Sloshing Motion in a Real-Scale Water Storage Tank under Nonlinear Ground Motion
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water
Article
Sloshing Motion in a Real-Scale Water Storage Tank
under Nonlinear Ground Motion
Heng Jin 1,2 , Ruiyin Song 1, * and Yi Liu 1,3
1 Ningbo Institute of Technology, Zhejiang University, Ningbo 315000, China; jinheng@nit.zju.edu.cn (H.J.);
liuyilulu@nit.zju.edu.cn (Y.L.)
2 Ningbo Institute of Dalian University of Technology, Ningbo 315016, China
3 Ningbo Shenglong Group Company Limited, Ningbo 315105, China
* Correspondence: sry@nit.zju.edu.cn
Received: 9 June 2020; Accepted: 18 July 2020; Published: 24 July 2020
Abstract: Water storage tanks in cities are usually large and are occasionally affected by earthquakes.
A sudden earthquake can cause pressure pulses that damage water containers severely. In this study,
the sloshing motion in a high filling level tank caused by seismic excitation is investigated by
the numerical method in a 2D model. Two well-studied strong earthquakes are used to analyze
the broadband frequency nonlinear displacement of the tank both in the longitudinal and vertical
directions. Based on careful experimental verification, the free surface motion and the elevations at
the side wall are captured, and the sloshing pressure response is examined. The results show that the
2D section of the cylindrical tank can be used to estimate the maximum response of the 3D sloshing,
and the water motions under the seismic excitations are consistent with the modal characteristics of
the sloshing. The time histories response of the water motion reflected that the sloshing response is
hysteretic compared with the seismic excitation. The anti-seismic ability of the damping baffle shows
that its effect on sloshing pressure suppression is limited, and further study on the seismic design of
water tanks in earthquake-prone regions is needed.
Keywords: sloshing; ground motion; water storage tank; damping baffle; response analysis
1. Introduction
Water is the source of life. It is an essential element in human production and life. Almost all
fields including agricultural irrigation, hydropower, climate, etc., are inextricably linked to water;
therefore, the management of the water resources is necessary. The utilization, environmental health,
and storage safety of the water resources all are part of the management of water. The water tanks,
as the container for water storage, are usually large and have higher safety standards. Therefore, it is
very important to ensure the safety of the water tank under external environmental loads such as an
earthquakes or wind.
In high seismic regions, the seismic design of water tanks has attracted more attention from
structural engineers as the urban water storage demand increases [1,2]. The design of a liquid container
is of particular concern when the sloshing frequency is close to the seismic excitation frequency
(i.e., resonance), because pressure pulses from the dynamic load may seriously damage the system [3,4].
Sloshing has been widely studied as a typical fluid motion phenomenon [5]. The free surface motion
and the resonance frequency location are the key factors determining the hydrodynamic load in the
container, and the movement of the water body in the tank is closely related to external excitations.
Therefore, a more careful evaluation of the hydrodynamics performance of a sloshing water storage
tank under external excitation is required.
Water 2020, 12, 2098; doi:10.3390/w12082098 www.mdpi.com/journal/water
Water 2020, 12, 2098 2 of 14
At first, researchers focused on the sloshing phenomena subject to a harmonic excitation [6–9].
As was shown, sloshing characteristics were influenced by tank parameters such as water depth [10],
tank shape [7,11,12], baffle type [9,11,13–15], external excitation [6], and so on [16]. A series of
analytical [15,17], numerical [10,18], and experimental [19,20] methods were established for solving the
sloshing problem under different conditions. However, Chen et al. [21] compared the response of liquid
tanks under harmonic excitations with those under the ground motion excitation, and they found that
simple harmonic excitation cannot replace nonlinear seismic excitation for the seismic analysis of liquid
tanks, because seismic excitation usually consists of a broadband frequency vibration, and harmonic
excitation only contains a single frequency. The primary frequencies of the ground motion decrease
with the increasing intensity of a seismic event. The frequencies of a strong earthquake are usually
distributed across a lower range, which is similar to the natural frequencies of many large industrial
tanks [22]. Therefore, the sloshing phenomena under various types of seismic excitation may differ.
