly in the higher frequency region, as it would be expected due to the higher acceleration levels and generated waves’ slope. A near to 10% discrepancy of the added mass in roll-to-roll relative to that of Sutulo et al. (2009) resulted for S10, though the difference reduces to virtually zero in roll-to-sway. The damping coefficient follows the same pattern, however augmented due to the known viscous effects discussed previously relative to eddy making. A 20% discrepancy is observed in the added mass results of section S10. This section contour at the waterline is far from adhering to the constant water plane assumption made in Sutulo et al. (2009), which may be contributing to this phenomenon. This section required the finest mesh to be applied to ensure convergence, with the damping coefficient in roll-into-roll reaching between three to four times that for potential flow. The difference is much lower, however significant, in roll-into-sway: the substantial eddy roll damping component is thus exposed. In a similar way to the square section, an additional calculation with increased width domain was carried out for S02 with oscillation period 13.83 s - results are listed in Table 5. The corresponding extended domain results differ approx. 2%, which is an acceptable precision.
other colours denote intermediate values. The evolution of pressure fields and vortices’ dynamics in the immediate vicinity of each section, for the same oscillation period, is presented in Figs. 7 through 9; the time interval depicted is from 7.82 to 11.73 s. For sections S02 and S10 the flow has a transient nature till the end of the second oscillation; a nearly repetitive sequence of pressure fields, similar to that illustrated in the figures, is seen from the third period onwards. For section S22 the transitory stage is longer, though, as it may be confirmed by comparing the first and last snapshots in Fig. 9. Analysis of the flow about S02 reveals that the roll motion of this section does not induce the generation of vortices. Even though this section has a very smooth geometry, effectively managing to avoid flow separation, still the damping coefficient is consistently higher than that for the other two sections, throughout the whole simulated frequency range. The wave generation is, thus, thought to be a significant component for this section's damping, with the validity of such a conclusion being probably extended to any typical bulbous bow section. For S10 two small vortices in way of the bilge arise when the section reaches the maximum angle of rotation, but these dissipate quickly. Incidentally, this behaviour had already been witnessed by Mohsin et al. (2014) in their study of a closely resembling section. Lastly, from Fig. 9, it is seen that for S22 significant vortex shedding occurs at the keel – this is to no surprise as the keel sharp edge of the contour would surely suggest this phenomenon. In Figs. 10 and 11, the computed values of added mass and damping coefficients, regarding the three ship sections, are presented and compared to the potential code based coefficients published by Sutulo et al. (2009). The non-dimensionalization follows the same ratios applied by those authors:
μ24nd = μ44nd =
μ24 3 ρπL ref μ44 4 ρπL ref
υ24nd = υ44nd =
5. Conclusions
υ24 3 ρπωL ref
The open source code OpenFOAM was applied to the calculation of small forced roll oscillations, in a wide range of frequencies, of four 2D
υ44 4 ρπωL ref
(17)
Fig. 9. Evolution in time of pressure fields and vortices dynamics for section S22; T=3.91 s and t0=7.82 s.
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Fig. 10. Roll-into-roll added masses and damping coefficients for sections S02, S10 and S22.
the added mass was observed throughout the whole frequency range. Nonlinear effects relative to the non-vertical hull shape at the water plane are thought to contribute to this discrepancy. The vortex creation throughout the periods of oscillation has been investigated to support these conclusions. Although based on single sections of each type, the following conclusions may be formulated: 1) bulbous bow sections seem to manage to avoid vortex shedding, leading to a potential flow approach being valid; 2) midship sections present a limited vortex shedding, making obvious the necessity of bilge keels or other type of stabilizers in long ships. 3) aft sections with wedge like shapes show a significant flow separation, so
sections: a rectangular one; and three sections of the container ship S175. An assumption that the flow is laminar has been made and the dynamic mesh utilities included in OpenFOAM have been used with success. The calculated hydrodynamic coefficients for the rectangular section have been shown to agree with the experimental data and other researchers’ results. The ship sections’ simulations resulted in good correlations with the results from the application of a linear frequency domain, potential flow, code. Damping coefficient discrepancies were witnessed for shapes and frequency values that were evidently prone to exhibit flow separation. For the same shapes, a 10–20% difference in 44
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Fig. 11. Roll-into-sway added masses and damping coefficients for sections S02, S10 and S22.
viscosity effects are significant and must be included. Table 5 Influence of the domain semi-width on added masses and damping coefficients regarding rolling of S02 section for T=13.83 s and λ=289 m. L1/2=600 m M3
Acknowledgments This work has been financed by the project SHOPERA: Energy Efficient Safe SHip OPERAtion (Grant Agreement number 605221) cofunded by the Research DG of the European Commission within the RTD activities of the FP7 Thematic Priority Transport (FP7-SST-2013RTD-1). The first author has been financed by Fundação para a Ciência e a Tecnologia, Portugal, through the grant SFRH/BPD/78137/2011.
