Challenges on the numerical prediction of slamming loads on LNG tank insulation panels

Abstract

The load assessment in water impacts problems involving complex geometries is generally a quite challenging task for numerical solvers. Indeed, the occurrence of large free-surface deformation, air cushioning and possible compressibility effects strain the standard Computational Fluid Dynamics (CFD) codes to their limits. In the present work, an enhanced Smoothed Particle Hydrodynamics (SPH) model is adopted to evaluate the pressures acting on the surface of smooth and corrugated panels impacting water and the numerical outcomes are compared to experimental data and analytical solutions. Specifically, the water entry of a flat panel at is firstly studied in order to highlight the main critical aspects underlying the numerical solution of water impacts with small deadrise angles. Then, the water impact of a Mark III type panel (a corrugated insulation panel) adopted in LNG tanks is considered. Experimental data involving wet drop tests of both flat and corrugated panels have been performed and the pressures during the impact have been measured at several points along the panel surface. Complex features of the flow, such as 3D effects and air-cushioning, have been addressed by a developing the numerical study in steps of increasing complexity.

Full text

Challenges on the numerical prediction of
slamming loads on LNG tank insulation
panels
S. Marrone a, A. Colagrossi a,∗, J.S. Park b, E.F. Campana a
aCNR-INSEAN, Natl. Research Council - Marine Tech. Research Inst., Rome,
Italy
bHyundai Heavy Industries Co. Ltd., South Korea
Abstract
The load assessment in water impacts problems involving complex geometries is
generally a quite challenging task for numerical solvers. Indeed, the occurrence of
large free-surface deformation, air cushioning and possible compressibility effects
strain the standard Computational Fluid Dynamics (CFD) codes to their limits. In
the present work, an enhanced Smoothed Particle Hydrodynamics (SPH) model is
adopted to evaluate the pressures acting on the surface of smooth and corrugated
panels impacting water and the numerical outcomes are compared to experimental
data and analytical solutions. Specifically, the water entry of a flat panel at 4◦
is firstly studied in order to highlight the main critical aspects underlying the
numerical solution of water impacts with small deadrise angles. Then, the water
impact of a Mark III type panel (a corrugated insulation panel) adopted in LNG
tanks is considered. Experimental data involving wet drop tests of both flat and
corrugated panels have been performed and the pressures during the impact have
been measured at several points along the panel surface. Complex features of the
flow, such as 3D effects and air-cushioning, have been addressed by a developing
the numerical study in steps of increasing complexity.
Key words: Smoothed Particle Hydrodynamics; Water impact; Slamming;
air-cushion effects; Trapped gas cavity; LNG tank;
∗Corresponding author: Tel.: +39 06 50 299 343; Fax: +39 06 50 70 619.
Email address: andrea.colagrossi@cnr.it (A. Colagrossi ).
https://doi.org/10.1016/j.oceaneng.2017.06.041

Fig. 1. Left: A membrane type LNG carrier. Right: Inside view of the Cargo
Containment System (CCS) of a LNG carrier, the surface of the tank is covered
by Mark III corrugation panels.
1 Introduction
In recent years the consumption of the Liquefied Natural Gas (LNG) fuel
has largely increased due to the commitment of nations to reduce noxious
emissions of standard fuels, notably NOx and SOx oxide. In order to reduce
the transportation cost as well as to meet the increased demand of LNG,
designers have considered larger LNG carriers. These vessels are designed and
constructed for safe transportation of natural gas at cryogenic temperatures.
Storage and transportation of LNG are critical issues in the design and safety
aspects of LNG carrier. In particular, the Cargo Containment System (CCS),
which is responsible for keeping the liquid at low temperature (-165◦C), is
a key component. The CCS is a composite structure of metal membrane,
polyurethane foam and plywood laminated insulator and its strength can be
characterized as the maximum load that the system can sustain before it fails.
The loads acting on the insulation system are due to the sloshing motion of
the liquid in the LNG tank and are typically the result of an impact flow
dynamics, which is generally characterized by an impulsive loading processes.
The design of CCS should adequately take into account such load features to
evaluate the strength and failure mode of the insulation system. This applies
a fortiori in the contemporary shipping context, in which the LNG tank is not
barely filled or empty (as it was the case in the past). Indeed, as remarked in
[19], due to the changed market demand, nowadays any filling level is allowed,
which introduces higher risks related to sloshing fluid motions.
Recently, deformations of primary corrugation membrane (Mark III type,
see right plot of figure 1) adopted in LNG tanks have been reported
(see, e.g.,[4]). As a consequence, the importance of the load assessment
on corrugated insulation panels has become increasingly relevant (see, e.g.,
[6, 5]). The present investigation is dedicated to the numerical simulation of
wet drop tests of a Mark-III insulation panel using the Smoothed Particle
Hydrodynamics (SPH) method and to the validation of the numerical results
against experimental data. The wet drop test consists in the water entry of the
2