In order to improve the computational efficiency and reduce the costs, a small-scale geometric
model was usually used on a tank whose characteristic length in the excitation direction was between
0.6 and 1 m [23,24]. However, the ground motion has not been proportionally scaled, along with an
overestimated sloshing response estimated. Meanwhile, the filling levels of the tank in the previous
study were usually selected to be at a shallow depth [25–27] (depth/tank’s length < 0.1) or near critical
depth (depth/tank length ≈ 0.337) [25,28] to study the nonlinear sloshing phenomenon. The limited
cases considering high filling level conditions focused on the problem of the impact of water on the
tank roof under the large displacement [29]. Jin et al. [19] tested the sloshing response in a high filling
level tank with filling depth(d)/tank length(L) = 0.5 by the experiments, and they found that the water
motion tended to be linear in such a deep tank, which differed from the violent nonlinear sloshing
in the shallow water tank. For city water storage tanks, the size is large, and the filling level is high.
The load from the sudden ground movement will be the main threat to its safety. Unlike the long-term
loads such as wind or waves in tuned liquid damper systems on high-rise buildings or the liquid
tanks in ships, the seismic wave is a non-periodic impulse vibration with a broadband frequency
characteristic. Under this condition, the motion response of the water in the container may differ from
the previous studies.
In this research, the liquid sloshing motion in a real-scale water storage tank was investigated under
several selected ground motions, using the computational fluid dynamic (CFD) software—OpenFOAM®
(Version: 1906, OpenCFD, Bracknell, UK) [30]. The high filling level, real-scale tank, and ground
motion were used to distinguish the present water motion from previous sloshing studies in other
fields. First, the numerical model and the governing equation were introduced. The problem of seismic
response of the water tank was described in detail. Then, the numerical model was used to discuss the
motion of the free surface in the water tank and the sloshing control effect by the damping plate after
rigorous numerical verification.
2. Numerical Method and Model
2.1. Governing Equation
Navier–Stokes equations [30] were used to describe the incompressible, viscous, water flows in
the present numerical model. In the sloshing problem, the sloshing flow was usually assumed to be a
laminar flow, and it can give an acceptable result when the free surface motion was small [10,16,31].
In this study, the laminar model was used to solve the governing equations below:
∂ui
=0 (1)
∂xi
∂ui ∂u 1 ∂p µ ∂ ∂ui
+ ui i = − + + fbody (2)
∂t ∂x j ρ ∂xi ρ ∂x j ∂x j
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The meaning of the variables in the above equation is as follows: ρ is the density of the water, ui
The meaning
represents of thecomponents
the velocity variables ininthethe
above equation
i direction, xi is
(i as follows:
= x, ρ is the densityand
y in two-dimensions of the water,
x, y, z in
u i represents the
three-dimensions), velocity components in the i direction, x
p is the pressure, μ is the dynamic viscosity, i (i = x, y in two-dimensions and x,
and fbody represents the body forces, y, z in
such as gravity. Thep Finite
three-dimensions), is the pressure, µ is the dynamic
Volume Method (FVM) was viscosity,
used for and f bodydiscretization
spatial represents theofbody forces,
the partial
such as gravity. The Finite Volume Method (FVM) was used for spatial discretization
differential equations. The quantities of interest, e.g., the pressure and velocity of fluid, were stored of the partial
differential equations.
at the centroid The volumes.
of control quantitiesThe
of interest, e.g., the pressure
volume-of-fluid and velocity
(VoF) method of fluid, to
was selected were storedthe
capture at
the
freecentroid
surface. of
Thecontrol
volumevolumes.