L1/2=1445 m 772,336 cells
μ44nd
ν44nd
μ44nd
ν44nd
0.1317
0.0145
0.1305
0.0148
45
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References
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Avalos, G.O.G., Wanderley, J.B.V., Fernandes, A.C., Oliveira, A.C., 2014. Roll damping decay of a FPSO with bilge keel. Ocean Eng. 87, 111–120. Bonfiglio, L., Brizzolara, S., 2013. Influence of viscosity on radiation forces: a comparison between monohull, catamaran and SWATH. In: Proceedings 23rd International Offshore Polar Eng. Conf. pp. 718–725. Chen, H., Liu, T., Huang, E.T., 2001. Time-domain simulation of large amplitude ship roll motions by a Chimera RANS method. In: Proceedings of the 11th International Offshore and Polar Engineering Conference. Stavanger, Norway, pp. 299–306. Datta, R., Rodrigues, J.M., Guedes Soares, C., 2011. Study of the motions of fishing vessels by a time domain panel method. Ocean Eng. 38, 782–792. Fernandes, A.C., Oliveira, A.C., 2009. The roll damping assessment via decay model testing (new ideas about an old subject). J. Mar. Sci. Appl. 8, 144–150. Frank, W., 1967. Oscillation of Cylinders in or Bellow the free surface of deep fluids. Naval Ship Research and Development Center. Report no. 2375. Washington, USA. Gadelho, J.F.M., Lavrov, A., Guedes Soares, C., 2014. Modeling the effect of obstacles on the 2D wave propagation with OpenFOAM. In: Guedes Soares, C., Peña, L. (Eds.), Maritime Transportation and Exploitation of Sea Resources. Taylor & Francis Group, London, UK, 1057–1065. Gadelho, J.F.M., Rodrigues, J.M., Lavrov, A., Guedes Soares, C., 2015. Determining hydrodynamic coefficients of a cylinder with Navier-Stokes equations. In: Guedes Soares, C., Santos, T.A. (Eds.), Maritime Transportation and Exploitation of Sea Resources. Taylor & Francis Group, London, UK, 1001–1007. Henning, H.L., 2011. Investigation of the Heave, Sway and Roll Motions of Typical Ship like Hull Sections Using Rans Numerical Methods (Msc Thesis). Stellenbosch University, South Africa. Hirt, C., Nichols, B., 1981. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201–225. Ikeda, Y., Himeno, Y., Tanaka, N., (1978). A prediction method for ship roll damping. University of Osaka Prefecture, Report No. 405. Inglis, R.B., Price, W.G., 1981. The influence of speed independent boundary conditions in three dimensional ship motion problems. Int. Shipbuild. Prog. 28, 22–29. Inok, F., Lavrov, A., Guedes Soares, C., 2014. Analysis of complex fluid flow test cases with OPENFOAM. In: Guedes Soares, C., Peña, L. (Eds.), Developments in Maritime Transportation and Exploitation of Sea Resources. Taylor & Francis Group, London, UK, 183–189. Issa, R.I., 1986. Solution of the implicitly discretised fluid flow equations by operatorsplitting. J. Comput. Phys. 62, 40–65. John, F., 1950. On the motion of floating bodies. II - simple harmonic motions. Commun. Pure Appl. Math. 3, 45–101. Kacham, B., 2004. Inviscid and viscous 2D unsteady flow solvers applied to FPSO hull roll motions. Environmental and Water Resources Engineering, Department of Civil Engineering, The University of Texas at Austin. Report No. 04-7. Austin, U.S.A. Kakar, K., 2002. Computational Modeling of FPSO hull roll motions and two-component marine propulsion systems. Environmental and Water Resources Engineering, Department of Civil Engineering, The University of Texas at Austin. Report No. 02-3. Austin, U.S.A. Kinnas, S.A., 2004. FPSO Roll Motions. Comprehensive status report: November 18, 2004. The University of Texas at Austin. Austin, U.S.A. Kinnas, S.A., Yu, Y., Vinayan, V., 2007. Numerical methods for the prediction of the bilge effects on the response of ship-shaped hulls. In: Proceedings of the 26th International Conference on Offshore Mechanics and Artic Engineering. San Diego, U.S.A. Kwon, S.H., Kim, B.J., Han, S.Y., Kim, Y.J., Ahn, K.S., Ren, D., Lu, L., Jiang, S.C., Chen, X.B., 2014. Study on roll damping around a circular cylinder. In: Proceedings of the 29th International Workshop on Water Waves and Floating Bodies. Osaka, Japan. Lewandowski, E.M., 2004. The Dynamics of Marine Craft - Maneuvering and Seakeeping. World Scientific Publishing Co. Pte. Ltd., Singapore. Lewis, F.M., 1929. The inertia of water surrounding a vibrating ship. SNAME - Trans. 37, 1–20. Lin, W.M., Yue, D.K.P., 1990. Numerical solutions for large-amplitude ship motions in the time domain. In: Proceedings of the 18th International Symposium on Naval Hydrodynamics. Ann Arbor, Michigan, U.S.A, pp. 41–66. Miller, E.R., 1974. Roll damping. NAVSPEC Tech. Rep., 6136–74-280. Mohsin A.R., Nallayarasu, I., Bhattacharyya, S.K., 2014. Experimental and Cfd Simulation of Roll Motion of Ship With Bilge Keel. In: International Conference on Computational and Experimental Marine Hydrodynamics, MARHY. Chennai, India, pp. 1–9. Nakos, D., Sclavounos, P.D., 1990. Ship motions by a three dimensional Rankine panel method. In: Proceedings of the 18th International Symposium Naval Hydrodynamics. Ann Arbor, Michigan, U.S.A, pp. 21–40. Nakos, D., Kring, D., Sclavounos, P.D., 1993. Rankine panel methods for transient free
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Calculation of hydrodynamic coefficients of ship sections in roll motion using Navier-Stokes equations
MARK
⁎
A. Lavrov, J.M. Rodrigues, J.F.M. Gadelho, C. Guedes Soares
Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisbon, Portugal
A R T I C L E I N F O
A BS T RAC T
Keywords: Nonlinear roll motion Roll damping Hydrodynamic coefficients Navier Stokes equations
The flow in the vicinity of 2D ship sections carrying out forced roll motions is simulated using the solution of the Navier-Stokes equations implemented in OpenFOAM. A hybrid Lagrangian-Eulerian adaptive mesh scheme is applied to solve forced motions on ship sections oscillating in calm water as well as the free surface capturing problem. Three sections of a containership, with differing geometrical properties, are investigated by comparing viscous unsteady results with potential flow solutions. The method presented in this study has been validated by comparing numerical results with experiments performed on a rectangular section. The influence of viscosity, particularly on vortex shedding, on the roll motion is investigated for an array of oscillation frequencies.