panel with a specific configuration and at a fixed velocity. This kind of test
case can be seen as analogous to the problem of a fluid wedge impacting
a vertical wall (see, e.g., [32]) and can give useful insights for the more
complex problem of a wave impacting a wall composed by Mark III panels.
With respect to the latter, in the wet drop test it is simpler to control and
reproduce (also numerically) the impact conditions, allowing for a detailed
comparison between physical model and numerical output. Because of the
small deadrise angle, the numerical simulation of such an impact is quite
demanding and requires suitable models. Among the available solvers, the SPH
method has been selected, being accurate and robust in treating violent free-
surface flows. A brief description of the method is given in section 3. In order
to address the capability of such a solver, a test using a flat plate is used in a
preliminary test. Indeed, for such a simple geometry solutions using potential
theory are available (being aware of the fact that these solutions are referred
to a symmetrical flow, see e.g. [33, 28, 8]) and, furthermore, experimental
data are also available. For the chosen impact angle the air phase can be
neglected, therefore allowing for a simple comparison between experimental
and numerical outcomes. The analytical references and the experimental data
allows the proper establishment of the time-space discretization for the other
test-cases investigated (see section 4). A critical discussion on the comparison
between analytical solution, numerical results and experiments will be given
by focusing the most relevant challenges of the numerical simulation of water
impact.
Regarding the actual focus of the present work, for the Mark-III insulation
panel the entrapping of the gaseous phase by the liquid one can play a
significant role on the time-space distributions of the local loads. Because
of the square geometry, formed by the corrugation stencil, 3D effects can
also be relevant, indeed the water impact jets can be deflected and focused
on specific zones of the panel. The effect of the surface tension can be also
important in the bubble-flow formation as a result of the gas trapped during
the impact stages. Because of the above considerations, numerical simulation
of the water impact against Mark-III insulation panel requires two-phase 3D
simulations with an accurate surface-tension model. To our knowledge this
kind of numerical investigation is not yet available in the literature, even in
the most recent works, mainly because of the high computation costs involved.
In a recent paper by [19] the initial stage of the impact of the Mark-III panel
is studied in 2D with null deadrise angle by means of a Wagner approach. In
that study also the air compressibility and the structure elasticity are taken
into account. Conversely, in the present work a dead-rise angle equal to 4◦
is considered and, hence, the role of the air can not be a priori assessed.
Therefore, in order to tackle the problem in steps with increasing complexity,
the analysis has been subdivided in three sets of simulations:
3

i) 2D single-phase simulations
ii) 3D single-phase simulations
iii) 2D two-phase simulations
Due to these three steps, it is possible separate the role of 3D effects and the
air cushion phenomena. Future perspectives and conclusions will wrap-up the
paper.
2 Experimental Set-up
Wet drop tests of the Mark-III insulation panel were performed in the
Slamming Research Lab. at Pusan National University. The maximum drop
height tested was 3.8 m. The incident angles were 4, and 6 degrees. In the
present work only the first condition has been considered, it being the most
critical.
The dimensions of the test specimens were 640x520 mm (LxB). The specimens
used in the drop test were: a real Mark III corrugation membrane (steel
SUS304L, of thickness 1.2 mm), a rigid flat plate (steel thickness 20 mm,
hereinafter referred as the flat panel) and a rigid Mark III corrugated panel
(steel, thickness 20 mm).
Fig. 2. Test specimen and setup for the drop test experiment.
4

Fig. 3. Flat (left) and corrugated (right) panels used in the drop tests.The holes
visible in both the pictures indicates the location of the pressure sensors.
The flat plate was tested in order to compare the pressure distribution
measurements with reference data available in the literature. The real
corrugated membrane was tested to find the relation between the drop height
and the deformation of the corrugation, facilitating the analysis of the buckling
strength of the membrane. However, in the present work the hydroelastic
problem is not considered and we focus our investigation only on the rigid
panels in order to study the pressure distributions developed during the water
entry.
The pressure sensors used were ICP type sensors whose maximum capacity is
70 bar. The size of the pressure gauge is 6.3 mm. The sampling rate of the
data acquisition was up to 100 kHz which is expected to be high enough to
capture the time histories of the pressure impacts, even for the small dead-rise
angles considered here (see e.g., [29, 18]).
Figure 2 shows a picture of the drop test facility used in the Slamming
Research Lab of Pusan National University; in figure 3 pictures of the rigid
panels are shown. The total weight of the support, the wedge and the
panels is about 1500 kg, which guarantees that during the impact stages the
deceleration of the wedge is negligible (as confirmed by the time history of the
wedge displacement, which was measured using accelerometers). This allowed
numerical simulations imposing a constant water entry velocity, thus avoiding
the possibility of errors due to changing body velocity during the drop test.
As noted in [33], the effects of changes in velocity are not always considered in
many of the works presented in the literature. In [33], the authors also remark
on the importance of the size of the pressure gauges; indeed as discussed in
the following section, this needs to be taken in to account when small deadrise
angles are considered.
5

3 The Riemann-ALE Smoothed Particle Hydrodynamics model
In the present work the Riemann-based solver described in [26] is adopted.
In that work Arbitrary Lagrangian-Eulerian formalism is used allowing for
maintaining a regular particle spatial distribution and smooth pressure fields
while preserving the whole scheme conservation and consistency of the classical
SPH scheme.
Both compressible and incompressible formulations of SPH can be found in
the literature. We focus here in its compressible form, addressing weakly-
compressible fluids. Therefore, the continuity equation is used together with
the Euler equation:























Dρ
Dt =−ρ∇ · u
Du
Dt =−∇p
ρ+g
Dr
Dt =u
(1)
where D/Dt represents the Lagrangian derivative, while r,u,pand ρare,
respectively, the position of a generic material point, its velocity, pressure and
density; and grepresents gravitational acceleration. The system (1) is written
in a Lagrangian form, which represents one of the main attractions of the
SPH method. Since the Reynolds number of the flow is quite high and only
the impact stage is simulated (short time-range regime) viscous effects can
be considered negligible. Moreover, the fluid is assumed to be barotropic and,
therefore, the following functional dependence between ρand pis adopted:
p(ρ) = c2
0ρ0
γ" ρ
ρ0!γ
−1#+p0,(2)
where ρ0and p0are constant, c0is the speed of sound, and γis a dimensionless
parameter greater than 1 (in all of the following examples γ= 7 is used).
In the SPH method, the speed of sound c0is chosen much lower than the
real one, but much higher than the reference velocity, Uref , to satisfy the
weakly-compressible assumption. When considering violent free-surface flows,
the proper identification of the reference velocity Uref is crucial, as is discussed
in the following sections. Considering the Mach number Ma = Uref /c0, the
constraint Ma <0.1 is taken into account to make compressibility effects
negligible. Additionally, during violent impact events (i.e., flat impacts) the
acoustic pressure p=ρ u c0can be reached, and in this case the pressure
6

