close

Вход

Забыли?

вход по аккаунту

?

1.4038242

код для вставкиСкачать
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
ed
ite
d
Investigation of Wave Characteristics in
Oscillatory Motion of Partially Filled Rectangular
M. Ozbulut∗
ot
Co
py
Tanks
tN
Piri Reis University
sc
rip
Engineering Faculty
Istanbul, Turkey
Ma
nu
Email: [email protected]
N. Tofighi
University of Victoria
Ac
ce
pt
ed
Department of Mechanical Engineering
Victoria, Canada
Email: [email protected]
O. Goren
Istanbul Tecnical University
Faculty of Naval Architecture and Ocean Engineering
Istanbul, Turkey
Email: [email protected]
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
M. Yildiz
Integrated Manufacturing Technologies Research and Application Center,
Sabanci University, Tuzla, 34956, Istanbul, Turkey.
Composite Technologies Center of Excellence,
ed
ite
d
Sabanci University-Kordsa,
Istanbul Technology Development Zone,
py
Sanayi Mah. Teknopark Blvd. No: 1/1B,Pendik, 34906 Istanbul, Turkey.
Co
Faculty of Engineering and Natural Sciences,
Sabanci University, Tuzla, 34956 Istanbul, Turkey.
sc
rip
tN
ot
Email: [email protected]
ABSTRACT
Ma
nu
Simulations of oscillatory motion in partially filled rectangular tanks with different tank
geometries, fullness ratios and motion frequencies are presented. Smoothed Particle Hydrodynamics (SPH) method is used to discretize the governing equations together with new velocity
ed
variance based free surface and artificial particle displacement algorithms to enhance the robustness and the accuracy of the numerical scheme. 2-D oscillatory motion is investigated for
pt
three different scenarios where the first one scrutinizes the kinematic characteristics in reso-
ce
nance conditions, the second one covers a wave response analysis in a wide range of enforced
Ac
motion frequencies, and the last one examines the dynamic properties of the fluid motion in detail. The simulations are carried on for at least 50 periods in the wave response analysis. It is
shown that numerical results of the proposed SPH scheme are in match with experimental and
numerical findings of the literature.
∗ Corresponding
author.
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
by ASME
1 2017
INTRODUCTION
The inner free-surface flows in partially filled tanks have significant effects on the general stability
and motion of the ships having large liquid cargo tanks such as LNG ships [1, 2]. The solution of
coupled sloshing-seakeeping problems posses a great importance on the preliminary and concept design
processes of ships [3] and new researches on this issue are still needed [4]. Depending on the enforced
ed
ite
d
frequency of the motion and fullness ratio of the tank, the fluid motion inside the tank may become
extremely complex and hence requires long term analysis to understand its characteristics. If the tank is
forced to surge around the lowest natural frequency, the resonant fluid motion inside the tank becomes
py
important. Therefore, enforced surge or pitch motion of liquids in a tank has attracted the attention of
Co
engineers and scientists in the field of hydrodynamics. To this end, one may find several theoretical,
experimental and numerical studies in the literature that have tried to explore the complicated physics
ot
behind the oscillatory motion inside a partially filled tank [5–10].
tN
Faltinsen and Timokha [11] modeled the surge and pitch excited resonant sloshing motion by adaptive multi-modal theory, assuming that the flow is irrotational and has no overturning waves. They stated
sc
rip
that direct numerical methods (finite difference, finite element and boundary element methods) experience some difficulties in volume and energy conservation as well as in accurate description of fluid
Ma
nu
impact on the tank walls during long term simulations. Their reported results are in a good agreement
with the experimental observations provided that the ratio of water depth to the tank beam is larger than
0.24. To further extend the scope of the theory, Faltinsen and Timokha [12] proposed an asymptotic
model approximation to examine nonlinear resonant waves for depth/breadth ratios between 0.1 and
ed
0.24. In addition to theoretical models, numerous studies involving a variety of numerical methods such
pt
as marker and cell (MAC), volume of fluid (VOF), level set (LS) and hybrid VOF-LS can be found in
ce
literature. Chen et al. [13] gave a list of numerical studies dedicated to excited sloshing motion of liquids
Ac
and discussed advantages and disadvantages of these schemes. They stated that the inability to capture
the topology of the free-surface region and low numerical accuracy in obtaining impact pressures and
forces are the main difficulties of these numerical techniques.
Beside mesh-based numerical methods, there are also relevant studies on the modeling of oscillatory
free-surface flows with mesh-free methods such as Smoothed Particles Hydrodynamics (SPH) [14, 15]
and Moving Particle Semi-implicit (MPS) methods [16]. Khayyer and Gotoh performed multi-phase
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
2017 by ASME
modeling
of sloshing flows using the MPS method [17]. Different surge and pitch enforced sloshing
problems were studied by SPH method in literature [18–22]. However, these studies are generally
limited to a few critical motion frequencies which do not fully represent the wave response of the fluid
motion. The construction of a wave response curve through performing simulations at various excitation
frequencies where higher order harmonic modes of fluid motion become effective is a very important
ed
ite
d
benchmark for testing the capabilities of numerical methods in terms of capturing long-term sloshing
phenomena under a wide variety of physical conditions.
py
The long term sloshing problem can exhibit different physical behaviors. For example, the fullness
Co
ratio of the tank can lead to shallow and deep water configurations which might have distinct sloshing
characteristic under a given excitation frequency. Moreover, in engineering applications, the higher
ot
order modes of sloshing in long-term excitations can be significant since these modes may overlap with
tN
the lowest resonance frequency [11] which induces a damping effect on the evolution of the fluid flow
sc
rip
and causes a remarkable change in the physical behavior of the fluid motion. Towards this end, in this
study, three different test cases are chosen carefully to assess the performance of the proposed numerical
scheme. The first case involves the comparison of free surface elevation time series on the side wall
Ma
nu
of the rectangular tank which is excited close to resonance frequency. The obtained results for two
different fullness ratios of the tank corresponding to shallow and relatively deep water configurations
are compared with the experimental and numerical solutions presented in [23]. Having showed that the
ed
presented weakly compressible SPH (WCSPH) scheme can handle the steady-state wave regimes for
both shallow and deep water sloshing in a rectangular tank around corresponding resonance frequencies,
pt
for completeness, it would be prudent to study in detail the higher order modes of sloshing phenomenon.
ce
To this end, the rectangular tank with a constant water depth is enforced to move with a series of
Ac
excitation frequencies and a quantitative comparison on the wave response curves with the results of [11]
is given. This part is generally missing in particle based numerical studies in the literature [18,19,21,22]
and this work would like to enclose the validation/verification examination of the presented numerical
modeling by such a challenging test. To examine the capability of the presented WCSPH scheme for
modeling dynamic characteristics of sloshing flows, as a final test case, pressure loads on some critical
points of the tank are calculated and compared with the results of the study in [13].