1. Introduction Several 3D boundary element methods, incorporating a potential flow approach, have been developed to solve the radiation (and diffraction) problem. The frequency domain approaches proposed by Inglis and Price (1981), Newman and Sclavounos (1988) and Nakos and Sclavounos (1990), and the time domain methods developed by Lin and Yue (1990) and Nakos et al. (1993) represent some of the most significant milestones in applying Boundary Element Methods for the solution of ship motions. Nonetheless, 2D methodologies to compute hydrodynamic coefficients are still very much in use nowadays. This undoubtedly comes from the high speed in computation, due to the simplification of the domain, but also one should bear in mind that 3D methods also pose specific problems. An example is the inclusion of forward speed effects in 3D methods similar to that by Papanikolaou and Schellin (1992). In their method, usage is done of zero speed Green functions formulations with the advance speed being accounted in a somewhat similar manner to that of strip theory (Salvesen et al., 1970). The method is valid, in all frequencies, for slender bodies where lengthwise hydrodynamic pressure variation is small relative to its girthwise variation. Another factor to take into consideration is that evaluating the finite depth Green's function for the zero speed 3D radiation-diffraction problem is extremely complicated, as is the evaluation of the forward speed frequency domain Green's function for the 3D ship motion problem (Datta et al., 2011). The earliest of these two dimensional methods were based on
⁎
conformal mapping. From the seminal works of Lewis (1929), Tasai (1959,1961) and Ursell (1949), several conformal mapping techniques have been proposed and benchmarked throughout the years (Ramos and Guedes Soares, 1997); examples of recent studies using multiconformal mapping are the ones by Rajendran et al. (2015a,2015b). However, conformal mapping methods present limitations regarding the geometry of the ships’ sections to be assessed: Frank's close fit method (Frank, 1967) surpasses those limitations but suffers from the phenomenon of irregular frequencies (John, 1950). The method by Yeung (1973) is immune to this problem while fulfilling an arbitrary section geometry requirement in addition to implementing the possibility of shallow water consideration. The method was further developed by Sutulo and Guedes Soares (2004), and applied to the study of a set of sections of the S175 hull in different depths and bottom geometries in Sutulo et al. (2009, 2010). All the above mentioned methods are based on the idea that the fluid is ideal and implement the potential flow paradigm, failing to account for the viscous damping in roll, which typically becomes substantial relative to the (small) wave damping contribution by slender, ship-like, bodies (Lewandowski, 2004). So, to include the viscous damping component it is necessary to resort to forced roll or roll decay model tests, as discussed, for example, by Uzunoglu and Guedes Soares (2015), whenever the application of semi-empirical methods, e.g. Miller (1974) and Ikeda et al. (1978), are not deemed accurate. Experimental tests consistently acknowledge the nonlinear nature of such a coefficient, with 2nd or 3rd order polynomial regressions being usually proposed. However, Fernandes and
Corresponding author. E-mail address: [email protected] (C. Guedes Soares).
http://dx.doi.org/10.1016/j.oceaneng.2017.01.027 Received 6 February 2016; Received in revised form 9 December 2016; Accepted 27 January 2017 0029-8018/ © 2017 Elsevier Ltd. All rights reserved.