peak intensity becomes proportional to 1/Ma. On the other hand, in such a
condition an incompressible constraint can induce singularities on the pressure
field (see [22] for a discussion on the difference between these two models in
impact situations). This is linked to the fact that for this kind of impacts the
presence of the air phase is generally crucial and the single-phase approach
can lead to incorrect pressure evaluations under the incompressible/weakly-
compressible hypothesis (for a deeper discussion see also [24]). Being aware of
these limits for a single-phase model, the results obtained in this paper have
been produced considering a possible Mach dependency.
The SPH model is based on the interpolation of a generic flow field fthrough
a convolution integral with a kernel function Wover its compact support Ω:
hfi(r) = ZΩf(r0)W(r0−r;h)dV 0(3)
The fluid domain is discretized into a finite number of particles representing
elementary fluid volumes, each with its own local mass miand other physical
properties. At the discrete level a generic field fevaluated at the position ri
of the i-th particle is approximated through a convolution sum. Equation (3)
thus becomes:
hfi(ri) = X
j
f(rj)W(ri−rj;h)Vj(4)
where Vjis the volume of the j-th particle. Hence any function or its gradient
can be interpolated in total absence of connectivity.
The use of the Riemann-based solver leads to an increased stability and
robustness of the scheme with respect to standard SPH schemes. The
formalism proposed by [30], which is inspired by compressible Finite Volume
techniques, is adopted. For each pair of particle-to-particle interactions (i, j),
a Riemann problem is solved in the direction (i, j) for Tait’s equation of
state, providing a solution for the flux that is then convolved using the
kernel gradient in order to solve for the expected flux divergence. Thanks to
the introduction of Riemann-solvers the fluxes between particles are upwind
oriented and the resulting scheme is characterized by good stability properties.
7

The discrete Euler equations are written as follows:











































Dri
Dt =v0i,
D Vi
Dt =ViX
j
(v0j−v0i)∇Wij Vj
D(Viρi)
Dt =−ViX
j
2ρE(vE−v0(rij ))∇Wij Vj
D(Viρivi)
Dt =−ViX
j
2 [ρEvE⊗(vE−v0(rij )] ∇Wij Vj
−ViX
j
2PEI∇Wij Vj+ωiρig
(5)
where ρE,PEand vEare the solutions of the Riemann problem at the
interface rij = (ri+rj)/2, between particles iand j. The particle transport
velocity v0is obtained as the summation of the particle velocity plus a small
perturbation which helps preserving a regular particle distribution (details
about the adopted model can be found in [26]).
The two-phase model is based on the work by Leduc et al. [21] where
an approximate Riemann solver without mass fluxes is introduced at the
interface between particles belonging to different phases. Specifically, the
scheme adopted in the present work is an extension of the one described in [16]
where the ALE formulation presented in [26] has been added. In particular,
at the fluid interface the purely Lagrangian form of the ALE equations 5 is
adopted for a strip of particles spanning a kernel radius. Readers can find also
more information on general weakly-compressible SPH multiphase models in
e.g. [15, 14, 12].
4 Preliminary test on a flat panel
In this section the results of the 2D single-phase simulations are described and
compared to the experimental data from a wet drop test performed with a flat
panel (panel length Lequal to 64 cm, see Section 2) impacting with an angle
of 4◦and a vertical impact velocity Uof 6.0 m/s.
As described in the theoretical work by [33], for this small angle a very thin,
high-speed jet of water is formed, and the time-spatial gradients of the pressure
field are extremely high. This makes the test conditions very demanding for
numerical solvers. From the potential flow theory by [33], the jet thickness
at model scale is about 0.1 mm. It is worth noting that the theory in [33]
is formulated for symmetric wedge impacts whereas in the present case the
8

impact of a inclined single plate is considered. More details about the reference
solution to be adopted are given in section 4.2.
On the base of this theoretical data, the 2D simulation has been conducted
using a very high spatial resolution, corresponding to a particle size ∆x=
15.6µm; by referring the latter to the panel length L, the ratio L/∆xis equal
to 41,000. The whole tank depth and width are, respectively, 3 m and 6 m.
In order to manage such a small particle size a multi-resolution technique has
been used. Specifically, the particle size gradually changes with a maximum
magnification factor of 3,200 between the most refined region and the lowest
resolution one (see figure 4). The total number of particles is about 3 million.
For small dead-rise angles water compressibility cannot be neglected when
calculating the speed of the water jet Ujet (see e.g. [20, 7, 9]). In the
experimental conditions the speed of sound in water is c?
0= 1481 m/s '247 U,
therefore the Mach number Ma = U/c?
0is about 4 ×10−3. Under such a
condition an estimate of the jet speed is Ujet ∼42U∼250m/s. As noted in
[20], this very thin, high-speed jet disintegrates at some distance from the root
due to interaction with both the surrounding air and the surface of the body.
Note that in such kind of instability surface tension plays an important role
on the jet evolution. This local and complex physics is not taken into account
in the present numerical method but it is not supposed to play a relevant role
on the pressure distribution.
Fig. 4. Flat panel impacting water. Colors are representative of the SPH velocity
intensity. Pressure probe number 12, which is the gauge used for the comparison
with the experimental data, is also depicted.
9