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017
ASME
Thisbystudy
presents a robust and accurate numerical WCSPH scheme capable of capturing sloshing
phenomena under a wide variety of physical conditions and contributes to the state of the art in the SPH
field in terms of: (i) suggesting a new velocity variance based free surface (VFS) and artificial particle
displacement (APD) algorithms to produce a robust and accurate scheme for numerical simulation of
violent free-surface flows, (ii) investigating a variety of oscillatory motion conditions and parameters,
ed
ite
d
and (iii) performing wave response analysis. The rest of this paper is structured as follows. In section
2, governing equations, SPH discretization schemes and the new free surface and APD algorithms are
succinctly explained. In section 3, the simulation results are presented and quantitatively compared with
py
the experimental and numerical findings in literature in terms of flow kinematics and dynamics. Finally,
2.1
GOVERNING EQUATIONS AND NUMERICAL MODELING
sc
rip
2
tN
ot
Co
the concluding remarks are drawn in section 4.
Field Equations with WCSPH Discretization
Ma
nu
The sway-sloshing problem, characterized as violent free surface flow, is solved using Euler’s equation of motion and continuity coupled with Lagrangian particle advection. Neglecting viscous effects
1
D~u
= − ∇P +~g
Dt
ρ
(1)
Dρ
= −ρ∇ ·~u
Dt
(2)
Ac
ce
pt
ed
and allowing for rotation of fluid elements, the governing equations can be written as
~u =
D~r
Dt
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
(3)
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
2017D/Dt,∇,
by ASME~u,~r, P and ρ are the material time derivative, nabla operator, velocity and position vecwhere
ed
ite
d
tors, pressure and the density of particles, respectively, while~g denotes gravitational acceleration vector.
The governing equations are discretized using WCSPH approach. WCSPH method uses an artificial
equation of state which couples pressure and density variations through an artificial velocity commonly
known as the speed of sound which is lower than the real one under the weakly-compressibility limit.
py
There are various forms of artificial equation of state used within the scope of WCSPH approach to be
Co
able to calculate the pressure for computing pressure gradient term in the equation of motion. The one
ρ
ρ0
γ
tN
−1 ,
(4)
sc
rip
ρ0 c20
p=
γ
ot
proposed in [24],
is used in this study where c0 is the reference speed of sound, γ is the specific heat-ratio of water and
Ma
nu
is taken equal to 7 while ρ0 is the reference density which is equal to 1000 (kg/m3 ) for fresh water.
The value of reference speed of sound is determined by Mach (M) number (a dimensionless quantity
representing the ratio of velocity of fluid to speed of sound). It is required that M should be in the
ed
vicinity of 0.1 in order to keep density variation within %1 of reference density [24]. As a Lagrangian
method, SPH represents the flow field by a finite number of moving particles. These particles carry
pt
the characteristic properties of the flow such as mass, position, velocity, momentum, and energy. SPH
ce
is basically an interpolation process where the fluid domain is modeled through the interactions of
Ac
neighboring particles using an analytical function widely referred to as the kernel/weighting function
W (rij , h). Here, h is the smoothing length and rij represents the magnitude of the distance vector given as
~rij =~ri −~rj for particle of interest and its neighboring particle, denoted using boldface subscripts i and
j, respectively while~ri and~rj are the position vectors for the particles. Although it is computationally
more expensive than some of the other kernel functions, the piecewise quintic kernel function is chosen
due to its better accuracy and stability characteristics [25]. The compactly supported, two dimensional
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
2017 by ASME
piecewise
quintic kernel function is given as
(5)
ed
ite
d



(3 − R)5 − 6 (2 − R)5 + 15 (1 − R)5 , 0 ≤ R < 1





 (3 − R)5 − 6 (2 − R)5 ,
1≤R<2
W (R, h) = αd


(3 − R)5 ,
2≤R<3





 0,
R≥3
where R = rij /h and αd is a coefficient dependent on the dimension of the problem [26]. In two di-
py
mensions, αd is equal to 7/(478πh2 ). The SPH method interpolates any arbitrary continuous function,
Z
(6)
tN
Ω
A ~rj W (rij , h)d 3~rij ,
ot
Ai ∼
= hA (~ri )i ≡
Co
A(~ri ), or concisely denoted as Ai in the following manner:
sc
rip
where the angled bracket hi denotes the kernel approximation, d 3~rij is the infinitesimally small volume
element inside the domain and Ω represents the total bounded volume of the domain. The governing
equations are discretized based on the above given SPH approximation where the integral operation
Ma
nu
over the volume of the bounded domain is replaced by the summation operation over all neighboring
particles j of the particle of interest i. The differential volume element d 3~rij is also replaced by mj /ρj .
As a result, the Euler’s equation of motion and the mass conservation may be discretized by the SPH
N
pi pj
d~ui
= − ∑ ( 2 + 2 + Πij )∇iWij ,
dt
ρj
j=1 ρi
(7)
N m
dρi
j
~ui −~uj · ∇iWij .
= ρi ∑
dt
j=1 ρj
(8)
Ac
ce
pt
ed
method to provide the following relations [24]:
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
2017mbydenotes
ASME the mass of particle j, ∇ is the gradient operator where the particle identifier i indicates
Here,
j
i
that the spatial derivative is evaluated at particle position i. ∏ij is the artificial viscosity term defined as
i


0,
j
(9)
~uij ·~rij ≥ 0
ed
ite
d
Πij =


 −αµij ci +cj , ~uij ·~rij < 0
ρ +ρ
(10)
ot
Co
~ui −~uj · ~ri −~rj
µij = h .
~ri −~rj 2 + θh2
py
where
(γ−1)/2
tN
where, the local speed of sound for a particle, included in Eq. 9 is computed according to ci = co (ρi /ρo )
Artificial viscosity term is inserted into the linear momentum balance equation for inviscid flow to im-
sc
rip
prove the stability of the numerical solution via the addition of a rather small amount of diffusion to
the fluid. Artificial viscosity term was initially introduced by [27] for finite difference schemes and has
Ma
nu
been extensively utilized in many SPH studies as well [2]. The level of the artificial viscosity added
to the fluid should be optimized to minimize its impact on the solution while preserving its stabilizing
effect. It should be noted as the spatial resolution increases (i.e. smoothing length parameter (h) goes to
zero), the effect of artificial viscosity goes to zero hence recovering the Euler’s equation of motion [28].
ed
Here, θ is a constant which is added to the denominator of the Eq. 10 to prevent any singularity and
pt
α parameter is the coefficient which determines the amount of the artificial diffusion in the numerical
ce
solution. The numerical value of α in this study is determined through referring to numerical solutions
Ac
of benchmark problems. Assigning α = 0.006 is found to provide satisfactory results and used in all of
the simulations presented in this work.