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for very small roll amplitudes: time domain methods were deemed inevitable for proper levels of accuracy. Yu (2008) obtained a good correlation of his NSEQ code's results with the hydrodynamic coefficients computed with FLUENT, considering laminar and turbulent flow, for ship-shaped hull sections without bilge keels – turbulent effect was found small. Results were not so good for low frequency oscillations when bilge keels are present, though. Also worth of mention is the investigation of roll damping decay of an FPSO, fitted with bilge keels, by Avalos et al. (2014) by means of the incompressible two-dimensional Navier-Stokes equations. A strong dependence of the damping coefficient on the bilge radius was reported and, more interestingly, the occurrence of the so-called damping coefficient saturation phenomenon. This means that a fixed value of this quantity, irrespective of the roll motion amplitude, was observed, in agreement with the conclusions of Oliveira and Fernandes (2011). In what relates to the application of distributed applications, Quérard et al. (2008) carried out RANS based simulations with ANSYS-CFX 11.0, and concluded on the superiority of their predictions compared to potential flow analysis. Four sections’ geometries were investigated: rectangular, triangular, chine and bulbous bow; however, the bulbous bow section did not exhibit an actual protruding bulb. In Quérard et al. (2010), the same tool was used to simulate the roll motion of 2D FPSO sections. Another important study is that of Henning (2011), where predictions of large motions of simple shapes (triangle, square and demi-circle) were carried out resorting to the proprietary program FLUENT. The latter compared worse with the experimental results by Vugts (1968), probably due to the large amplitudes tested. Bonfiglio and Brizzolara (2013), have carried out comparisons between monohull, catamaran and SWATH midship sections, using OpenFOAM. However, only the heave motion is considered. More recently, when OpenFOAM was used by Kwon et al. (2014) to simulate the forced roll motion of a circular cylinder, a mesh with approx. 200 thousand cells resulted in moment amplitudes with errors ranging from 30% to 80%. In the following year, a study on forced motion of a circular cylinder, was carried out by Gadelho et al. (2015), where the potential flow damping coefficient was meant to be estimated by measuring the height of the radiated waves. OpenFOAM was used to calculate the flow for the heave and sway motions of the body, which provided good results for lower frequencies. But divergences with Sutulo et al. (2009) were witnessed for higher frequencies, probably due to the high wave slope of the corresponding radiated waves which violate the linear free surface assumption made. From the above, it is clear that research on forced roll motions of 2D sections, by using CFD techniques, has been mainly focused on, but not limited to, the effect of bilge keels on midship sections of FPSO hulls. In the present paper, three typical mono-hull sections, with clearly distinct geometrical topologies, are considered: bulbous bow, midship and skeg. OpenFOAM is used to determine the moment amplitude of forced roll motions of each, with comparisons between viscous unsteady results with potential flow solutions being carried out. The methodology applied is described and validated by comparing numerical results with experiments performed on a rectangular section. A laminar flow is considered while solving the Navier-Stokes equations using OpenFoam's interDyMFoam solver in an incompressible isothermal fluid; at each time step, the mesh is morphed, according to the motion lay imposed at the hull surface, following a hybrid Lagrangian-Eulerian approach. The ultimate objective of the present paper is the preliminary assessment of the need for CFD computations of roll motion hydrodynamic coefficients and its dependence on the sections topology. These are to be applied to methods resorting to 2D formulations of the coefficients, where such a formulation is regarded as sufficiently accurate for the specific problem being solved in the first place. Applying a full 3D viscous approach, with its much heavier computa-
Oliveira (2009) and Oliveira and Fernandes (2011) question the ability of the application of polynomials to properly account for the viscous damping in large roll amplitudes, even though the traditional quadratic approach is seen to remain accurate for V shaped hulls. A (nonpolynomial) bi-linear approach, which acknowledges the existence of two vortex shedding regimes separated by a roll amplitude threshold, has been developed and tested by the same authors (Oliveira and Fernandes, 2013). Still, the time and cost necessary to carry out such tests may be significant: Computational Fluid Dynamics (CFD) based calculations become attractive - e.g. Wanderley et al. (2007). CFD methods are, thus, being increasingly used nowadays for roll damping estimation in an effort to improve, or substitute entirely, the application of traditional semi-empirical approaches to it. Few CFD based studies on the roll radiation problem consider a domain incorporating the whole vessel. The complexity of the domain discretization, problem setup and the cumbersome computational burden result in a strip-wise approach being the standard. Chen et al. (2001) carried out time domain simulations of the entire ship using Reynolds-Averaged-Navier-Stokes equations (RANS) and applying a Chimera domain decomposition approach with approx. 65,000 cells. In their study, forced large amplitude roll motions of a ship fitted with bilge keels with a hull presenting a skeg were simulated. The authors realized the high viscosity and low wave damping generated by the bilge keels, while both damping effects were seen significant as a result of the skeg. Also using RANS, the roll motion of a surface combatant ship was numerically predicted by Wilson et al. (2006), but a much larger number of cells was part of the domain: approx. 2.3 million. In both studies, the strong viscous effect of bilge keels was evident; furthermore, it was shown that such methods are accurate for practical geometries. Studies dealing with 2D roll motion using CFD typically carry out simulations concerning the effect of bilge keels fitted amidships of vessels with a long midbody having a close to rectangular sectional shape. In Sarkar and Vassalos (2000) a RANS-based technique for simulation of the flow near a 2D rolling rectangular cylinder was applied; only approx. 