4.1 Selection of the speed of sound for the SPH model
Before starting the SPH simulations the speed of sound needs to be specified.
As mentioned in section 3, the numerical Mach number usually adopted within
the SPH simulations is Ma = U/c0= 0.1, which guarantees the weakly-
compressible regime (i.e., compressibility plays a negligible role). Nonetheless,
for this kind of impact the reference speed for the Mach number can not be the
impact velocity U. Indeed, considering that the intersection point between the
horizontal undisturbed free-surface and the wedge surface has a speed equal
to Uinters =U/sin(α)'14 U. The water jet formed during the impact has a
speed higher than Uinters . Therefore, if one chooses the speed of sound using
the wedge speed U, i.e., c0= 10 U, the jet would not form at all, its speed
being in the supersonic regime. This is a clear example where the weakly-
compressible rule Ma ≤0.1 needs to be enforced in a proper way, considering
the specific problem at hand.
In the present case the reference speed should be the water jet speed Ujet
which, however, is an unknown of the problem. Using the theory in [20], the
estimate Ujet = 40Ucan be used. Note that, using the latter constraint,
the speed of sound in water would result even higher than the real one,
c0= 400Uversus c?
0= 247U. This means that for the adopted model scale
water compressibility effects are not negligible, at least inside the jet region.
However, considering that in the jet zone the pressure is close to the ambient
pressure, water compressibility effects should not play a relevant role on the
local impact loads.
In order to satisfy the weakly-compressible assumption, at least in the impact
region, the reference velocity used is Uref =qPmax/ρ, where Pmax is an
unknown of the problem and needs to be estimated. Then, an ex-post facto
verification of the SPH simulations is required. Using the Wagner theory
(which is valid for small deadrise angles as far as the air presence is negligible)
the maximum pressure predicted is:
Pmax =1
2ρ U2π2/4
[tan(α)]2
α=4◦
'252 ρ U2(6)
corresponding to about 91 bar in our experiment.
The water-hammer pressure for this impact is ρ c?
0U= 88 bar which is the
maximum pressure level that can be physically reached in the experiments.
The Pmax predicted by potential flow theory is higher than the acoustic
pressure, and this is a further indication that water compressibility cannot
be neglected for this problem. Thus, considering Pmax ∼80 bar, the reference
velocity is Uref =qPmax/ρ ∼15 Uwhich is smaller than Ujet = 40U.
10

Fig. 5. Flat panel impacting water: colors are representative of the SPH pressure
field. Red solid line is the free surface extracted from [33].
Taking this into account, a good compromise for the SPH speed of sound
can be obtained by adopting a reference velocity Uref = 10Uwhich implies
a speed of sound c0= 100 U(i.e., Mach number Ma = U/c0= 0.01). The
resulting time step is equal to ∆t= 1.5 ∆x/c0'0.04 µs which is 250
times smaller than the sampling rate of the pressure probes used in the
experiments (i.e., the SPH sample frequency is 25 MHz versus the 100 kHz
of the experimental pressure probes). This aspect is expected to influence
the observed pressure peak which, for the considered configuration, needs a
high time/space resolution to be captured. Further, in order to verify the
appropriateness of such a choice, also a simulation using c?
0= 247Uhas been
run, the results will be shown at the end of section 4.2.
4.2 Comparison between SPH results, analytical solution and experimental
data
Figure 5 shows the pressure field predicted by the SPH for the flat panel
impact. In the same plot the free surface deformation evaluated by the
potential flow theory by [33] is reported. Left plot of figure 6 shows an enlarged
view of the flow velocity predicted by the SPH in the area of the highest
pressure levels. A thin water jet is formed with a thickness of about 0.1 mm
corresponding to about ten particles, thus justifying the high L/∆xratio
needed to properly solve such a flow.
In the right plot of the same figure an enlarged view of the pressure field is
shown (in this case the displayed pressure range is enlarged too). From this
plot it is seen that the high-pressure region is limited to an area of 1 mm2with
a pressure peak of about 60 bar. In the same figure the size of the probe used
in the experiments is depicted. Clearly, the pressure sensor has a size much
larger than the pressure bulb formed below the water-jet root. This aspect
will be discussed in the following section.
11

Fig. 6. Flat panel impacting water: enlarged view of the impact zone. Left: contours
of the velocity module. Right: contours of the SPH pressure field. The size of the
pressure gauges used in the experiments is also depicted.
In figure 7 the time histories of the pressure measured at probe P12 are shown.
This probe is positioned 23.6 cm from the left-hand edge of the panel (see
figure 4). The SPH solution is compared with the experiments and with the
pressure peak predicted by Wagner theory. The SPH prediction is between the
two reference data sets, and is characterized by high-frequency components
due to the fragmented jet of water that pass over the numerical pressure
probe. The latter has a dimension of 0.125 mm (which is the size of the SPH
kernel support) and the SPH sampling rate is 25 MHz. Both the SPH and
the analytical predictions overestimate the experimental data for which the
maximum pressure recorded is 34 bar (in the SPH solution the pressure peak
reaches 70 bar whereas the analytical prediction is 91 bar). It has to be noted
that, even though the Wagner theory is referred to the case of a symmetric
wedge entry, the value of the pressure peak should substantially the same of
the case of an oblique flat panel impact (see e.g. [13, 27]). Conversely, regarding
the entire pressure signal, it is not possible to compare to classical analytical
solutions, such as in [33], since they are all formulated for the symmetric wedge
entry case.
As mentioned above, air entrainment is expected to play a minor role for
deadrise angles greater than 3◦[33, 27]. Notwithstanding that, most of the
experimental measurements available in the literature for angles close to 4◦
exhibit pressure peaks much smaller than the one predicted by potential theory
(cf., [11, 25, 27, 29]) and the values measured in the present study are in fair
agreement with previous experiments by [27, 11]. As for possible 3D effects,
these have been checked by comparing the pressure values on gauges aligned
at the same distance from the piercing edge. For the probes positioned in
the most central region no relevant differences have been observed. However,
the maximum pressure impact measured in the experiment for small dead-rise
angles can be also affected by:
(1) Changes of the body velocity during the impact stage
(2) Rotations of the body during the impact stage
12

Fig. 7. Flat panel impacting water: SPH pressure time histories at probe P12
compared with experimental data and with analytical solution Wagner theory.
Fig. 8. Flat panel impacting water: SPH pressure time histories versus experimental
data. The SPH signal is filtered with a running average filters (MAF) at 100 kHz
(dashed-dotted line).
(3) Deformations of the wedge surface
(4) Sampling rate of the pressure signals
(5) Size of the pressure gauge
Thanks to the experimental setup adopted (see Section 2) the first three points
can be neglected, while the last two can play an important role. In order to
take into account the experimental sampling rate, the SPH signal has first
been filtered using a moving average filter (MAF) reducing the numerical
sampling rate from 25 MHz to 100 kHz. The result is illustrated in figure 8.
The reduction of the SPH peak due to this filtering procedure is not enough
to get a good agreement with the experimental data.
As a further step the SPH pressure has been measured integrating on a circular
13