Time marching of the numerical solution is achieved via a predictor-corrector time integration scheme
while free-slip wall boundary condition is applied to all bounding walls through ghost particle technique. The dynamic free-surface condition on the free surface is satisfied by setting the pressure of free
surface particles to zero or, equivalently, setting the densities of the free surface particles to the reference
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
.
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
2017 byFurther
ASME details of the utilized numerical scheme and boundary implementations can be found
density.
in our previous study [29].
2.2
Corrective Numerical Treatments
Being a Lagrangian numerical method, SPH needs suitable corrective numerical treatments to re-
ed
ite
d
duce particle clustering and noisy pressure field which leads to random and rapid pressure oscillations
thereby causing reduced numerical accuracy and stability or even break down of the numerical simulations. Here, in addition to well-known density correction algorithm, velocity variance based free surface
(VFS) and artificial particle displacement (APD) algorithms are incorporated into the numerical scheme
py
of WCSPH in a hybrid manner (namely, VFS+APD) to circumvent particle clustering induced numeri-
Co
cal problems.
ot
In WCSPH approach one of the main difficulties arises from the occurrence of highly noisy pressure
tN
field due to the acoustic components derived from the use of an artificial speed of sound. In the recent
sc
rip
years, several studies were addressed to smooth out the density field at each time step using a diffusive
term in the continuity equation [18, 30]. A good comprehensive review on limits and capabilities of
diffusive terms in literature can be found in [31,32]. In our previous work [29], the effect of APD which
Ma
nu
provides a homogeneous particle distribution on the problem domain and hence helps to prevent acoustic
components derived from the use of an artificial speed of sound which leads to a noisy and in accurate
pressure field was presented. The proposed numerical scheme is further tested for violent sway-sloshing
ed
flows together with a variety of different fullness ratio of tank and under the effects of variable enforced
pt
motion frequencies. Due to the extreme computational costs of density diffusion algorithm [33] and on
ce
the light of the benchmark simulation results, it is decided to utilize the basic Shepard density smoothing
algorithm together with the APD treatment which brings a satisfactory accuracy in terms of kinematic
Ac
and dynamic flow characteristics. The density correction treatment is applied through
b
ρi = ρi − σ
∑Nj=1 ρi − ρj Wij
∑Nj=1 Wij
,
(11)
where b
ρ is the corrected density, N is the number of neighbor particles for particle i and σ is a constant
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
2017is
byset
ASME
which
to unity in this study thereby leading to the well-known “Shephard” interpolation [34].
N
δ~ri =
,
bi =~ui − εδ~ui
~u
py
∑Nj=1 Wij
~rij
∑ r3 ro2uc f f ∆t,
(12)
(13)
ot
j=1 ij
Co
δ~ui =
∑Nj=1 ~ui −~uj Wij
ed
ite
d
The hybrid VFS+APD treatment is formulated as
tN
where ~ubi is called the corrected velocity used to replace the velocity of the free-surface particle calcu-
sc
rip
lated based on the solution of momentum balance equation at each time step, ε is a constant parameter
which is sufficient to assign a value between 0.05-0.1 times the initial particle spacing (dx) and a value of
0.003 is chosen for the cases simulated in the present work. ro is an average distance calculated for each
Ma
nu
particle as ro = ∑j rij /N and uc f f is the velocity based APD coefficient. In the hybrid approach, VFS
is applied only to free-surface particles while APD is used to regularize the arrangement of particles
with fully populated influence domain, which are identified via the kernel truncation. Namely, particles
ed
with fully populated influence domain are considered to be internal particles while those whose neighbor numbers are below certain threshold compared to the average neighbor number of all particles with
pt
fully populated compact support are regarded as free-surface particles. We have experienced that parti-
ce
cles with neighbor numbers fewer than (60-70)% of the average neighbor particle number demarcates
Ac
the free-surface accurately. If this ratio is taken lower than 60% we observed that the free surface was
not demarcated totally and discontinuities occurred along the free surface profile. On the other hand,
if this ratio is assigned more than 70%, the free-surface thickness increases and it leads to unphysical
free surface tension forces. Figure 1(a) shows a snapshot of test case 1-b (the simulation parameters are
given in the following section Tab. 1) and illustrates the regions where the VFS and APD treatments are
effective.
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
The APD formulation employed in this study is different from that in literature [29] such that it uses
the magnitude of the velocity variance computed for each particle as a velocity based APD coefficient
(herein, referred to as the local velocity coefficient, uc f f = δ~ui ) rather than the magnitude of the maximum velocity in the problem domain (referred to as the global velocity coefficient, uc f f = βumax ) and
ed
ite
d
hence eliminates the need for the problem dependent calibration coefficient β in the APD formulation.
The existing APD formulation in literature has been reported to be quite successful in yielding homogeneous particle distribution for problems in confined domain. Nevertheless, we observed that once used
py
in violent free-surface flows such as dam-break and sloshing problems, it leads to the increase in the
Co
total volume of the water reserve since in these problems, the variation in the velocity field of the flow
is extremely high, especially upon and after the impact of the water reserve on the tank walls. Hence,
ot
utilizing global velocity coefficient in APD requires tailoring the amount of allowable artificial particle
tN
displacement carefully to avoid the gradual increase of the volume due to the movement of particles
toward free-surface. To ensure the conservation of total volume, a long-term simulation of dam-break
sc
rip
problem is performed utilizing exactly the same problem parameters reported our previous work [29].
Figure 1(b) indicates that the employed corrective algorithm (VFS and the new APD algorithm) con-
NUMERICAL RESULTS
ed
3
Ma
nu
serves the total volume during the simulation.
In this work, sway-sloshing problem is simulated for three different test cases where the first and
pt
the second test cases address the kinematic characteristics of the sway-sloshing motion while the third
ce
case investigates the dynamic properties of the fluid motion. The harmonic sway motion of the tank is
Ac
induced by a sinusoidal function such that x(t) = A sin(ωt) where x is the horizontal position of the tank,
A is the amplitude and ω is the angular frequency of the enforced motion. The representative geometry
of the problem is given in Fig. 2, and dimensions, amplitudes and angular frequencies of all test cases
are tabulated in Tab. 1 where frequency values in test case 2 indicates the minimum and maximum of
the investigated frequency range. The theoretical nth natural frequency at a given water depth (D) and
tank width (L) is given as [13, 23]
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
r
(14)
ed
ite
d
ωn =
nπ ngπ
tanh
D .
L
L
For all test cases, the speed of sound, reference density, gravitational acceleration and the smoothing
length parameter are taken as c = 40 (m/s), ρ = 1000 (kg/m3 ), g = 9.81 (m/s2 ) and h = 1.33∆x (m),
respectively. For the sake of comparison, the initial conditions are taken the same as the reference
Co
py
studies [13, 23] where all particles are at rest and have only hydrostatic pressure at t = 0 (s).