8000 cells were required for successful comparisons with Vugts (1968) experiments and Yeung et al. (1998) inviscid calculations. A relatively accurate RANS based numerical prediction of the time history of the roll motion of a rectangular cylinder, fitted with bilge keels, was achieved by Yeung et al. (2001). However, a total of 40,000 cells were necessary for a good comparison with experimental data, despite the slight over estimation of the motion amplitude. Kakar (2002) solved the Euler equations regarding the inviscid flow around a 2D rectangular section, with and without bilge keels, under forced roll motion. Grids with 37, 74 and 118 thousand elements were tested and the results revealed a convergence to the damping coefficients obtained by Yeung et al. (1998), though much higher values of added mass were observed, especially in low frequency range. On the other hand, Vugts (1968) reports the less accurate experimental results of the hydrodynamic coefficients in roll, due to the mechanical difficulties of the tests, which adds to the smaller order of these values relative to translational modes. Later, Kacham (2004) resorted to Navier-Stokes equations (NSEQ) to solve the same problem, using three grids with 17, 21 and 25 thousand cells. Comparisons with Kakar (2002) and Vugts (1968) confirmed the over prediction of added mass coefficients experienced by Kakar (2002) for large periods and revealed some inaccuracies in the damping coefficient as well. Finally, Kinnas (2004) further developed the code used by Kacham (2004) and Kakar (2002) and managed to achieve good results for all frequency values. Vinayan et al. (2005) studied the combined effect of the free surface and flow separation at the bilge keels and Kinnas et al. (2007) found that in the case of roll motion nonlinear effects are considerable, even 37
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→ ⎯ ∇(U ) = 0
tional burden, would thus be counterproductive, albeit the more realistic results. A hybrid approach to the radiation problem can thus be formulated, where CFD based solutions are only applied to zones of the hull where these are deemed necessary, with the potential formulation still being applicable to the remaining sections. Although past investigations have provided general remarks on this issue, the methodology of the present study and the specific consideration of ship sections in forced roll motion, is, to the best of the authors’ knowledge, a novelty. In Section 2, the problem formulation for the forced roll motion, the fluid's governing equations and their numerical implementation are presented. The process to obtain the hydrodynamic coefficients is described in Section 3. The validation of the scheme is done in Section 4, by comparing its results to those by Vugts (1968) and Kacham (2004), regarding a rectangular section; results for the three ship sections are also presented and compared with those by Sutulo et al. (2009). Finally, some conclusions are drawn in Section 5.
→ ⎯ → ⎯ → ⎯ ∂ρU + ∇(ρU ⊗ U ) = − ∇τ , ∂t
(4) ⎯ → where ρ represents the fluid density, U stands for the velocity vector, with components u1, u 2 , u3 in the x1, x 2 , x3 directions, respectively, and the tensor τij is the stress tensor in a Newtonian fluid, with general form:
⎛ ∂u ∂uj 2 ∂u ⎞ τij = −δijp + μ⎜⎜ i + − δij k ⎟⎟ , i , j , k = 1, 2, 3 ∂xi 3 ∂xk ⎠ ⎝ ∂xj ∂u
2.1. Problem formulation
∂ωq ∂t
The case of a bi-dimensional semi-submerged body, forced to perform small angular oscillatory motion with prescribed amplitude and period, is to be investigated and the sectional hydrodynamic coefficients determined. The motion, φ , may be described by the following kinematic equation:
+ ui
∂ωq ∂xi
= 0 , q = 1, 2
(6)
The governing equations are evaluated on the domain bounded by the bottom, a wall on each side, the oscillating body contour and a top limit ceiling representing the atmosphere. The boundary condition for the first four boundaries sets the normal pressure gradient to be null:
(1)
∂p =0 ∂n
where φ0 is the amplitude of the motion, ω its circular frequency of oscillation and t represents time. The radiation problem is formulated by not considering body mass inertial effects or restoring forces, while imposing a single mode oscillation to the body. By doing so, only the forced motion forces and the fluid hydrodynamic reaction forces are present. The fluid forces can in turn be decomposed between forces in phase with the acceleration and those in phase with the velocity. These forces correspond to added masses and damping coefficients multiplied by the acceleration and velocity, respectively. The single mode of oscillation in roll will have fluid reactions in more than one direction; considering only the nonzero components, the following equations of motion hold:
(7)
The atmosphere boundary has the following boundary conditions:
∂U = 0, ∂n
∂p = 0, ∂n
ωI = 0
(8)
2.3. Numerical implementation A schematic view of the geometry of the computational domain is presented in Fig. 1. The inertial coordinate system is located at the intersection of the section symmetry plane and the undisturbed water surface. The domain must have sufficient horizontal extent to prevent too early reflection of the water waves on the lateral walls. In addition, the depth must be such so as to prevent any significant shallow water effects on the body and on the free surface, i.e. it must ensure the validity of the dispersion relation: k =ω2 / g , where k is the wave number and g the standard gravity acceleration. Both these issues are more significant to lower frequencies. On the other hand, too big a domain will force the presence of a large number of cells. The level of mesh refinement in way of the free surface does not allow for sufficient mesh coarsening in areas away from the body, in order to lessen the increase number of cells as would otherwise be possible. This concern becomes more serious in higher oscillation frequencies as the wave slope becomes significant. The minimum domain half-width, L1/2 , may be determined by the total number of oscillations, N , without interference from reflected waves, at some point located at a distance ∆x from the centre of the domain:
μ24 φ + η24φ = Fx sin(ωt + ϕ24 ) μ34 φ + η34φ = Fysin(ωt + ϕ34 ) μ44 φ + η44φ = Mzsin(ωt + ϕ44 )
(5) ∂uj
In the abovementioned equation, p is the pressure, ∂xi + ∂x the j i u the normal stress and μ denotes the shear stress, ∂uk /∂xk = ∇⋅→ dynamic viscosity coefficient. The Volume of Fluid method (VoF) (Hirt and Nichols, 1981) is used for capturing the water-air interface. If the concentrations of water and air in each cell are defined by ω1 and ω2 , respectively, these sum to one at all times and are determined through the continuity equation:
2. Numerical method
φ = φ0sin(ωt ),
(3)
(2)
where μi4 and ηi4 are the sectional added mass and damping coefficients, respectively, in the ith direction, due to the roll motion, Fx , Fy and Mz represent the sway force, heave force and roll moment amplitudes, respectively, with their corresponding phase angles: ϕ24 , ϕ34 , and ϕ44 ; phases are relative to the motion. The subscripts x , y and z denote alignment with the base components x1, x 2 and x3 of a direct coordinate system with origin at the intersection of the section vertical mid line and the undisturbed waterline. Evidently, the force Fy is zero for a body symmetric about the x 2 axis. From the knowledge of the right hand sides of Eq. (2), and the prescribed motion characteristics included in Eq. (1), the hydrodynamic coefficients may be determined. These forces represent the action of the fluid on the body and are calculated from the equations which govern the flow. 2.2. Flow governing equations A laminar flow regime is assumed, as in Inok et al. (2014). The Navier-Stokes equations for the time-depended unsteady incompressible flow in Cartesian coordinates are:
Fig. 1. Sketch of the computational domain (not to scale).