fractionThe volume-of-fluid
of the fluid was defined(VoF) in method was selected
the transport equation:to capture the free
surface. The volume fraction of the fluid was defined in the transport equation:
ρ = ρα=w ater ρ ρ + +(1(1−−ααw ater ))ρρ air, ,v v==α α w ater
α w ater
v + (1 − α w) ater ) v air，
+ (1 − α
v w ater v ,
(3)
(3)
water water water air water water water air
water is
where αwater isthe
thevolume
volumefraction
fractionof
of water
water and
and vv is
is the kinematic viscosity. The volume fraction of
following equation:
water was expressed by the following equation:
∂ α∂α ∂ui
++ αα ∂ui = =0 (4)
∂ t∂t ∂ xi i 0
∂x
(4)
The variable
The variable α represents water
α represents water when
when its value is
its value equal to
is equal 1 and
to 1 and the
the interface
interface of
of water
water and
and air
air
when α ≈ 0.5.
when α ≈ 0.5.
2.2. Numerical Model
The sloshing model was established based on OpenFOAM (Version: 1906), and the results were
solved through the interFoam solver. The solution process of the new interFoam solver is shown in
Figure 1.1.The
ThePIMPLE
PIMPLE algorithm,
algorithm, a combination
a combination of (Pressure
of PISO PISO (Pressure
ImplicitImplicit with of
with Splitting Splitting of
Operator)
Operator) and SIMPLE (Semi-Implicit Method for Pressure-Linked Equations), in
and SIMPLE (Semi-Implicit Method for Pressure-Linked Equations), in Figure 1 was applied to solve the Figure 1 was
applied to solve the
pressure–velocity pressure–velocity
coupling relationship.coupling relationship.
More detail Morethe
of the solver, detail of the conditions,
boundary solver, the boundary
and other
conditions,
setups and other
can refer to thesetups can study
previous refer to[18].
the previous
The meshstudy near [18]. Thesurface
the free mesh nearwasthe free surface
refined was
to improve
refined to improve the accuracy of free surface capture. The 6 degree-of-freedom
the accuracy of free surface capture. The 6 degree-of-freedom (6DoF) tank movements were imposed (6DoF) tank
movements
through were imposed
a formatted throughofa time
displacement formatted
series displacement
data. of time series data.
Figure 1.
Figure Solution process
1. Solution process of
of the
the interFoam
interFoam solver.
solver.
2.3. Problem Description
2.3. Problem Description
To investigate the water motion in a real-scale tank, which was subjected to the ground motions,
To investigate the water motion in a real-scale tank, which was subjected to the ground
the present numerical model was validated first with experimental tests. The experimental water tank
motions, the present numerical model was validated first with experimental tests. The experimental
was 1 m (L) × 0.8 m (H) × 0.1 m (B) as shown in Figure 2, and the filling depth d/L was 0.5. The amplitude
water tank was 1 m (L) × 0.8 m (H) × 0.1 m (B) as shown in Figure 2, and the filling depth d/L was 0.5.
of horizontal displacement A/L was 0.0025, and the excitation frequency ω/ω1 = 0.996 in Figure 2b
The amplitude of horizontal displacement A/L was 0.0025, and the excitation frequency ω/ω1 = 0.996
and ω/ω1 = 0.603 in Figure 2c (ω1 is the first-order natural frequency of the liquid tank). The detailed
in Figure 2b and ω/ω1 = 0.603 in Figure 2c (ω1 is the first-order natural frequency of the liquid tank).
experimental setup and the arrangement of the experimental system were the same as the author’s
The detailed experimental setup and the arrangement of the experimental system were the same as
previous experimental device, and there is no baffle installation in the validation test [19]. Then, the free
the author’s previous experimental device, and there is no baffle installation in the validation test
surface shape and the flow motion were compared carefully in Figure 2b,c. The heights of the free
[19]. Then, the free surface shape and the flow motion were compared carefully in Figure 2b,c. The
surface curves and the time history of the free surface elevation along the left tank wall are also shown
heights of the free surface curves and the time history of the free surface elevation along the left tank
in Figure 3 and validated based on the experimental result and theory model [32]. The uncertainty
wall are also shown in Figure 3 and validated based on the experimental result and theory model
analysis was conducted to obtain the standard error and confidence level of the experimental result.