Fig. 9. Flat panel impacting water: SPH pressure time histories versus experimental
data . The SPH signal recorded using the size of the experimental pressure gauges
and filtered at 100 kHz is also shown.
Fig. 10. Flat panel impacting water: experimental data for the pressure time histories
on a row of probes (not equispaced). The time history from the first and last probes
on the panel impacting the water are colored in red. The time history in blue is
from probe P12 , used in the previous figures for comparison with the SPH solution.
area equal to the size of the experimental pressure probes and then filtered at
100 kHz. This result is shown in figure 9. In this case the SPH output is much
closer to the pressure recorded in the experiment.
Summarizing, according to the analytical solution very narrow pressure peaks
are expected for this kind of impact. Using pressure gauges with a size of few
millimeters and a sampling rate of 100 kHz it is not possible to record such
a localized event. Having in mind such limits, through SPH it is possible to
get predictions close to the experimental data if the pressures are integrated
over the experimental gauge area, even when using a speed of sound c0smaller
than the real one c?
0.
14

Fig. 11. Flat panel impacting water: SPH pressure time histories on a row of probes.
The original SPH signals have been filtered with a MAF at 100 kHz as in the
experimental signals. The time history from the first probe on the panel impacting
the water is colored in red.
In figure 10 the experimental pressure time histories recorded on a sequence of
pressure probes is reported. Even if the experimental pressure peaks present
some fluctuations, very similar pressure evolutions are recorded with an
almost constant time shift at each probe. The SPH results show quite good
repeatability of the pressure peaks along the wedge surface as well (see figure
11). In this regard, it is worth comparing the propagation velocity of the
pressure peak along the plate, UP. Indeed, in water entry flows the value of
the pressure peak should correlate to U2
P(see e.g. [17, 3]) as:
CP=Pmax
1
/2ρ U2
P
= 1.(7)
In figure 12 the calculated values of UPfor both SPH and experimental results
are reported for several probes. Only the probes far from the panel edges have
been considered to avoid influences of either the initial impact stage or the
final stage, for which the flow self-similarity is not applicable. In the same
figure the values of UPobtained from a simulation adopting the real speed
of sound c?
0= 247U(Ma=0.004) are also reported. The difference between
the simulation with Ma=0.01 and Ma=0.004 is very small (about 2% of the
average value), confirming that within the weakly-compressible regime the
Mach number effect is limited. For the sake of completeness in the same plot
also the analytical value of UP(valid for a wedge entry problem) is reported,
i.e.:
UP=π
2Ucot(α).(8)
In Table 1 The average values of UPare reported. For the simulation at
Ma=0.01 the average value of UPis about 117 m/s corresponding to a pressure
coefficient CP= 1.03 which is close to the expected value (7). On the other
15

Fig. 12. Flat panel impacting water: Peak propagation velocity UPcalculated on
several positions along the panel for the experiment (triangles), SPH simulation
at Ma=0.01 (circles) and Ma=0.004 (squares), and analytical value from potential
flow theory (dashed line). The origin of the x-axis is set at the left-hand edge of the
panel.
UP(m/s) 1/2ρ U 2
P(bar) Pmax (bar) CP
Exp. 104 54 30 0.56
SPH Ma=0.01 117 68 70 1.03
SPH Ma=0.004 119 71 75 1.06
Table 1
Average of the peak propagation velocity UPplotted in fig. 12. The corresponding
pressure peak 1/2ρ U 2
P, the actual average pressure peak measured at the probes
Pmax and the related pressure coefficient CP(7) are also reported.
hand since the value of UPpredicted by the SPH is smaller than the analytical
one this implies that the maximum pressure of the SPH cannot be equal to
the one predicted by the Wagner theory as shown above. This difference is
essentially due to the fact that Wagner theory is referred to a symmetric
wedge entry, while the present case is asymmetric (see section 4.3).
When considering the experimental pressure probes an average value of UP
equal to 104 m/s is obtained (with a standard deviation σ= 3.1). This
propagation velocity of the pressure peak corresponds to a pressure coefficient
CPequal to 0.56. This confirms that the measurement system adopted is not
able to record the real pressure peaks which are therefore underestimated as
shown in this section.
16

4.3 Comparison between symmetric and asymmetric water entry
In the last subsection it has been shown that the pressure peak predicted
by SPH is consistent with the water impact theory. Indeed, the calculated
peak propagation velocity UPand the pressure peak Pmax give a pressure
coefficient CPclose to unity (see eq. 7). Nonetheless, it still remains a
significant discrepancy between SPH results and the analytical solution for
this impact angle. In order to investigate the source of this incongruity a
further simulation has been performed. As already mentioned, most of the
analytical solutions available in the literature refer to a wedge entry problem.
Therefore, the comparison between asymmetric (i.e. single flat panel entry)
and symmetric (i.e. wedge entry) problem is useful to clarify the main cause
behind the difference between SPH and analytical solutions.
The symmetric problem has been set by retaining the initial particle
configuration of the asymmetric case and closing the fluid domain at the left
edge of the panel with a vertical wall (see figure 13). The simulation has been
run with Ma=0.01.
In figure 14 the peak propagation velocity for both symmetric and asymmetric
problems are shown. It is clear that in the symmetric configuration the SPH
solution is now much closer to the potential flow prediction, the computed
average value being about 131 m/s (the theoretical value is 135 m/s). Because
of this increase in the value of UP, a larger pressure peak is expected on the
panel surface. The recorded pressure at 0.16 m from the left-hand edge of the
panel is shown in figure 15 for both symmetric and asymmetric SPH solutions.
Consistently with the observed value of UP, the pressure peak of the symmetric
solution is about 88 bar. In the same figures the analytical solutions from [31]
and [33] are also reported. Evidently, the symmetric SPH solution is again
much closer to the analytical prediction and the remaining difference can be
Fig. 13. Flat panel impacting water: fluid domain adopted in the symmetric problem
configuration. Colors refer to the module of the velocity field.
17