Before starting systematical investigations on the kinematic and dynamic characteristics of oscil-
ot
latory flows in rectangular tanks, a convergence test is performed in order to determine correct par-
tN
ticle resolution bearing in mind the trade off between the numerical accuracy and the computational
sc
rip
time. Test case 1-a is chosen for the convergence study, and three different particle resolutions, namely,
H/dx=15, H/dx=30 and H/dx=60 are simulated. The free-surface elevation, ξ(t), and corresponding
frequency domain analysis for each particle resolution is compared with the experimental data of [23]
Ma
nu
which is given in Fig. 3. It can be seen from Fig. 3 that the free-surface elevations at the left wall are
in alignment with experimental measurements, especially when the resolution is higher than H/dx=15.
The standard deviations of each particle resolutions with respect to experimental measurements are cal-
ed
culated as 0.072, 0.061 and 0.054 for the spatial resolutions H/dx = 15, 30, 60, respectively. On the other
hand, the normalized computational time costs for our serial in-house code on an Intel(R) Xeon(R) CPU
pt
E5-2690 v3 @2.60GHz processor are approximately, 0.13, 1.0 and 13.3 accordingly. Considering the
ce
computational cost and the sufficient accuracy of the results, we decided to use the particle resolution
Ac
of H/dx=30, which approximatively corresponds to the initial particle spacing values ∆x = ∆y = 0.01
(m), and the time step value is chosen as ∆t = 0.00015 (s) for the free-surface elevation simulations
performed in test cases 1 and 2. Unlike the examination of free-surface elevations on the tank walls,
the calculation of impact pressures on the certain points of tank requires more higher resolution on the
problem domain. To capture the pressure variations on the tank walls accurately in test case 3, both
spatial and temporal resolutions of the simulation are doubled with respect to cases 1 and 2.
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
by ASME
3.12017Examination
of the Free-Surface Elevations on the Side Wall
Investigating the variation of the free-surface profiles during the motion sheds light on the physics
behind the oscillatory motion of the fluid in the tank. The first case models sway sloshing with two
different tank fullness ratio and is dedicated to the study of kinematic characteristics of the oscillatory
ed
ite
d
motion. The Case 1-a is regarded as a shallow water condition where the water depth to tank width ratio
D/L=0.173 is lower than the value of 0.24 [12]. For both Case 1-a and 1-b, the angular frequency of the
enforced harmonic sway motion is chosen close to the first theoretical natural frequency of the sloshing
motion (ω1 in Tab. 1) following the study of [23]. As the tank is enforced to move in an harmonic
py
manner with a constant frequency, the fluid content is also expected to respond with approximately the
Co
same frequency for the free-surface elevation on the side walls. The results of the present simulations
are compared with the findings of [23] obtained using conventional WCSPH method which employs
ot
periodical density re-initialization procedure to increase the accuracy of the pressure field, particle re-
tN
finement algorithm at the tank corners, XSPH numerical treatment and Runge-Kutta time integration
sc
rip
scheme for the time marching of the flow.
Fig. 4(a) and Fig. 5(a) present time histories of the water level elevation comparison for two dif-
Ma
nu
ferent cases, namely, Case 1-a and Case 1-b at the given water depth, amplitude and frequency of the
enforced motion. It may be noticed that the periods of the fluid motion are in good agreement with both
experimental and numerical results of reference study. These results of both test cases are also compared
ed
with literature in frequency domain, which are given in Fig. 4(b) and Fig. 5(b). Except for a few instances, the amplitudes of the harmonic motion match with the experimental data. These discrepancies
pt
may be attributed to inherent differences in the determination of free-surface location (on vertical walls
ce
via particle positions) compared to the data acquired by the probe used in experiments. An additional
Ac
level of complexity arises when the water splashes after its impact on the walls. In the present work,
the numerical data sampling-rate frequency is 11.11 (Hz) and the free-surface is obtained as the average
elevation of the free-surface fluid particles within x = −0.805 (m) to x = −0.825 (m), an interval chosen
to encompass the physical region covered by the wave probe (x = −0.815 (m)) in the reference study. It
should be noted that the specified horizontal positions in the determination of the free surface are given
for the initial position of the tank (t = 0(s)).
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
by ASME
3.22017Examination
of Wave Responses Due to Enforced Motion Frequencies
The response of higher order modes of sloshing phenomenon should also be investigated where
these modes can be significantly important when they overlap with the lowest resonance frequency [11].
Higher order modes and especially the second harmonics of the motion can be effective in long-term
excitations and may change the physical behavior of the fluid motion. To test the capabilities of the
ed
ite
d
proposed numerical scheme further, the rectangular tank with constant water depth is enforced to move
with a series of excitation frequencies.
Excitation frequencies are 6.26, 5.63, 5.21, 4.81, 4.54, 4.32, 4.19, 4.08, 3.98, 3.78, 3.67, 3.62, 3.50
py
(rad/s). The simulation results are presented and compared with literature [11] in the form of wave
response curve as shown in Fig. 6 where T and T1 denote the excitation period and the first theoretical
Co
natural period of the motion, respectively. Our results are in better agreement with the experimental
ot
measurements of reference work in the range between 0.8 < T /T1 < 1.1 than the simulation results pre-
tN
sented within the same study. In the frequency interval of 1.1 < T /T1 < 1.25, the secondary resonance
mode becomes effective leading to a notable drop in wave amplitude which is captured with a slight
sc
rip
delay in our simulation. As the excitation frequency increases, the higher order modes become prevalent thereby further decreasing the wave elevation which is computed within the same order of accuracy
3.3
Ma
nu
with the numerical results of the reference study in comparison to the experimental data [11].
Examination of the Impact Pressures on the Side and Top Surfaces of the Tank
ed
Having obtained simulation results in mesh with those reported in [23] and [11] for kinematic characteristics of the sway-sloshing motion, the ability of the proposed scheme is tested if it can capture
pt
the more sensitive dynamic of the flow. The results are compared with level set simulations of [13].
ce
The impact pressure time series of the water reserve are calculated at four different points of the tank,
Ac
designated as P1 through P4 in Fig. 7.
As the impact pressure on the walls may change in small fractions of a second due to passing waves
and splashes, simulation results are saved every 0.0075 (s) which corresponds to a data capture frequency of 133.33 (Hz). Pressure values at the given sensor points of the tank are interpolated by using
the same quintic kernel function utilized during the simulations. However, the smoothing length is set
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
by ASME
to 2017
be four
times bigger than the value used in simulations to increase the contributing particle number,
hence accounting for the pressure sensor’s diameter and filter out local oscillations in pressure. Ghost
particles generated during the implementation of the wall boundary conditions are also included in the
interpolation processes at all given pressure sensor points on the tank surface.
ed
ite
d
The comparison of the full time history of pressures with the experimental and numerical results of
the reference data is shown in Figs. 8 and 10 for pressure points P1 and P4 , respectively. Table 2 provides
the consecutive maximum pressure values due to the harmonic motion of the fluid for pressure points
py
P2 and P3 .