[32]. The uncertainty analysis was conducted to obtain the standard error and confidence level of the
The numerical result fell within the 95% confidence interval. The theoretical results were a bit biased
experimental result. The numerical result fell within the 95% confidence interval. The theoretical
results were a bit biased after 14 s, which may be because it did not consider the viscosity and
Water 2020, 12, 2098 4 of 14
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afterWater
14 s,2020,
which may
12, x FOR beREVIEW
PEER because it did not consider the viscosity and compressibility of the
4 of 14 fluid.
In general, the error of the experimental tests was acceptable, and the result of
compressibility of the fluid. In general, the error of the experimental tests was acceptable, and the
the numerical model
compressibility
wasresult
credible. of the fluid. In general, the error of the experimental tests was acceptable, and the
of theFurthermore,
numerical model the ability of the present
was credible. modelthe
Furthermore, to ability
simulate strong
of the nonlinear
present modelwater motion
to simulate
was result
also of the numerical
verified with themodel was credible.
experimental testFurthermore,
of Faltinsen the
etability
al. of the
[33] in present4.
Figure model
The to simulate
tank’s internal
strong nonlinear water motion was also verified with the experimental test of Faltinsen et al. [33] in
strong nonlinear water motion was also verified with the experimental test of Faltinsen et al. [33] in
dimensions
Figure were
4. The 1.0 m
tank’s × 0.98dimensions
internal m × 0.1 m were 1.0 ×
(length × 0.98 m× ×breadth),
m height 0.1 m (lengththe d/L was 0.4,
× height and thethe
× breadth), A/L
Figure 4. The tank’s internal dimensions were 1.0 m × 0.98 m × 0.1 m (length × height × breadth), the
was 0.01. The
d/L goodthe agreements wereTheobtained in two sets were
of tests. The present numerical using a
d/Lwas
was0.4,0.4,and
and the A/L
A/L was
was 0.01.
0.01. The good agreements
good agreements obtained
were obtained in in two
two sets
sets of of tests.
tests. TheThe
laminar
presentassumption
presentnumerical
can
numericalusing
be used
usingaalaminar
to analyze
laminar assumption
the
assumption can
sloshing
canbebeusedproblem.
usedtotoanalyze
analyze thethe sloshing
sloshing problem.
problem.
Figure 2. Free
2. 2.Free surface shapeofofsloshing
sloshing water in a rectangular tank under horizontal harmonic
Figure
Figure Freesurface
surfaceshape
shape of sloshing water
water in
in aa rectangular tank
rectangular tank under
under horizontal
horizontal harmonic
harmonic
excitation, (a) data acquisition system, (b) Resonant sloshing, ω/ω1 = 0.996, (c) non-resonant sloshing,
excitation, (a)
excitation, data acquisition system, (b) Resonant sloshing, ω/ω 1 = 0.996, (c) non-resonant sloshing,
(a) data acquisition system, (b) Resonant sloshing, ω/ω1 = 0.996, (c) non-resonant sloshing,
ω/ω1 = 0.603.
ω/ωω/ω= 0.603.
1 1 = 0.603.
Figure 3. The
Figure validation
3. The validationofof
thethepresent
presentnumerical modelwith
numerical model withexperimental
experimental teststests
[19] [19]
and and theoretical
theoretical
solution
solution ω/ωω/ω
[32],[32], 1 = 10.996,
= 0.996, A/L==0.0025,
A/L 0.0025, d/L = 0.5.(a)
= 0.5. (a)Uncertainty
Uncertainty analysis
analysis of three
of three experimental
experimental tests, tests,
theFigure
error bars
the error indicated
bars a
indicated Standard
a Error
Standard of
Error time
of (X-error)
time and
(X-error) andheight (Y-error)
height (Y-error) of the
of the
3. The validation of the present numerical model with experimental tests [19] and theoretical free
free surface
surface wave.
wave.