38
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L1/2 =
λN +2Δx 4
the values listed in Table 2 were set for the ship sections simulations. Eq. (9) gives a close to eight cycles of pure radiation oscillations of the ship sections when measurements are taken at the body surface, for a period of 13.83 s (the longest considered in the present study). For the same period, the minimum depth would be about 150 m. But the d=95 m value implemented was shown to result in the bottom presence not influencing the flow about the sections tested significantly. In addition, the results for depths equal to or higher than 95 m are virtually the same in Sutulo et al. (2009). Four mesh levels of increasing cell density were used and denominated M1 through M4. The total number of cells (hexahedra and prisms), for each section, is listed in Table 3 - only hexahedral cells are used for the rectangular section case. Meshes for the ship sections mostly include hexahedral cells; a small number of prismatic cells is situated near the section's contour. Only Section S22 required using the finest mesh, for it has a complicated shape and generates a lot of vortices. For the M3 mesh, the dimensions of the cells in the x1, and x2 directions, εx and εy, near the contour of the rectangular and S02 sections are εx/B≈εy/B≈0.003 and εx/B≈εy/B≈0.008, respectively; other ship sections have mesh elements’ dimensions of the same order of magnitude. All calculations are conducted using Courant number equal to 0.5, while the time step varied between 8*10−4 s and 10−2 s, depending on the section and the mesh. The number of cycles, for which measurements at the body surface are free of reflection effects, drops to two for the case of the semisubmerged rectangular section oscillating at the maximum period considered, T=3.59 s. The practical procedure adopted for obtaining the hydrodynamic coefficients was to calculate the value of μ and ν from the evaluation of Eqs. (12) and (13) in the second and remaining periods of oscillation, which are not affected by reflected waves. For the particular case of the rectangular section with T=3.59 s, this means that only one period, the second, was analysed, as the first is the most subjected to transient effects as depicted in Fig. 3 for T=0.9 s. Still regarding the rectangular section simulations’ setup, the minimum depth, as per the deep waters approximation, would be about 10.1 m. From Table 1 it is seen that these limits are not respected in the simulations carried out, but one should recall that this minimum depth decreases for all the other (higher) frequencies simulated. An Intel Core i7-4770K CPU @3.5 GHz×8 processor was used for calculations. The computational time varied between 10 and 100 h, depending on the mesh applied and the period considered.
(9)
where λ=2πg / ω2 is the wave length. If the evaluation of the quantities of interest is to be done on the section contour then Δx ≅B /2 , with B denoting the sectional breadth. Domain discretization for the case of rectangular sections may be carried out by resorting to OpenFoam's utility BlockMesh, while more complicated geometries require OpenFoam's SnappyHexMesh tool. The latter manipulates an initial grid created with BlockMesh to adapt it to the actual shape of a section in its contour's vicinity. The SnappyhexMesh tool is more suitable to work with full 3D meshes, so OpenFoam's tools ExtractMesh and CreatePatch are used to create meshes with a single cell in the direction perpendicular to the sections. Smooth element size transition was applied to the mesh: a finer mesh is present in way of the section contour, getting progressively coarser at farther locations. Finally, the interDyMFoam solver of OpenFOAM is used for solving Navier-Stokes equations, which are evaluated on the aforementioned adaptive mesh. It implements the PIMPLE algorithm, which is based on the integration of the PISO algorithm of Issa (1986) and the SIMPLE algorithm of Patankar and Spalding (1972). 3. Determination of the hydrodynamic coefficients A precise way of determining the hydrodynamic coefficients is by substituting Eq. (1) into Eq. (2) and solving it for both coefficients, e.g. for roll into sway, of a symmetric contour, one has:
Fx
μ24 =
φ0ω2
cosϕ24
(10)
Fx sinϕ24 φ0ω
υ24 =
(11)
The force amplitudes may be obtained directly from the force time record, taken when the force oscillation has reached a steady phase, whereas the phase angles may be found from the lagging of the force up-zero crossings relative to the motion up-zero crossings. Nevertheless, a more general and robust method is to take the cosine and sine Fourier transforms of the force or moment, e.g. for the rollinto-roll case one has:
μ44 =−
ν44 =
1 πφ0ω
1 πφ0
∫0
∫0
T
Mz(t )sin(ωt )dt
(12)
4.1. Square section
T
Mz(t )cos(ωt )dt
The square section corresponds to the one experimentally investigated by Vugts (1968) and whose hydrodynamic coefficients were numerically calculated by Kacham (2004). The time history of Mz , with different mesh density levels is shown in Fig. 3, relative to a period T=0.9 s: a successful convergence is evident. The comparison between the results of the present study, calculated from Eqs. (12) and (13), with those from the aforementioned two authors is shown in Fig. 4, where the following non-dimensionalizations are applied:
(13)
where T is the period of oscillation. 4. Results The forced roll motion of 2D floating bodies is investigated, where the hydrodynamic coefficients are calculated for a set of oscillations with given amplitude and frequency. Four semi-submerged sections are considered as case studies: one semi-submerged rectangular section, with bilge radius 2.5 mm, and three semi-submerged sections – S02, S10 and S22 – of the container ship S175. Each section's breadth (B) and draft is listed in Table 1 and their geometries are shown in Fig. 2. From previous experience acquired in Gadelho et al. (2014,2015),