Co
The first pressure point P1 remains below free-surface level during the course of flow evolution,
yielding a continuous and more reliable pressure data acquisition point from both numerical and exper-
ot
imental points of view. On the other hand, measuring inherently noisy pressure values in experimental
tN
studies generally requires a tedious procedure of post-processes like signal processing or data filtering
in order to obtain reasonable analyses. This situation is also reported in the referenced study [13] such
sc
rip
that the authors processed the given face pressure values because the experimental set-up does not give
any information whether there is a signal processing nor data filtering over the input data. Keeping this
Ma
nu
uncertainty on the experimental results in mind, continuous pressure time history of the present simulation provided in Fig. 8, is able to capture the wiggly nature of the pressure data collected by the P1
sensor. The obtained results are also in match with period and magnitude of the pressure peaks of the
pt
engineering applications.
ed
experimental data which constitute critical loads for structural optimization and the point of interest for
ce
The pressure points P2 , P3 and P4 do not sense the fluid pressure continuously as P2 lies just above
Ac
the initial free-surface level and P3 and P4 are close to the top right corner of the tank as shown in Fig.
7. Unlike the continuously varying pressure profile obtained for P1 location, P2 , P3 and P4 locations
will register pressure data as periodical pressure beats which last over very small time intervals. The
comparative results for the data points P2 and P3 are given as consecutive maximum pressure values in
Tab. 2.
It is observed from Tab. 2 that the standard deviation of the experimental measurements of the ref-
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
2017 by
ASME
erence
data
is higher than the numerical results, which is expected due to the difficulty in collecting
precise data from the pressure sensors. If mean value and standard deviation of these consecutive impact pressures are taken into account together, one may practically infer that the SPH impact pressures
are compatible with those of the experimental measurements. The standard deviations of both numerical
results with respect to the experimental measurements are also performed and found as follows: For the
ed
ite
d
hit point P2 , SPH results have a standard deviation of 1.55 while [13] has 2.71. For the hit point P3 ,
SPH results have a standard deviation of 4.89 while referenced study has 6.54. To show the consecutive
pressure hits at sensor locations P2 and P3 clearly, Fig. 9 is also provided where the horizontal lines
py
denotes the mean values of the pressure beats. The consecutive pressure beats of the present SPH study
Co
is between the numerical and experimental results of [13].
ot
To illustrate that the current SPH algorithm is also able to capture the pressure beats in extreme
tN
sensor locations on the tank, the pressure time history at P4 sensor location is displayed together with
the corresponding experimental and numerical results of the associated literature in Fig. 10. As there
sc
rip
are sudden and rapid impacts which occur less than 0.025s in real time, it requires very sensitive and
accurate data processing on sensor points P3 and P4 and lead to high memory necessities and rise in
Ma
nu
computational time costs. Despite of utilizing a high data-sampling value of 133.33Hz, it should be
noted that there may have some missing instants during the small impact time interval while reading
pressure time series. Nevertheless, if the periodicity and amplitude of pressure beats of water reserve
are taken into account, the obtained results of the present study are compatible with those reported in
pt
ed
numerical and experimental results of reference study.
ce
A closer survey on the evolution of a single pressure beat on the left wall is also performed in
Ac
order to observe the complex dynamics of the fluid motion. The instantaneous pressure beats on the
wall may have a prominent importance in the strength of the solid tank because the superposition of
these pressure beats determines the magnitude of the critical loads on the structure. Towards this end,
a period of single pressure beat which is calculated at P1 is divided into four separate instants, namely,
t1 = 10.54(s),t2 = 10.64(s),t3 = 10.86(s) and t4 = 11.17(s) which were shown in Fig. 8(b). To explain
the fluid flow between these time fractions in a more clear way, the free-surface profiles and the pressure
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
2017ofbythe
ASME
fields
whole domain at the given instants are also shown in Fig. 11. It can be said from the Fig. 8
that the proposed SPH method is able to capture the primary and secondary pressure peaks on a given
main pressure peak as observed in the experimental data. The main peak in the pressure occurs when
a period of the harmonic motion is about to finish (t1 ) and the velocity of the tank is converging to
zero while at the same time the fluid particles are attacking to the vertical wall with a very high speed
ed
ite
d
which yields a violent impact on the wall. As soon as the tank changes its direction a sudden drop in
pressure occurs (t2 ) which is then followed by a secondary pressure peak (t3 ) when the velocity of the
rigid tank reaches its maximum velocity in the opposite direction. Eventually, the water reserve moves
py
to the right wall with an increasing velocity, the pressure starts to reduce on the left wall and finally gets
CONCLUSION
ot
4
Co
its minimum value at t4 when the water level at the left wall is minimum.
tN
In this paper, a modeling study for two-dimensional oscillatory motion of partially filled rectangular tanks is presented where the test cases are chosen to target different physics in sloshing; namely,
sc
rip
(i) the resonance conditions of two different fullness ratios, (ii) wave response analysis based on the
series of excitation motion frequency, and (iii) dynamic analysis of fluid behavior by probing pressure
Ma
nu
field at extreme locations of tank. Euler’s equation of motion and continuity equation are discretized
using WCSPH approach and integrated in time via a predictor-corrector scheme. Density correction and
hybrid VFS+APD numerical treatments are incorporated into the conventional WCSPH numerical solu-
ed
tion procedures to improve the accuracy and robustness of the numerical solution. Long-term dam-break
simulations are carried out using a new APD algorithm with the local velocity coefficient computed for
pt
each particle and it is shown that the total volume of the fluid domain is exactly conserved even in the
Ac
ce
long-term simulation of dam-break problem.
Having showed the effectiveness of the local velocity coefficient in the APD algorithm in terms
of giving uniform particle distribution while conserving total volume, the kinematic and dynamic flow
characteristics of sway-sloshing motion are investigated for three different cases. The first case has
two benchmark problems which are modeled for the same tank geometry with different water depths
and motion frequencies. The free-surface elevations on the left wall of the tank are compared with
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
2017 by ASME
experimental
and numerical solutions available in literature, showing good agreement in both time and
frequency domains. The second case tests the capability of the presented SPH scheme for a wide range
of enforced motion frequencies which yields the response of fluid motion in different harmonic modes.