(b)solution
Red (b)
shallow Red shallow
area area represented
represented the the prediction
prediction limitlimit
of of
the the average
average experimental
experimental result
result
[32], ω/ω1 = 0.996, A/L = 0.0025, d/L = 0.5. (a) Uncertainty analysis of three experimental tests, at
at aa 95%
95%
confidenceconfidence
the errorlevel. level.
bars indicated a Standard Error of time (X-error) and height (Y-error) of the free surface
wave. (b) Red shallow area represented the prediction limit of the average experimental result at a
95% confidence level.
Water 2020, 12, x FOR PEER REVIEW 5 of 14
Water 2020, 12, 2098 5 of 14
Water 2020, 12, x FOR PEER REVIEW 5 of 14
Figure 4. Flow motion through the vertical slot baffle in Faltinsen’s experimental tests [33], τ is the
porosity of the baffle, (a) τ = 0.085, ω/ω1 = 1.328, Time = 14 s, (b) τ = 0.527, ω/ω1 = 0.96, Time = 46 s.
Figure 4.4. Flow
Figure Flow motion
motion through
through the
the vertical
vertical slot
slot baffle
baffle in
in Faltinsen’s
Faltinsen’s experimental
experimental tests
tests [33],
[33], ττ is
is the
the
porosity
porosityof ofthe
thebaffle, (a)ττ== 0.085,
baffle,(a) 0.085, ω/ω
ω/ω11 =
= 1.328, Time = 14 s,
= 14 (b) ττ ==0.527,
s, (b) ω/ω1 1= =
0.527,ω/ω 0.96,
0.96, Time= =
Time 4646 s. s.
Then, a real-scale water tank in Haroun [34] was selected as the baseline structure for this study.
This tankThen, was located in Los Angeles, which is a city with potential the earthquakes risk [35]. The study.
tank
Then, aa real-scale
real-scale waterwater tank tank in in Haroun
Haroun [34] [34] was
was selected
selected as as the baseline structure
baseline structure for this
for this study.
geometry
This is shown in Figure 5, with 18 m (Diameter (D)) × 18.25 m (Height′(H′)). The filling depth d
This tank
tank waswas located
located in inLosLosAngeles,
Angeles, which which isisaacity citywithwithpotential
potentialearthquakes
earthquakes risk risk[35].
[35]. TheThe tank
tank
was 10.8 m (60% D), and then the tank can be regarded as a slender [36] or deep
0 (H )).water
0 tank (2d/D >d
geometry
geometry is is shown
shown in inFigure
Figure5,5,with with18 18mm(Diameter
(Diameter(D)) (D))××18.2518.25mm(Height
(Height′(H′)). The
The filling
filling depth
depth d
1). This filling level was larger than the value of d/D = 0.4, and therefore, the wave motion resembles > 1).>
was
was10.810.8m (60%D),
m(60% D),and andthen thenthe thetank
tankcan canbebe regarded
regarded as as
a slender
a slender [36][36]
or deep
or deep water
water tank (2d/D
tank (2d/D
aThis
standing
filling wave
level with
was the
largerlargest
than free
the surface
value ofelevations
d/D = 0.4, atand
the tank walls the
therefore, [37].wave
The Froude
motion number
resembles is
1). This filling level was larger than the value of d/D = 0.4, and therefore, the wave motion resembles
the only dimensionless
aa standing number
largesttofree be considered if thereatisthe a scaled model[37]. used. TheIn CFD analysis,
standing wave wave with with the the largest free surface elevations
surface elevations tankwalls
at the tank walls [37]. The Froude
Froude number
number is
simplified
is the only assumptions
dimensionless and
number models to be are usually
considered if required
there is a to
scaled makemodel the calculation
used. In CFD process
analysis,
the only dimensionless number to be considered if there is a scaled model used. In CFD analysis,
economically
simplified viable. Therefore, a two-dimensional (2D)to Oxymakesection of the cylindrical tank was
simplifiedassumptions
assumptions andand models models are usually required
are usually required the
to calculation
make the process calculation economically
process
extracted
viable. to build
Therefore, a numerical
a two-dimensional water tank. The size
(2D) Oxy section (2D) of the numerical
of theOxy cylindrical tank was the same as the reala
economically viable. Therefore, a two-dimensional section tank of thewascylindrical
extracted to build
tank was
water
numericaltank in X
water and Y
tank. directions.