μˆ = μ44 /4ρQb 2
νˆ = ν44
ωˆ = ω44
Table 1 Main parameters of the investigated sections. Section:
Rectangular
S02
S10
S22
Breadth (m) Draft (m)
0.4 0.2
3.06 9.5
24.84 9.5
9.0 9.5
b /4ρQb 2 g b g
(14)
(15)
,(16)
where Q is the submerged area of the rectangular section and b represents the draft. The amplitude of the forced roll motions is 0.05 rad and the period of the oscillations ranges between 0.6 s and 3.59 s. A good agreement with the experimental values, regarding the damping coefficient, for both the present study and the one by Kacham (2004) is observed, while a divergent trend is seen for the lower 39
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Fig. 2. Sectional shapes and mesh views: a) rectangular section; b) S02; c) S10; d) S22; e) mesh for rectangular section; f) mesh for S02 (not to the same scale). Table 2 Domain semi-width, water and air height. Section
Semi-width L1/2 (m)
Water height d (m)
Air height a (m)
Rectangular S02, S10, S22
12 600
2 95
0.6 10
Table 3 Number of cells (hexahedra and prisms) in the simulations. Meshes Section
M1
M2
M3
M4
Rectangular S02 S10 S22
58,584 251,932 251,476 251,906
118,262 327,726 319,448 327,320
234,604 454,876 419,818 430,332
589,902
Fig. 4. Dependence of added mass and damping on frequency for roll of rectangular section.
frequencies. This divergence is much more pronounced in the results of Kacham (2004), though. The same behaviour is seen regarding the added mass, but with not so good correlations with the experimental values. The larger deviation of the computed values from the experi-
Fig. 3. Time history of moment Mz applied on the rectangular section, with three different meshes; Period of oscillation is T=0.9 s.
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Table 4 Influence of the domain semi-width on added masses and damping coefficients regarding rolling of rectangular section for T=3.59 s and λ=14.2 m. L1/2=12 m M3
μˆ44 0.0523
L1/2=71 m 926,738 cells
νˆ44 0.00318
μˆ44 0.0532
νˆ44 0.00311
Fig. 6. VoF snapshots for section S02 during a roll half-cycle with T=3.91 s, using mesh M3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
mental ones, for lower frequencies, is probably a consequence of the shallow water effect in those longer wavelengths. To confirm that the chosen value of L1/2 allows getting results that are not exposed to the influence of the reflected wave, an additional calculation for the rectangular section with period 3.59 s was carried
Fig. 5. Time history of moment Mz applied on sections S02, S10 and S22, with three different meshes; Period of oscillation is T=13.83 s.
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roll with an amplitude φ0=0.056rad , for eight different oscillation periods between 3.91 s and 13.83 s, during 60 s. The time series of the roll moment, for an oscillation period T=13.83 s, is shown in Fig. 5. Convergence is manifest for all three sections and it is seen that the moment amplitude is significantly higher for S02, with values in excess of 1.75 times that of S10 and S22. VoF snapshots for section S02 during a roll half-cycle, for the smallest period and using mesh M3, are shown in Fig. 6. In each cell, red corresponds to ω1=1 and blue to ω2=1;
out with a wider domain; corresponding parameters and results are listed in Table 4. It is seen that hydrodynamic coefficients resulting from this variation differ approx. 2%, which allows for the L1/2 domain width to be seen as suitable. 4.2. Ship sections In the simulations carried out, each ship section has been forced to
Fig. 7. Evolution in time of pressure fields for section S02; T=3.91 s and t0=7.82 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Evolution in time of pressure fields and vortices dynamics for section S10; T=3.91 s and t0=7.82 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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where L ref is the maximum of the draught and the half breadth at the waterline, for each contour. Due to symmetrical geometry of the sections, the roll-into-heave coefficients are null and are not presented. A good agreement is manifest for section S02 roll-into-roll and rollinto-sway results, for both coefficients, strengthening the conviction that potential flow solvers suffice for accurately predicting the flow about similar bulbous bow sections, due to their streamlined geometric nature. It is but for the coarsest mesh that the results diverge from the potential ones, mainly in the higher frequency region, as it would be expected due to the higher acceleration levels and generated waves’ slope. A near to 10% discrepancy of the added mass in roll-to-roll relative to that of Sutulo et al. (2009) resulted for S10, though the difference reduces to virtually zero in roll-to-sway. The damping coefficient follows the same pattern, however augmented due to the known viscous effects discussed previously relative to eddy making. A 20% discrepancy is observed in the added mass results of section S10. This section contour at the waterline is far from adhering to the constant water plane assumption made in Sutulo et al. (2009), which may be contributing to this phenomenon. This section required the finest mesh to be applied to ensure convergence, with the damping coefficient in roll-into-roll reaching between three to four times that for potential flow. The difference is much lower, however significant, in roll-into-sway: the substantial eddy roll damping component is thus exposed. In a similar way to the square section, an additional calculation with increased width domain was carried out for S02 with oscillation period 13.83 s - results are listed in Table 5. The corresponding extended domain results differ approx. 2%, which is an acceptable precision.
