Greco et al. (2011) tried to identify the proper numerical choices for the development of a three-dimensional Domain- Decomposition (DD) strategy. It aims to study the violent interaction of FPSO ships with head-sea regular waves. The present work represents a contribution in this direction

Colicchio, Giuseppina

Greco, Marilena

Lugni, Claudio

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Assessment study of a Domain-Decomposition strategy for marine applications.
M. Greco1,2,3 G. Colicchio1,2 C. Lugni1,2
m.greco@insean.it g.colicchio@insean.it c.lugni@insean.it
1 INSEAN, Italian Ship Model Basin, Roma – Italy.
2 Centre for Ships and Ocean Structures (CeSOS), NTNU, Trondheim – Norway.
3 Department of Marine Technology, NTNU, Trondheim – Norway.
Greco et al. (2011) tried to identify the proper numerical choices for the development of a three-dimensional Domain-
Decomposition (DD) strategy. It aims to study the violent interaction of FPSO ships with head-sea regular waves. The
present work represents a contribution in this direction. The under development DD solver combines a Boundary Element
Method (BEM) for the global linear seakeeping analysis in frequency domain with a Navier-Stokes (NS) method for the
flow investigation in an inner sea region containing the forward portion of the vessel. In this region, water shipping and
slamming events can be caused by the liquid-structure interactions, while it is assumed that the rest of the fluid domain
can be suitably described by the linear potential-flow solver. Within the NS solver, a Projection method is adopted with
a finite-difference scheme on an Eulerian grid and a Predictor-Corrector scheme for the time evolution. The solver is
accurate to the second order. The evolution of the free surface is captured by means of a Level-Set (LS) technique
updating in time the normal distance (with-sign) from the water-air interface. Moreover the influence of the air on the
water is neglected and the velocity field is smoothly extended from the liquid to the gas domain. The non-slip body-
boundary condition is enforced through a hybrid Eulerian-Lagrangian approach combining the body LS function φbody
(positive in the fluid) with point markers moving with the body and initially defined on a uniform grid four times finer
than the minimum mesh size in the computational grid and within a band across the body surface six times larger than the
maximum mesh size of the computational grid. The body points evolve in time carrying with them their distance from the
body surface and this allows a more accurate estimate of φbody on the Eulerian grid by interpolation from the markers.
The coupling between the outer and inner solvers is implemented as a weak and a strong strategy. In the former case,
the information travels from the outer to the inner solver but not vice versa; in the latter, the information travels back
and fourth. The NS solver needs initial and boundary conditions in terms of velocity, pressure and free surface elevation
in the fluid and the body-boundary condition along the solid surface. The seakeeping solver feels the nonlinearities and
possible viscous effects predicted in the inner domain through local and global loads acting on the ship portion inside this
region. The rest of the ship hull is assumed to be subjected to the loads provided by the potential-flow outer solver. For a
freely-floating vessel, the inner-domain global loads are introduced in the body-motion equations, so that the ship motion
can be affected by the inner-flow features that can alter the flow pattern in the inner domain in return.
The study in Greco et al. (2011) examined the application of the DD strategy to a patrol ship tested at CNR-INSEAN
without forward speed and highlighted numerical challenges. They are connected with the boundary conditions of the
inner domain and with the prediction of the body loads by the NS solver. Within that investigation, the loads on the
-0.6 -0.4 -0.2 0 0.2 0.4
-0.4
-0.2
0
0.2
0.4
Ly2
Lz
D
ξ4
B
Ly1
waves ξ3(t)z
y
Figure 1: Weak coupling. Left: forced-heave problem for a patrol ship. Vertical force on the ship portion inside the inner
domain as estimated by extrapolation and interpolation with marching-cube algorithm. Oscillation period corresponding
to a wavelength λ ' 1.25L and motion amplitude ξ3a/D = 0.1. ∆x/L = 0.006. Right: definition of the examined 2D
problem. B/D = 2.
body surface were approximated as a parabolic extrapolation from the loads estimated at the three iso-surfaces φbody =
0.5∆x,∆x and 1.5 ∆x outside of the ship. This approach proved to be not optimal for moving bodies because of
numerical oscillations induced by errors in extrapolating the pressure not exactly normally to the body surface. Therefore
another strategy has been identified: the body pressure is interpolated on the vertices of the triangles that identify the
body surface, then it is integrated along each triangle. The triangles are identified at any needed time instant through the
marching-cube scheme searching for the intersections of each grid cell with the body surface. This avoids the temporal
oscillations in the integrated loads, the back side of the medal is given by greater CPU-time requirements. Left plot