In light of results of the wave response analysis, it is shown that the proposed SPH algorithm captures the
general characteristics of the oscillatory motion and shows a good agreement with the experimental and
ed
ite
d
computational results found in the literature. The last case analyzes and compares the impact pressures
on four points of the tank available in literature data. Comparison of fully submerged sensor point’s
pressure values shows satisfactory match with experimental data. As for the points that have occasional
py
contact with fluid, the mean and standard deviations of pressure fields of this study are compatible with
Co
experimental data.
ot
Considering the results obtained fr all three test cases, it is possible to infer that the utilized WCSPH
tN
scheme and related corrective numerical treatments are able to capture the physics behind the evolution
of oscillatory motion in partially filled rectangular tanks. On the enlightenment of the simulation results,
sc
rip
it is also possible to input obtained hydrodynamics loads acting on walls for the structural design and
optimization studies in future, which can be useful for practical engineering applications like ships
5
ACKNOWLEDGMENT
Ma
nu
having large liquid cargo tanks and/or liquid containers on board.
The authors gratefully acknowledge financial support provided by the Scientific and Technological
Ac
ce
pt
ed
Research Council of Turkey (TUBITAK) for project number 112M721.
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
2017 by ASME
References
[1] Lee, S., Kim, M., Lee, D., Kim, J., and Kim, Y., 2007. “The Effect of LNG Tank Sloshing on the
Global Motions of LNG Carriers”. Ocean Engineering, 34, pp. 10–20.
[2] Delorme, L., Iglesias, S., and Perez, S., 2005. “Sloshing loads simulation in lng tankers with sph”.
In Proceedings of International Conference on Computational Methods in Marine Engineering,
ed
ite
d
MARINE, pp. 1–10.
[3] Delorme, L., Colagrossi, A., Iglesias, S., Rodriguez, R., and Botia-Vera, E., 2009. “A Set of Canonical Problems in Sloshing, Part I: Pressure Field in Forced Roll Comparison Between Experimental
py
Results and SPH”. Ocean Engineering, 36, pp. 168–178.
[4] Servan-Camas, B., Cercos-Pita, J. L., Garcia-Espinoza, J., and Souto-Iglesias, A., 2016. “Time
Co
Domain Simulation of Coupled Sloshing Seakeeping Problems by SPH FEM Coupling”. Ocean
ot
Engineering, 123(7), pp. 383–396.
tN
[5] Faltinsen, O. M., and Timokha, A. N., 2009. Sloshing. Cambridge University Press.
[6] Okamoto, T., and Kawahara, M., 1990. “Two Dimensional Sloshing Analysis by Lagrangian Finite
sc
rip
Element Method”. International Journal of Numerical Methods in Fluids, 11, pp. 453–477.
[7] Wang, C., Teng, J., and Huang, G., 2011. “Numerical Simulation of Sloshing Motion Inside a
Two Dimensional Rectangular Tank by Level Set Method”. International Journal of Numerical
Ma
nu
Methods for Heat & Fluid Flow, 21(1), pp. 5–31.
[8] Guorong, Y., Subhask, R., and Kamran, S., 2009. “Experimental Study of Liquid Slosh Dynamics
in a Partially-Filled Tank”. ASME J. Fluids Eng., 131(7), pp. Article ID 071303, 14 pages.
ed
[9] Shin, H. R., 2005. “Unstructured Grid Based Reynolds-Averaged Navier-Stokes Method for Liquid
pt
Tank Sloshing”. ASME J. Fluids Eng., 127, pp. 572–582.
ce
[10] Gotoh, H., Khayyer, A., Ikari, H., Arikawa, T., and Shimosako, K., 2014. “On Enhancement of
Ac
Incompressible SPH Method for Simulation of Violent Sloshing Flows”. Applied Ocean Research,
46, pp. 104–115.
[11] Faltinsen, O., and Timokha, A., 2001. “An Adaptive Multimodal Approach to Nonlinear Sloshing
in a Rectangular Tank”. Journal of Fluid Mechanics, 432, pp. 167–200.
[12] Faltinsen, O., and Timokha, A., 2002. “Asymptotic Modal Approximation of Nonlinear Resonant Sloshing in a Rectangular Tank with Small Fluid Depth”. Journal of Fluid Mechanics, 470,
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017
ASME
pp.by319–357.
[13] Chen, Y., Djidjeli, K., and Price, W., 2009. “Numerical Simulation of Liquid Sloshing Phenomena
in Partially Filled Containers”. Computers & Fluids, 38, pp. 830–842.
[14] Monaghan, J., and Gingold, R., 1977. “Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars”. Monthly Notices of The Royal Astronomical Society, 181, pp. 375–
ed
ite
d
389.
[15] Lucy, L., 1977. “Numerical Approach to Testing the Fission Hypothesis”. Astronomical Journal,
82, pp. 1013–1024.
py
[16] Koshizuka, S., and Oka, Y., 1996. “Moving Particle Semi-implicit Method for Fragmentation of
Incompressible Fluid, nuclear science and engineering”. Nuclear Science and Engineering, 123,
Co
pp. 421–434.
ot
[17] Khayyer, A., and Gotoh, H., 2013. “Enhancement of Performance and Stability of MPS Mesh-
Computational Physics, 242, pp. 211–233.
tN
free Particle Method for Multiphase Flows Characterized by High Density Ratios”. Journal of
sc
rip
[18] Antuono, M., Colagrossi, A., Marrone, S., and Molteni, D., 2010. “Free Surface Flows Solved by
Means of SPH Schemes with Numerical Diffusive Term”. Computer Physics Communications,
181, pp. 532–549.
Ma
nu
[19] Rafiee, A., Cummins, S., Rudman, M., and Thiagarajan, K., 2012. “Comparative Study on the
Accuracy and Stability of SPH Schemes in Simulating Energetic Free Surface Flows”. European
Journal of Mechanics B/Fluids, 36, pp. 1–16.
ed
[20] Chowdhurry, S., and Sannasiraj, S., 2014. “Numerical Simulation of 2D Sloshing Waves Using
pt
SPH with Diffusive Terms”. Applied Ocean Research, 47, pp. 219–240.
ce
[21] Cao, X., Ming, F., and Zhang, A., 2011. “Sloshing in a Rectangular Tank Based on SPH Simula-
Ac
tion”. Computer Physics Communications, 182, pp. 866–877.
[22] Meringolo, D., Colagrossi, A., Marrone, S., and Aristodemo, F., 2017. “On the Filtering of Acoustic Components in Weakly-Compressible SPH Simulations”. Journal of Fluids and Structures, 70,
pp. 1–23.
[23] Pakozdi, C., 2008. “A Smoothed Particle Hydrodynamics Study of Two-Dimensional Nonlinear
Sloshing in Rectangular Tanks”. PhD thesis, Norwegian University of Science and Technology,
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017
by ASME University of Science and Technology.
Norwegian
[24] Monaghan, J., and Kos, A., 1999. “Solitary Waves on a Cretan Beach”. Journal of Waterway Port
Coastal and Ocean Engineering-Asce, 125(3), pp. 145–154.