other colours denote intermediate values. The evolution of pressure fields and vortices’ dynamics in the immediate vicinity of each section, for the same oscillation period, is presented in Figs. 7 through 9; the time interval depicted is from 7.82 to 11.73 s. For sections S02 and S10 the flow has a transient nature till the end of the second oscillation; a nearly repetitive sequence of pressure fields, similar to that illustrated in the figures, is seen from the third period onwards. For section S22 the transitory stage is longer, though, as it may be confirmed by comparing the first and last snapshots in Fig. 9. Analysis of the flow about S02 reveals that the roll motion of this section does not induce the generation of vortices. Even though this section has a very smooth geometry, effectively managing to avoid flow separation, still the damping coefficient is consistently higher than that for the other two sections, throughout the whole simulated frequency range. The wave generation is, thus, thought to be a significant component for this section's damping, with the validity of such a conclusion being probably extended to any typical bulbous bow section. For S10 two small vortices in way of the bilge arise when the section reaches the maximum angle of rotation, but these dissipate quickly. Incidentally, this behaviour had already been witnessed by Mohsin et al. (2014) in their study of a closely resembling section. Lastly, from Fig. 9, it is seen that for S22 significant vortex shedding occurs at the keel – this is to no surprise as the keel sharp edge of the contour would surely suggest this phenomenon. In Figs. 10 and 11, the computed values of added mass and damping coefficients, regarding the three ship sections, are presented and compared to the potential code based coefficients published by Sutulo et al. (2009). The non-dimensionalization follows the same ratios applied by those authors:
μ24nd = μ44nd =
μ24 3 ρπL ref μ44 4 ρπL ref
υ24nd = υ44nd =
5. Conclusions
υ24 3 ρπωL ref
The open source code OpenFOAM was applied to the calculation of small forced roll oscillations, in a wide range of frequencies, of four 2D
υ44 4 ρπωL ref
(17)
Fig. 9. Evolution in time of pressure fields and vortices dynamics for section S22; T=3.91 s and t0=7.82 s.
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Fig. 10. Roll-into-roll added masses and damping coefficients for sections S02, S10 and S22.
the added mass was observed throughout the whole frequency range. Nonlinear effects relative to the non-vertical hull shape at the water plane are thought to contribute to this discrepancy. The vortex creation throughout the periods of oscillation has been investigated to support these conclusions. Although based on single sections of each type, the following conclusions may be formulated: 1) bulbous bow sections seem to manage to avoid vortex shedding, leading to a potential flow approach being valid; 2) midship sections present a limited vortex shedding, making obvious the necessity of bilge keels or other type of stabilizers in long ships. 3) aft sections with wedge like shapes show a significant flow separation, so
sections: a rectangular one; and three sections of the container ship S175. An assumption that the flow is laminar has been made and the dynamic mesh utilities included in OpenFOAM have been used with success. The calculated hydrodynamic coefficients for the rectangular section have been shown to agree with the experimental data and other researchers’ results. The ship sections’ simulations resulted in good correlations with the results from the application of a linear frequency domain, potential flow, code. Damping coefficient discrepancies were witnessed for shapes and frequency values that were evidently prone to exhibit flow separation. For the same shapes, a 10–20% difference in 44
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Fig. 11. Roll-into-sway added masses and damping coefficients for sections S02, S10 and S22.
viscosity effects are significant and must be included. Table 5 Influence of the domain semi-width on added masses and damping coefficients regarding rolling of S02 section for T=13.83 s and λ=289 m. L1/2=600 m M3
Acknowledgments This work has been financed by the project SHOPERA: Energy Efficient Safe SHip OPERAtion (Grant Agreement number 605221) cofunded by the Research DG of the European Commission within the RTD activities of the FP7 Thematic Priority Transport (FP7-SST-2013RTD-1). The first author has been financed by Fundação para a Ciência e a Tecnologia, Portugal, through the grant SFRH/BPD/78137/2011.
L1/2=1445 m 772,336 cells
μ44nd
ν44nd
μ44nd
ν44nd
0.1317
0.0145
0.1305
0.0148
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References
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