mainly attributed to fluid compressibility. It is therefore clear that in the
asymmetric problems the differences are not only limited to the pressure at
stagnation (i.e. far from the pressure peak), which is expected to be lower
due to the uprising jet at the left edge, but also the pressure peak intensity is
affected (at least in the considered test conditions). This is in agreement with
the analytical work in [28]. In that work it is shown that, generally, asymmetric
impacts induce smaller pressure peaks with respect to the symmetric case. This
effect is emphasized when the deadrise angle is smaller (in [28] the smallest
angle considered is 10◦).
Fig. 14. Flat panel impacting water: Peak propagation velocity UPcalculated
on several positions along the panel for the asymmetric (circles) and symmetric
(triangles) SPH simulations, and analytical value from potential flow theory (dashed
line). The origin of the x-axis is set at the left-hand edge of the panel.
Fig. 15. Flat panel impacting water: pressure time histories of symmetric (solid line)
and asymmetric (dashed) solutions measured at 0.16 m from the left-hand edge of
the panel. Analytical solutions by [31] (dash-dot line) and by [33] (dash dot-dot
line) are also reported.
18

5 Wet drop test of Mark III type insulation panel
In the present section the results of the numerical simulations for the impact
of the Mark III type insulation panel are described. The impact velocity U
and the impact angle are again equal to 6.0 m/s and 4◦, as for the case studied
in the previous section. The geometry of the Mark-III insulation panel, used
for wet drop test, was initially described by an Initial Graphics Exchange
Specification (iges) file. The latter needed to be manipulated and transformed
into an unstructured surface mesh in order to be a suitable input for the SPH
code.
The geometrical impact configuration is given by the sketch in figure 16. The
angle αhas been set equal to 4◦, this condition resulting as a critical one in
the experiments. Some numerical pressure probes have been set in order to
record pressure time histories during the simulation. The probe labels adopted
for the experiments have been maintained in order to compare the numerical
data with the reference one without any ambiguity.
Figure 17 shows bottom and lateral views of the numerical reconstruction of
the Mark-III insulation panel along with the pressure probe positions. The
present section is subdivided into three subsections addressing, respectively:
i) 2D Simulation with a single-phase model
ii) 3D Simulations with a single-phase model
iii) 2D Simulation with a two-phase model.
5.1 2D Simulation with a single-phase model
The 2D profile of the corrugated panel has been extracted from central section
of the panel. For this simulation a multi-resolution technique has been used
in order to change the particle size with a magnification factor of 100 from
the most refined region up to the lowest resolution one; the particle size
Fig. 16. Sketch of the impact configuration for the Mark-III insulation panel.
19

Fig. 17. Bottom and lateral views of the Mark-III insulation panel obtained using an
unstructured mesh. On the left-hand plot labels for the pressure probes are shown.
Fig. 18. 2D single-phase model: particle-size distribution.
distribution is shown in figure 18. In the area involved in the impact, the
spatial resolution is ∆x= 0.25mm, the length Lof the panel being 64 cm
and the ratio L/∆xbeing equal to 2560. Three different simulations with
resolution ratios equal to L/∆x= 640,1280,2560 have been performed and
figure 19 shows the pressure time histories at probe P40 recorded with the
three different resolutions. The initial time t= 0 is set at the instant when
the left dent touches the free-surface. The same rule has also been adopted for
the following numerical test cases.
The plot depicted in figure 19 shows rather limited discrepancies for the
20

Fig. 19. 2D single-phase model: pressure-time evolution measured at probe P40 .
Fig. 20. 2D single-phase model: time evolution of the pressure field for six different
time instants.
21

Fig. 21. 2D single-phase model: pressure time histories for six different probes.
Comparison between experimental data and 2D single-phase SPH solution.
22

pressure peaks, hence for the present problem it is not necessary to reach
the L/∆x= 10,000 used with the flat panel (see section 4.2) Thanks to this
first analysis, we decided to use the ratio L/∆x= 640 for the 3D simulation
(see the next section), and for conformity the results presented in this section
will be compared with 3D ones using this spatial resolution.
Figure 20 depicts the pressure field during the impact stage. For this simulation
the Mach number Ma = U/c has been set equal to Ma = 0.1 since, as shown in
the following, the pressure peaks are less intense relative to the ones recorded
during the impact of the flat panel. At time t= 6.5ms (first plot of figure 20)
the left dent has completely entered in the fluid while the second one is only
partially immersed. At this stage no relevant pressure peaks are observed.
The corresponding recorded pressure time histories are shown in figure 21
where the 2D single-phase SPH solution is compared to the experimental data
for six different probes (the position of the probes is shown in the first plot
of figure 20). At time t= 9ms a pressure shock forms on the left part of the
panel. From these first two pictures it is clear that a cavity is formed between
the two dents and the free surface. Therefore, due to the absence of the air,
this cavity is going to collapse inducing high pressure shocks (see time instants
t= 9ms and t= 12ms). This explains also the two consecutive peaks shown
in figure 19: the first peak at t= 11.5ms is due to the passage of the left jet
propagating towards right; the second peak, at about t= 12.5ms, is due to
the complete collapse of the cavity resulting from the encounter between the
two jets.
As a consequence, in the time range t= 11 −14ms the maximum of the
pressure is reached between probes P40 and P47 where peaks of 16 bar are
measured. This pressure is linked to the water-hammer pressure ρ c Ucavity
where Ucavity '27 m/s is the relative velocity of the water fronts during the
cavity collapse (an analogous impact has been described in [23] for the collapse
of a plunging jet). The pressure calculated by the SPH model depends therefore
by the specific speed of sound adopted (i.e. the specific Mach number used).
However even adopting a Mach number of 0.1 the SPH pressure is four times
higher than the pressure recorded in the experiments (4 bar versus 16 bar)
and the experimental time history is much smoother resembling a sinusoidal
function. This is a clear indication that air cushion effects cannot be neglected
in the present problem.
The agreement between the numerics and the experimental data is better for
probe P49 apart from for a slight delay of the SPH data. At time t= 15ms a
third pressure impact is recorded on the right-hand part of the panel reaching a
pressure level of 8.5 bar at probe P53 where the experiments recorded a peak of
10.5 bar. In principle, in this region of the flow field, we do not expect relevant
three-dimensional effects. Further, in this part of the panel air cushion effects
23