[25] Shadloo, M., Zainali, A., Sadek, H., and Yildiz, M., 2011. “Improved Incompressible Smoothed
Particle Hydrodynamics Method for Simulating Flow Around Bluff Bodies”. Computational Meth-
ed
ite
d
ods in Applied Mechanical Engineering, 200, pp. 1008–1020.
[26] Liu, M., and Liu, G., 2010. “Smoothed Particle Hydrodynamics: An Overview and Recent Developments”. Arch. Comput. Methods Eng., 17, pp. 25–76.
py
[27] von Neumann, J., and Richtmyer, R., 1950. “A Method for the Numerical Calculation of Hydrodynamic Shocks”. Journal of Applied Physics, 21, pp. 232–247.
Co
[28] Antuono, M., Colagrossi, A., Marrone, S., and Lugni, C., 2011. “Propagation of Gravity Waves
ot
Through an SPH Scheme with Numerical Diffusive Terms”. Computer Physics Communications,
tN
182, pp. 866–877.
[29] Ozbulut, M., Yildiz, M., and Goren, O., 2014. “A Numerical Investigation into the Correction
Mechanical Sciences, 79, pp. 56–65.
sc
rip
Algorithms for SPH Method in Modeling Violent Free Surface Flows”. International Journal of
[30] Antuono, M., Colagrossi, A., and Marrone, S., 2012. “Numerical Diffusive Terms in Weakly-
Ma
nu
Compressible SPH Schemes”. Computer Physics Communications, 183, pp. 2570–2580.
[31] Meringolo, D., Colagrossi, A., Aristodemo, F., and Veltri, P., 2015. “SPH Numerical Modeling of
WavePerforated Breakwater Interaction”. Coastal Engineering, 101, pp. 48–68.
ed
[32] Meringolo, D., Colagrossi, A., Aristodemo, F., and Veltri, P., 2015. “Assessment of Dynamic
pt
Pressures at Vertical and Perforated Breakwaters Through Diffusive SPH Schemes”. Mathematical
ce
Problems in Engineering, 2015, p. 10 pages. Article ID 305028.
Ac
[33] Nomeritae, N., Daly, E., Grimaldi, S., and Bui, H., 2016. “Explicit Incompressible SPH Algorithm
for Free-Surface Flow Modelling: A Comparison with Weakly Compressible Schemes”. Advances
in Water Resources, 97, pp. 156–167.
[34] Shepard, D., 1968. “A two dimensional interpolation function for irregularly-spaced data”. In
Proceedings of the 23rdACM National conference, pp. 517–524.
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Ac
ce
pt
ed
Ma
nu
sc
rip
tN
ot
Co
py
ed
ite
d
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
by ASME
6 2017
TABLES
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Co
py
ed
ite
d
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
H (m) D (m) L (m)
ot
Dimension, amplitude and frequency values for the simulated test cases
A (m)
ω (rad/s)
ω/ω1
0.08
3.488
1.173
0.08
3.696
1.032
tN
Table 1.
1.05
0.30
1.73
Case 1-b
1.05
0.50
1.73
Case 2
1.00
0.35
1.00
0.05
3.50-6.26
0.70-1.26
Case 3
0.50
0.35
0.80
0.02
5.820
1.000
Ac
ce
pt
ed
Ma
nu
sc
rip
Case 1-a
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
Comparison of consecutive impact pressure values at points P2 and P3 with the experimental and numerical results of [13]
Numerical Results (kPa)
P3
P2
P3
1st
0.77
9.80
0.46
4.72
2nd
4.30
7.01
0.54
8.01
3rd
8.26
9.11
1.68
6.22
4th
3.28
8.12
2.64
2.93
5th
4.62
10.42
2.78
1.65
6th
2.82
10.83
1.98
2.72
7th
3.91
13.33
1.27
8th
2.50
17.84
1.37
9th
3.13
5.24
1.73
10th
5.11
13.35
1.86
11st
2.75
13.45
12nd
3.88
3.36
13rd
4.30
14th
P2
P3
ed
ite
d
P2
0.67
4.49
1.11
5.03
4.14
5.78
2.63
4.54
3.57
5.19
2.37
6.92
3.70
6.32
4.31
5.15
5.35
3.92
2.59
5.41
3.59
2.72
8.27
3.29
2.66
6.11
1.78
3.63
3.35
3.68
5.36
1.92
3.55
3.57
5.89
3.66
8.17
1.97
3.85
2.88
5.73
15th
4.83
7.08
2.07
3.85
3.29
4.70
16th
4.09
9.78
2.03
3.64
3.00
9.14
17th
3.91
4.79
1.97
-
4.55
6.97
18th
4.23
5.09
1.80
-
4.64
5.35
19th
3.66
8.70
2.00
-
3.10
-
3.42
-
-
-
3.19
-
3.70
-
-
-
3.34
-
3.86
8.99
1.78
4.00
3.15
5.83
Std. Deviation
1.38
3.68
0.57
1.44
1.05
1.35
Max. Value
8.26
17.84
2.78
8.01
5.15
9.14
ce
21st
Ac
Average
1.93
sc
rip
Ma
nu
ed
20th
ot
4.05
pt
Co
Consecutive Impacts
SPH Results (kPa)
py
Experimental Results (kPa)
tN
Table 2.
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c)
by ASME
7 2017
FIGURES
1.2
1
APD Particles
VFS Particles
Boundary Particles
z (m)
0.8
0.6
0.4
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x (m)
(a)
1
Fluid Particles
Boundary Particles
Final Volume Level
z(m)
0.6
Co
0.4
0.2
0
0.5
1
1.5
x(m)
2.5
3
tN
(b)
2
ot
0
Fig. 1.
1.8
py
0.8
ed
ite
d
0.2
The view of free surface (red) and fully populated (blue) regions during the evolution of sway-sloshing motion (at t ∼ 20(s) (a), and
Ac
ce
pt
ed
Ma
nu
sc
rip
the conservation of final volume at t = 25(s) for the dam-break problem with the local velocity coefficient in the APD treatment (b)
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
The representative geometry for the sway-sloshing simulations
Ac
ce
pt
ed
Ma
nu
sc
rip
tN
ot
Co
py
Fig. 2.
ed
ite
d
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
0.35
Experiment
H/dx=15
H/dx=30
H/dx=60
0.3
0.25
(ζ(t)−D)/L
0.2
0.15
0.1
0.05
−0.1
15
20
25
30
35
40
45
50
55
60
t(g/L)0.5
(a)
180
70
py
140
Co
120
|FFT(ζ(t)/L)|
65
Experiment
H/δx=15
H/δx=30
H/δx=60
160
100
80
ot
60
20
2
3
4
5
6
tN
40
0
1
ed
ite
d
0
−0.05
7
8
9
10
11
12
sc
rip
ω*(L/g)0.5
(b)
Fig. 3.