The size The
of the Z direction
numerical in
tankthe numerical
was the same model
as thewasreala symmetry
water tank plane.
in X The
and Y
extracted to build a numerical water tank. The size of the numerical tank was the same as the real
mesh dependence
directions. The Z was tested
direction in to ensure
the numerical thatmodel
the present
was a mesh
symmetry size can
plane.capture
The the sloshing
mesh dependence surface
was
water tank in X and Y directions. The Z direction in the numerical model was a symmetry plane. The
well
testedin toTable
ensure1. Finally,
that thethe medium
present mesh size with Δx =the 0.15 m, Δy =surface
0.05 mwell wasin selected,
Table 1.and the
mesh dependence was tested to mesh
ensuresize thatcan thecapture
present mesh sloshing
size can capture the sloshing Finally,
surface
mesh near the free water surface was refined locally. Two well-studied ground motions, the 1940
the
wellmedium
in Tablemesh size with
1. Finally, the ∆x medium= 0.15 mesh
m, ∆ysize = 0.05withm Δx was= selected,
0.15 m, Δy and the mesh
= 0.05 m was near the freeand
selected, waterthe
Imperial
surface Valley
was (IV) and
refined 1971Two
locally. San well-studied
Fernando (SF)ground earthquakes motions, [38],thewere
1940 selected
Imperial forValley
the following
(IV) and
mesh near the free water surface was refined locally. Two well-studied ground motions, the 1940
analysis.
1971 San They
Fernandotransferred
(SF) a nonlinear
earthquakes vibration
[38], were(SF) with the
selected for broadband
the following frequency
analysis. to the transferred
water tank.a
Imperial Valley (IV) and 1971 San Fernando earthquakes [38], were selectedThey for the following
The strongvibration
nonlinear vibrationwith of the theIV earthquake lasted a to long time, while
tank.the SFstrong
earthquake hadofstrong
analysis. They transferred a broadband
nonlinear vibrationfrequency withthe thewater
broadband The
frequency vibration
to the water the IV
tank.
instantaneous
earthquake vertical displacement and short duration. Both the ground motions in the longitudinal
The strong lasted a long
vibration of time,
the IV while the SF earthquake
earthquake lasted a long had time,
strongwhile
instantaneous vertical displacement
the SF earthquake had strong
(X
and direction)
short and vertical
duration. Both (Y direction)
the ground directions
motions in thewere considered,
longitudinal (X instead ofand
direction) the vertical
condition (Y in which
direction)
instantaneous vertical displacement and short duration. Both the ground motions in the longitudinal
only longitudinal
directions were displacement
considered, was applied in previous studiesonly[23,24]. By the way, the locationwas of
(X direction) and vertical (Y instead direction) of directions
the condition wereinconsidered,
which longitudinal
instead displacement
of the condition in which
the SF
applied earthquake is close
in previousdisplacement to
studies [23,24]. the water tank in Haroun. Therefore, the water motion under these
only longitudinal wasByapplied
the way, inthe location
previous of the [23,24].
studies SF earthquake
By the way, is close thetolocation
the water of
different
tank in excitations
Haroun. might the
Therefore, performwater differently.
motion under Each
these of the ground
different motionsmight
excitations was scaled
perform to differently.
the same
the SF earthquake is close to the water tank in Haroun. Therefore, the water motion under these
peak
Each ground
of the acceleration
ground motions (PGA) wasofscaled0.5 g so to that
the samethey represented the same earthquake intensity. The
different excitations might perform differently. Eachpeak of theground
ground acceleration
motions was (PGA)scaled of 0.5
to g sosame
the that
information
they of the two scaled earthquake events is listed in Table 2, and the scaled ground motions
peakrepresented the same (PGA)
ground acceleration earthquake of 0.5intensity.
g so that The theyinformation
representedofthe thesame
two earthquake
scaled earthquake intensity. events
The
are
is shown
listed in in Figure
Table 2, 6. the scaled ground motions are shown in Figure 6.
and
information of the two scaled earthquake events is listed in Table 2, and the scaled ground motions
are shown in Figure 6.