are not expected to be dominant and the flow resembles the one discussed for
the flat panel (see Section 4). From figure 21 it is possible to observe that the
duration of the peak registered at probe P53 is about 6 ms which is very close
to the experimental one.
5.2 3D simulation with a single-phase model
After the 2D single-phase simulations a 3D simulation has been performed.
Because of the large number of particles needed to accurately solve the water
impact, a computing cluster has been used.
The computational resources needed to carry out the final 3D simulation were
300 cores running for about 180 hours. A resolution of L/∆x= 640 is used,
as in the 2D simulations presented in the previous section, and also the Mach
number has been kept equal to Ma = 0.1.
Figure 22 shows the fluid domain used in the simulation. The size of the
domain has been chosen large enough to avoid any possible acoustic wave
reflection within the selected time interval. As in the 2D cases a multi-
resolution technique is used to get high spatial resolution only in the region
of the water impact.
The pressure fields for three instants of time t= 7.7,12.7 and 17.7ms during
the water impact are reported in figures 23 and 24. At time t= 12.7ms the
Fig. 22. 3D single-phase model, wet drop test of Mark III type insulation panel:
fluid domain used for the 3D simulation.
24

Fig. 23. 3D single-phase model, wet drop test of Mark III type insulation panel:
planar and 3D views for two different time instants: 7.7, 12.7. The color contours
are representative of the pressure fields. At time t = 12.7 ms a focusing effect is
quite visible due to the closure of the entrapped cavity which induces a sudden rise
of the pressure.
25

Fig. 24. 3D single-phase model, wet drop test of Mark III type insulation panel:
planar and 3D views at time instant 17.7 ms. The color contours are representative
of the pressure fields.
water jets entrapped in the cavity panel impact each other giving rise to a
focusing effect. At this time the pressures reach values close to those calculated
in the 2D simulations.
Figure 25 shows the time histories recorded by six different pressure probes:
SPH 3D results are compared with experimental data. Similar to the 2D single-
phase simulation (see previous section), the highest pressure levels are recorded
at probes P40 and P47 . With respect to the 2D cases the highest pressure region
has moved rightward from P40 toward probe P47 .
The impact pressure measured at probe P49 in the 3D simulation remains
similar to the 2D case in terms of intensity while it occurs earlier, producing
better agreement with the experiments.
As in the 2D case, large differences appear for probes P40 to P47 relative to
the experimental measurements, confirming that these discrepancies are more
likely related to air-cushion effects than to 3D effects.
26

Fig. 25. 3D single-phase model: pressure time histories for six different probes.
Comparison between experimental data and 3D single-phase SPH solution.
27

5.3 2D simulation with a two-phase model
The numerical investigation described in this work is concluded by performing
2D simulations with air and water. The air is treated as a compressible gas
using the classical polytropic state equation:
p=P0 ρ
ρ0air !γair
(9)
where ρ0air is the air density at rest, γair the air adiabatic index equal to 1.4,
and P0the ambient pressure.
As opposed to the single-phase model, the speed of sound c0air cannot be
adjusted for computational convenience as is for c0water, since the former is
directly linked with the Euler number used in the experiments (Eu'17):
Eu = P0
ρwater U2,
P0=c2
0air ρ0air
γair
,
w
w

c0air =Uv
u
u
u
tEu γair 

ρ0water
ρ0air 

(10)
Therefore, maintaining Eu number similitude and correctly simulating the air-
cushion effects can result in quite high CPU costs, even in a 2D framework.
We underline that with an Euler number of order 10 the air compressibility
cannot be neglected, because density variations on the order of 10% are
expected. This can represent a restriction for commercial software where air
is treated as an incompressible medium, like the liquid phase. Moreover, Eu
is of order 10 at model scale, while for full scale, it may reduce to order 1,
thus further increasing the role of air compressibility. This latter aspect can
be very important in the correct design of insulation panels.
The second point that makes the simulations with two phases more critical
and complex with respect to the single phase model concerns the treatment
of the fluid domain (reported in figure 26). The left-hand image shows the
domain at the beginning of the simulation, while the right-hand image shows
the configuration just before water impact. At this latter instant in time the
intensity of the air velocity field is shown. Because of the complexity of the
28

Fig. 26. 2D two-phase model, wet drop test of Mark III type insulation panel: fluid
domain used for the simulation.
air velocity field, we cannot start the simulation of the water impact ignoring
it. On the other hand the domain depicted in the left-hand image of figure 26
is too large and it will incur significant CPU costs even in a 2D framework.
To avoid such a complexity the following strategy has been adopted: a first
simulation from t= 0.0 to t= 1.02s is performed considering the air-water
interface as a solid boundary so only the air phase is simulated; at this stage
the air can be treated as incompressible or weakly-compressible and a fictitious
air speed of sound can be used in order to avoid too small time steps. Once
the time t= 1.02s is reached the air-water simulation is started using as input
the air field from the previous simulation. In this second simulation the real
air speed of sound is used and Eu number similitude is guaranteed. Figure 27
shows three time instants during the water impact stage.
In the left-hand column the air bubble entrapped by the panel is quite visible,
while in the right-hand column the pressure contour levels are depicted. As for
the previous simulations, the initial time for this case is set at the instant when
the left dent touches the free-surface, that occurs 1.07 ms after the beginning
of the air-water simulation. At t= 5.23ms air cushioning effects occur and
the air pressure starts rising. At t= 8.33ms the pressure in the cavity is quite
uniform with values of about 2 bar whereas a violent impact occurs at the
left-hand corner of the panel.
At time t= 11.03ms (see figure 28) an impact event is recorded close to
the right-hand corner of the panel in the region around probe P53, reaching
a pressure level of 8 bar. At the same time the pressure in the panel cavity
29