Comparison of the three different particle resolutions with experimental results (a) time histories of the free-surface elevations and
= −0.815 (m) for Case 1-a
Ac
ce
pt
ed
Ma
nu
(b) corresponding frequency analysis at x
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
Pakozdi Experiment
Pakozdi SPH Results
Present SPH Study
0.4
0.35
0.3
0.25
(ζ(t)−D)/L
0.2
0.15
0.1
0.05
0
−0.1
20
30
40
50
60
70
80
t(g/L)0.5
(a)
150
ed
ite
d
−0.05
90
Co
100
ot
|FFT(ζ(t)/L)|
py
Present SPH Study
Pakozdi Experiment
Pakozdi SPH Results
1
2
3
4
5
6
7
ω(L/g)0.5
sc
rip
0
tN
50
8
9
10
11
12
(b)
Fig. 4.
Comparison of the (a) time histories of the free-surface elevations and (b) corresponding frequency analysis at
Ac
ce
pt
ed
Ma
nu
Case 1-a
x = -0.815 (m) for
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
0.5
Pakozdi Experiment
Pakozdi SPH Solution
Present SPH Study
0.4
(ζ(t)−D)/L
0.3
0.2
0.1
ed
ite
d
0
−0.1
−0.2
0
10
20
30
40
t(g/L)0.5
50
60
Present Study SPH Solution
Pakozdi Experiment
Pakozdi SPH Solution
Co
80
70
ot
50
tN
|FFT(ζ(t)/L)|
60
40
20
10
2
3
4
5
Ma
nu
Fig. 5.
1
sc
rip
30
0
80
py
(a)
70
6
ω (L/g)0.5
7
8
9
10
11
12
(b)
Comparison of the (a) time histories of the free-surface elevations and (b) corresponding frequency analysis at the left wall at
Ac
ce
pt
ed
-0.815 (m) for Case 1-b
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
x=
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
1
Model Test (Faltinsen & Timokha, 2001)
Computational Results (Faltinsen & Timokha, 2001)
Present SPH Study
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.7
1
1.1
T/T1
1.2
1.3
1.4
1.5
The comparison of wave elevations near wall with D/L=0.35, A/L=0.05
Ac
ce
pt
ed
Ma
nu
sc
rip
tN
ot
Co
Fig. 6.
0.9
py
0.8
ed
ite
d
Wave elevation near the wall (m)
0.9
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Positions where pressure time series are calculated; dimensions are given in (mm)
Ac
ce
pt
ed
Ma
nu
sc
rip
tN
ot
Co
py
Fig. 7.
ed
ite
d
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
5
4.5
Pressure (kPa)
4
3.5
3
2.5
2
1.5
2
4
6
8
10
12
Time (s)
14
16
(a) Experimental results reported in [13]
5
t1
t
3
Co
3.5
3
t
2
2.5
t4
2
4
6
8
tN
2
1.5
0
20
py
4
18
ot
Pressure (kPa)
4.5
ed
ite
d
1
0.5
0
10
Time (s)
12
14
16
18
20
sc
rip
(b) Results of the present study
4.5
Ma
nu
Pressure (kPa)
4
3.5
3
2.5
2
Fig. 8.
4
6
8
10
12
Time(s)
14
16
18
20
(c) Numerical results reported in [13]
Comparison of the pressure time histories at P1
Ac
ce
pt
1.5
0
ed
2
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
9
Model Test (Chen et.al. 2009)
Computational Results (Chen et.al. 2009)
Present SPH Study
8
7
Pressure (kPa)
6
5
4
3
1
0
0
2
4
6
8
10
12
14
16
Number of Consecutive Hits of Water Reserve
18
(a) Consecutive pressure beat at point P2
22
py
18
Co
16
14
12
10
ot
Pressure (kPa)
20
Model Test (Chen et.al. 2009)
Computational Results (Chen et.al. 2009)
Present SPH Study
20
8
tN
6
4
2
4
sc
rip
2
0
0
ed
ite
d
2
6
8
10
12
14
Number of Consecutive Hits of Water Reserve
16
18
(b) Consecutive pressure beat at point P3
Fig. 9.
Quantitative comparison of the consecutive hits at pressure points P2 and P3 . Straight lines with the same color indicate the mean
Ac
ce
pt
ed
Ma
nu
impact pressure values of the corresponding result
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
20
Pressure (kPa)
15
10
5
0
2
4
6
8
10
12
Time (s)
14
16
18
(a) Experimental results reported in [13]
8
py
Pressure (kPa)
6
Co
4
2
4
6
8
10
Time (s)
12
14
16
18
20
tN
2
ot
0
−2
0
20
ed
ite
d
−5
0
(b) Results of the present study
sc
rip
12
10
Pressure (kPa)
8
4
2
0
−2
0
Ma
nu
6
2
4
6
8
10
12
Time (s)
14
16
18
20
ed
(c) Numerical results reported in [13]
Comparison of the pressure time histories at P4
Ac
ce
pt
Fig. 10.
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Journal of Fluids Engineering. Received November 23, 2016;
Accepted manuscript posted October 23, 2017. doi:10.1115/1.4038242
Copyright (c) 2017 by ASME
3.5
8
0.5
0.5
3
7
2
5
0.3
4
3
0.2
2.5
0.4
6
z (m)
z (m)
0.4
0.3
1.5
1
0.2
0.5
2
0.1
0.1
1
0
0
0
−0.5
0
−1
0
0.1
0.2
0.3
0.4
x (m)
0.5
0.6
0.7
−1
0.8
0
0.1
0.2
0.3
0.4
x (m)
0.5
0.6
0.7
0.8
0.5
3.5
0.5
ed
ite
d
(a) Free-surface profile and pressure field at t1 = 10.54(s) (b) Free-surface profile and pressure field at t2 = 10.64(s)
3
0.4
2.5
2
1.5
0.2
1
0.5
0.1
0.3
0.2
0.1
0
0
−0.5
0.1
0.2
0.3
0.4
x (m)
0.5
0.6
0.7
0
0.8
(c) Free-surface profile and pressure field at t3 = 10.86(s)
Fig. 11.
0.1
0.2
0.3
0.4
x (m)
0.5
3
2.5
2
1.5
1
0.5
0
−0.5
−1
0.6
0.7
0.8
Co
0
0
py
0.3
z (m)
z (m)
0.4
3.5
(d) Free-surface profile and pressure field at t4 = 11.17(s)
The snapshots of the free-surface profiles and pressure field between the time interval 10.54-11.17(s). The pressure values in color
Ac
ce
pt
ed
Ma
nu
sc
rip
tN
ot
bar are given in kPa.
Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Документ
Категория
Без категории
Просмотров
2
Размер файла
1 643 Кб
Теги
4038242
1/--страниц
Пожаловаться на содержимое документа