Thegeometry
Figure5.5.The
Figure geometryofofthe
thewater
waterstorage
storagetank.
tank.
Figure 5. The geometry of the water storage tank.
Water 2020, 12, x FOR PEER REVIEW 6 of 14
Table 1. Grid dependence of the free surface elevation at the right side wall under the Imperial
Valley (IV) earthquake.
Water 2020, 12, 2098 6 of 14
Maximum Calculation Averaged Free Surface Peaks at Relative
Mesh Description
Size (x × y) Time (s) Right Side Wall (x = 9 m) Difference
1Table 1. Grid
Fine 0.075 × 0.025 3391 0.1839 m -
dependence of the free surface elevation at the right side wall under the Imperial Valley
2(IV) earthquake.
Medium (used) 0.15 × 0.05 464 0.1746 m 5%
3 Coarse 0.3 × 0.1 88 0.1580 m 14%
Maximum Calculation Averaged Free Surface Peaks Relative
Mesh Description
Table 2. Information of twoSize (x × y)groundTime
strong (s)
motions. at Right
PGA: peakSide
ground = 9 m)
Wall (xacceleration,
Difference
SF: San
Fernando.
1 Fine 0.075 × 0.025 3391 0.1839 m -
2 Medium (used) 0.15 × 0.05 464 0.1746 m 5%
3 Coarse 0.3 × 0.1 88 0.1580 Strong Motion Duration
Earthquake Event Scaled PGA (g) Scale Factor Scaled PGD* (m) m 14%
(s)
Longitudinal: Longitudinal:
Table 2. Information of two strong ground motions. PGA: peak ground acceleration, SF: San Fernando.
IV 0.5zhiweiVertical: 1.78 0.154zhiweiVertical: 30
Earthquake Event 0.317
Scaled PGA (g) Scale Factor 0.048
Scaled PGD* (m) Strong Motion Duration (s)
Longitudinal:
Longitudinal: 0.5 Longitudinal:
Longitudinal: 0.154
SF IV 0.5zhiweiVertical: 1.78
0.41 0.160zhiweiVertical: 30 18
Vertical: 0.317 Vertical: 0.048
0.282
Longitudinal: 0.5 0.120
Longitudinal: 0.160
SF 0.41 18
Vertical: 0.282 Vertical: 0.120
PGD*: peak ground displacement.
PGD*: peak ground displacement.
Figure 6. Acceleration and displacement of two selected earthquakes, (a) the scaled acceleration and the
displacement of the Imperial Valley 1940 earthquake, (b) the scaled acceleration and the displacement
of the San Fernando 1971 earthquake.
Water 2020, 12, x FOR PEER REVIEW 7 of 14
Figure 6. Acceleration and displacement of two selected earthquakes, (a) the scaled acceleration and
the displacement
Figure of and
6. Acceleration the displacement
Imperial Valley 1940
of two earthquake,
selected (b) the
earthquakes, (a) scaled acceleration
the scaled andand
acceleration the
displacement of the San Fernando 1971 earthquake.
the displacement of the Imperial Valley 1940 earthquake, (b) the scaled acceleration and the
Water 2020, 12, 2098 of the San Fernando 1971 earthquake.
displacement 7 of 14
3. Results and Discussion
3. Results and Discussion
3.3.1.