Fig. 27. Two-phase 2D model, wet drop test of Mark III type insulation panel:
views of liquid surface configuration (left) and pressure field (right) for time instants
t=−1.07ms, t= 5.23ms and t= 8.33ms.
remains uniform with levels of about 2.5 bar. At t= 14.33ms the pressure field
starts to decrease consequently to the air escaping under the right dent. This
feature may also be responsible for the inhibition of the air pocket pressure
oscillations, in a manner analogous to the air leakage effect observed in [1].
At the final instant depicted in figure 28, an almost uniform pressure of 1 bar
acts on the panel surface.
In figure 29 the SPH pressure time histories for 6 different probes are compared
with the experimental data. From this figure three relevant observations can
be drawn:
•The two-phase SPH pressures inside the cavity panel, probes P37 to P47,
show a similar time behavior of the experimental pressure, thanks to the
30

Fig. 28. Two-phase 2D model, wet drop test of Mark III type insulation panel:
views of liquid surface configuration (left) and pressure field (right) for time instants
t= 11.03ms, t= 14.33ms and t= 19.13ms.
correct simulation of the air-cushion phenomenon. However the two-phase
SPH produces an underestimation of the pressure value (2.5 bar against
4 bar). This is an indication that in the experiments the air bubble is
possibly smaller and more compressed.
•The SPH prediction of the pressure at probe P53 (close to the right-hand
corner of the panel) is fair,
•All the numerical pressure peaks over-predict the time duration of the
pressure impacts, which could be due to the 2D framework where the impact
energy is more confined relative to the 3D one.
31

Fig. 29. 2D two-phase model: pressure time histories for six different probes:
comparison between experimental data and 2D two-phase SPH solution.
32

Fig. 30. Wet drop test of Mark III type insulation panel: pressure time histories
for probes P40 and P53 . Comparison among the different SPH simulations and the
experimental data.
6 Perspective
From the analysis presented in this paper, the numerical simulations highlight
the relevance of air-cushion effects when considering water impact of the Mark
III type insulation panel with small impact angle. Thanks to the action of the
air phase the pressure loads are drastically reduced with respect to those
measured on a flat panel under the same impact conditions.
However, in our opinion, the present analysis needs to be extended to produce
a more complete panorama of the problem. For example, if different impact
angles are considered we expect that for larger angles air-cushion effects will
not occur and significant loads would be experienced near the corrugation
dents.
An analysis from a structural point of view could also be important. Indeed
in the the work by [6] it is shown that the corrugation dents can be deformed
during a violent wave impact event. Further, in this work we consider a simple
wet-drop test. However, the final application shall be a wave impact against
a wall covered with Mark III panels which represents a more interesting test
from an engineering point of view. In fact, there are already some articles
in the literature where preliminary investigations under these conditions are
presented [see e.g. 5, 4].
In the present paper we stress the relevance of 3D effects together with air-
cushioning. The main difficulties in tackling the problem including both of
these aspects is linked to the high computational costs of the SPH solver. A
further element to be considered in future is the influence of surface tension
on peak duration. Incidentally, considering the ambient conditions inside an
33

LNG tank, the natural gas is close to its boiling temperature (see e.g. [2])
which makes the extension from model to full scale more complex [see e.g. 10].
7 Conclusions
In this work a numerical investigation of wet drop tests of insulation panels
has been conducted. The numerical model adopted is the Riemann Smoothed
Particle Hydrodynamics model. Simulations have been conducted using 2D
and 3D single phase models, whereas the air-water model simulations were
only conducted in a 2D framework, because of the greater complexity and the
high CPU costs needed to take air compressibility phenomena into account.
The water impact case studied is a 4◦angle of impact, both flat and corrugated
(Mark III type) insulation panel are considered. Because of the small angle of
impact the selected test case is characterized by very high space-time gradients
making the numerical simulations quite challenging.
For the flat panel only a 2D single-phase model has been used, since in the
literature the air entrapment is said to play a minor role at this impact angle.
Comparisons with theoretical and experimental data have been provided.
The validation has regarded several parameters ranging from the free-surface
deformation up to the consistency of the peak propagation velocity. A critical
discussion addressing different key aspects of this simulation, such as the choice
of the speed of sound and influence of the pressure measurement system, has
been given.
For the Mark III type insulation panel, because of its geometrical form at 4◦
angle of impact both air cushioning and 3D effects can play relevant roles. To
better investigate such aspects different simulations have been conducted. The
comparisons reported in figure 30 summarize the results obtained at INSEAN;
the two most critical probes, P40 (between the two dents) and P53 (close to
the panel edge), are considered.
The most accurate comparisons have been obtained with the two-phase SPH
model for both the probes (even for P53 where the air cushioning is not
expected to be essential).
On the other hand all the 2D simulations present time scales too large with
respect to the experiments. The 3D simulations, even with single phase, show
the complexities of the impact for this geometry, different water jets are formed
by the interaction between the free surface and the four dents of the panel.
Summarizing, a further investigation with a 3D two-phase model would allow
34

further inspection of the phenomenon, providing more accurate results in time-
space pressure distribution. Further, a remarkable improvement of the solution
could be provided by the addition of a surface tension model in order to
simulate the multiple bubble break-up and, consequently, to achieve a more
realistic evolution of the air cushioning.
Acknowledgements
The research activity was supported by the project SinSEOn “Sloshing
SPH Environment for long-time Oscillation simulation”, (CNR-INSEAN Prot.
N.17, 8 January 2016) within the Memorandum of Understanding (MOU)
2015-2018 between HTCI-HHI and CNR-INSEAN
The research leading to these results has partially received funding from the
European Union’s Horizon 2020 research and innovation programme under
grant agreement No 724139.
The SPH simulations performed under the present research have been
obtained using the SPH-Flow solver, software developed within a collaborative
consortium composed of Ecole Centrale de Nantes, NextFlow Software
company and CNR-INSEAN.
The research activity has been developed within the Project Area “Applied
Mathematics” of the Department of Engineering, ICT and Technology for
Energy and Transport (DIITET) of the Italian National Research Council
(CNR).
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