of figure 1 shows a typical comparison between the results from the two methods in terms of the vertical force for
a forced-heave case of the patrol ship considered in Greco et al. (2011). The results show that the two approaches
are associated with a different buoyancy contribution (included in the vertical force) and confirm a behavior free from
fictitious oscillations when using the developed interpolation method. A major change relative to the preliminary work in
Greco et al. (2011) concerns the boundary conditions for the inner domain bounded within a parallelepiped. A careful
analysis has shown that the most robust and effective strategy is an inflow condition for the free-surface elevation, the
pressure p and the velocity u spread in, respectively, six, two and six cells closest to the vertical boundaries. The first two
variables are sharply enforced from the BEM while the velocity passes linearly from the BEM to the NS solution. If the
boundary is crossed by the body, a mixed condition is applied for the velocity: the inflow condition is modified smoothly
into an outflow condition going close to the body surface to avoid inconsistency between the free-slip condition from the
BEM and the no-slip condition from the NS solver. On the bottom boundary the BEM p and u are sharply enforced on
the first cell.
To help the assessment of the numerical choices for the body-load predictions and for the enforcement of the inner-
domain boundary conditions, the 2D geometry in the right of figure 1 has been examined in the case of diffraction of
regular incident waves and radiation caused by periodic forced heave ξ3 and roll ξ5 motions assuming beam B=0.4 m and
draft D=0.2 m. For this problem, the DD is used as a weak-coupling strategy: the 3D BEM solver is applied to a cylinder
long L = 5B and provides the initial and boundary conditions to the NS-LS solver describing the flow in the central
cross-section plane. The inner domain has Ly1 = 1.5B, 2.15B Ly2 = 0, 1.5B, 2.15B, Lz = 4.3B, and was studied
using ∆y = 0.025B. The frequency ω is set as 0.6, 0.8, 1 and 1.2
√
2g/B. In the diffraction problem, the incident
wave amplitude is set as A = 0.1D and, in the radiation problems, the oscillation amplitudes ξ3a = 0.05D = 0.01m and
ξ4a = 0.05 rad are considered. The results are given in figure 2 using Ly1 = Ly2 = 2.15B. These choices lead to linear
conditions which can be challenging for CFD methods because the results are very sensitive to the numerical accuracy.
The solid and dashed lines in the plots represent the potential-flow solution; in particular, the 3D and 2D data are obtained
integrating, respectively, along the whole BEM cylinder with L = 5B and along a central strip wide 1.25B. But for b44,
the DD results are globally consistent with the 2D BEM results and confirm a dominant linear potential-flow behavior for
the studied conditions. Viscous effects connected with friction along the body and with the vortex shedding caused by the
small radius of curvature at the bottom edges of the cross-section can partially explain the larger values of the DD results
for b44. The instantaneous vorticity field in the case of forced roll is examined in figure 3 as a function of the oscillation
frequency. As ω increases more intense vortical structures are shed from the body and the previous oscillation vorticity
is still visible in the fluid field. An exception is ω = 0.8
√
2g/B which is associated with the lowest vorticity. The 2D
numerical results in figure 2 are close to the experimental data by Vugts (1968), also given in the figure, with the largest
discrepancies in the region of low ω. There, the model tests are quite consistent with the 3D BEM results. Though the
physical cylinder was much longer (L = 10.475B) than in the BEM simulations, this suggests possible 3D effects in the
experiments. The model tests are associated with an added-mass in roll much lower than the numerical results. This is
possibly due to experimental errors in calculating the roll added mass, as pointed out by Vugts (1968). In the experiments
it was difficult the properly estimate the moment of inertia and this led to a44 smaller than in reality. For the heave motion,
a larger amplitude ξ3a = 0.15D = 0.03m is also examined in figure 2. The added-mass and damping coefficients from
the experiments and the DD appear consistent and show that a33 is dominated by linear effects while b33 shows some
nonlinearities as ω increases.
To assess the numerical choices made in the case of a body crossing the inner-domain boundary, Ly2 has been set to
zero so that only half of the body is inside the NS-LS domain in its mean configuration. The results are given in the top
plots of figure 4 for the case of heave with amplitude ξ3a = 0.15D and ω = 0.6
√
2g/B. The vertical force on half of
the domain is consistent with half of the force acting on the whole body (left plot). On the other hand, as expected, near
the right overlapping region between the BEM and the NS-LS domains the pressure field deviates from the viscous-flow
solution. The results indicate that, within a strong-coupling analysis of a seakeeping problem, to be on the safe side the
loads from the NS-LS solver should be obtained integrating on a portion of the body sufficiently inside the inner domain,
say with a distance greater than the overlapping thickness (6 cells) plus four extra cells. A part from this area near the
boundary, the pressure fields in the two cases are consistent. Similar agreement is also observed in the case of forced roll
motion with ξ5a = 0.05 rad and ω = 0.6
√
2g/B, as shown in the bottom-left plot of figure 4. The pressures near the right
boundary of the narrower domain are more consistent for the two simulations than in the case of the forced heave. This can
be intuitively understood because the roll gives a motion normal to the boundary while the heave leads to a motion parallel
0.6 0.9 1.20
0.2
0.4
0.6
0.8
2D potential flow
2D DD strategy
Vugts experiments
3D potential flow
ω[B/(2g)]0.5
F3/(ρ g B A)
0.6 0.9 1.20
0.6
1.2
1.8
2.4
2D potential flow
2D DD strategy
Vugts experiments
3D potential flow
ω[B/(2g)]0.5
F4/(ρ g B3 kA/12)
0.6 0.9 1.20
0.5
1
1.5
2
2D potential flow
2D DD strategy
Vugts experiments
2D DD strategy
Vugts experiments
3D potential flow
ξ3=0.03 m
a33/(ρ Ar)
ω[B/(2g)]0.5
ξ3=0.01 m
0.6 0.9 1.20
0.04
0.08
0.12
2D potential flow
2D DD strategy
Vugts experiments
3D potential flow
ω[B/(2g)]0.5
a44/(ρ Ar B 2)
0.6 0.9 1.20
0.25
0.5
b33/(ρ Ar)[B/(2g)]0.5
ω[B/(2g)]0.5
ξ3=0.03 m
ξ3=0.01 m
2D potential flow
2D DD strategy
Vugts experiments
2D DD strategy
Vugts experiments
3D potential flow
0.6 0.9 1.20
0.02
0.04
2D potential flow
2D DD strategy
Vugts experiments
3Dpotential flow
b44/(ρ Ar B 2)[B/(2g)]0.5
ω[B/(2g)]0.5
Figure 2: Top: heave (left) and roll (right) wave exciting loads. Center: added-mass in heave (left) and roll (right).
Bottom: damping in heave (left) and roll (right). Ar = 0.9992BD is the cross-section area. (Ly1 = Ly2 = 2.15B)
to the boundary and so with larger sensitivity to the NS-LS and BEM description of the flow-boundary solution (velocity
and pressure must be furnished in an area seen as inside the body by the linear potential solution). The bottom-right plot
of the figure examines the influence of the domain horizontal dimensions for the same forced-roll conditions. In this case
the body is inside the NS-LS domain in both simulations. The narrower-domain solution shows some unphysical behavior
of the pressure near the left boundary but it is quite consistent with the wider-domain solution elsewhere. Instead, the
velocity field is more sensitive to the closeness of the boundary and appears somewhat different than the wider-domain
solution even besides the body surface. One must then identify the minimum Ly1 and Ly2 to ensure a physical solution
near the body. It should be noted that the difference in the velocity is not much felt by the roll moment acting on the
body (not shown here). An even more critical parameter is Lz . On the bottom, the inconsistency between the irrotational
potential flow and the rotational NS-LS can act like a porous boundary, influencing the detachment and evolution of the
shed vorticity. This must be avoided either by dynamically enlarging the inner-domain extension or by introducing a
numerical diffusivity of the vorticity near the domain boundary and sufficiently far from the body so that the physical
phenomena relevant for the liquid-structure interactions are correctly described. A parameter analysis is ongoing, aiming
to obtain the proper dimensioning of the inner domain as a function of the type and amplitude of motion and will be
presented at the workshop. The details of the developed DD solver are described in Greco et al. (2012) and will be also
discussed at the workshop.
This research activity is supported by the Centre for Ships and Ocean Structures (CeSOS), NTNU, Trondheim, within the
ω[B/(2g)]0.5=0.6
ω[B/(2g)]0.5=1.0
ω[B/(2g)]0.5=0.8
ω[B/(2g)]0.5=1.2
Figure 3: Forced roll motion with ξ5a = 0.05 rad: vorticity contours for ω = 0.6, 0.8, 1.0 and 1.2
√
2g/B. Snapshot
corresponding to maximum ξ5. (Ly1 = Ly2 = 2.15B)
8.5 9 9.5
-0.01
-0.005
0
0.005
0.01
F3/(ρAξ3)
t/T
Figure 4: Top: forced heave motion with ξ3a = 0.15D and ω = 0.6
√
2g/B. Vertical force (left) and pressure field
(right) using Ly2 = 0 (half left) and Ly2 = 2.15B (half right). Bottom: forced roll motion with ξ5a = 0.05 rad and
ω = 0.6
√
2g/B. Pressure field (left) using Ly2 = 0 (half left) and Ly2 = 2.15B (half right) and pressure and velocity
fields (right) using Ly1 = Ly2 = 1.5B (half left) and Ly1 = Ly2 = 2.15B (half right).
”Violent Water-Vessel Interactions and Related Structural Loads” project.
References
GRECO, M., G. COLICCHIO, AND C. LUGNI (2011). Development of a 3D Domain-Decomposition strategy for
violent head-sea wave-vessel interactions: Challenges. In 26th Int. Workshop of Water Waves and Floating Bodies,
Athens, Greece.
GRECO, M., G. COLICCHIO, C. LUGNI, AND O. FALTINSEN (2012). 3D Domain Decomposition for Violent Wave-
Ship Interactions. Manuscript in preparation.
VUGTS, J. (1968). The hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface. Int.
Ship-building Progr. 15.


Assessment study of a domain-decomposition strategy for marine applications

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Assessment study of a domain-decomposition strategy for marine applications

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