Results and Discussion
Comparison of Water Motion between Two-Dimensional and Three-Dimensional Results
3.1. Comparison
3.1. In order to
Comparison of Water
of make
Waterthe Motion betweenprocess
calculation
Motion between Two-Dimensional
economical,
Two-Dimensional andthe
and Three-Dimensional
Oxy section wasResults
Three-Dimensional Results
extracted to study the
water sloshing
In order
order in thethe
to make
make tank under seismic
calculation process excitation.
economical,However,
the Oxy
Oxy in section
case thewas 2D extracted
simulationtocan be used
study the
In to the calculation process economical, the section was extracted to study the
instead
water of the
sloshing real
in the3D status,
tank the
under comparison
seismic of water
excitation. motion
However, between
in case the the
2D 2D and
simulation3D problems
can be was
used
water sloshing in the tank under seismic excitation. However, in case the 2D simulation can be used
conducted.
instead theAreal
of the similar
real trend the
3D status,
status, can comparison
be seen in Figure 7. The
of water
water free between
motion surface atthe the2DOxyand(Z3D3D= 0) section in
problems wasthe
instead of 3D the comparison of motion between the 2D and problems was
3D simulation
conducted. A coincided with the 2D result. Meanwhile, it can be seen that the free surface elevation
conducted. A similar
similar trend
trend can be seen
can be seen in in Figure
Figure 7.
7. The
The free
free surface
surface at at the
the Oxy
Oxy (Z (Z = = 0) section in
0) section in the
the
3Ddecreased
simulation along the Z direction
coincided with the from
2D 0 to ±9
result. m (Oyz section
Meanwhile, it can of the
be seen tank)
that in the
the free 3D model,
surface which is
elevation
3D simulation coincided with the 2D result. Meanwhile, it can be seen that the free surface elevation
different from
decreased alongthe theprevious
Z directionresearch
direction from 0in0 toa cube
to ±9 m tank [10]. They showed a similar sloshing
model,response isin
decreased along the Z from ±9 m (Oyz section
(Oyz section of the tank)
of the tank) in the
in the 3D3D model, which is
which
the Z direction.
different from
from the This is
the previous because
previous research we
research inused a cylindrical
in aa cube
cube tank
tank [10]. tank,
[10]. They and
They showed the side wall
showed aa similar of the
similar sloshing tank was a
sloshing response curved
response inin
different
surface.
the Z Further
direction. investigation
This is because of the
we usedtwo a models is supplied
cylindrical tank, and inthe
Figure
side 8. of the tank was a curved
wall
the Z direction. This is because we used a cylindrical tank, and the side wall of the tank was a curved
surface. Further
surface. Further investigation
investigation of of the
the two
two models
models is is supplied
supplied in in Figure
Figure 8. 8.
Figure 7. Free surface shape of the sloshing in the water tank under the IV earthquake, (a) 2D case, t =
Figure s,
10.25
Figure 7. (b)
7. Free
Free3D case, tshape
surface
surface = 10.25
shape s.the
ofof thesloshing
sloshingininthe
thewater
water tank
tank under
under thethe
IVIV earthquake,
earthquake, (a) (a)
2D2D case,
case, t=
= 10.25
t10.25 s, (b)
s, (b) 3D 3D case,
case, t = 10.25
t = 10.25 s. s.
Figure 8. Free surface elevations at the right side wall.
Figure 8. Free surface elevations at the right side wall.
The maximum free surface Figure elevation appeared
8. Free surface at the
elevations Oxyright
at the section
side of the tank, as seen in Figure 8.
wall.
In thisThe maximum
study, the freefree surface
surface on elevation appeared
the Oxy section wasat extracted
the Oxy section
to showof the
thetank,
water assloshing
seen in Figure
in the8.
In this
tank The study,
under the free surface
seismic excitation.
maximum free surfaceThison the Oxy section
resultappeared
elevation was extracted
usually indicated
at the Oxythe to show
most of
section the water
dangerous sloshing
the tank, section in the
of Figure
as seen in tank
the tank.8.
under
Therefore,seismic
in excitation.
order to save This
time result
and usually
calculation indicated
costs, the the
watermost dangerous
movement in section
the
In this study, the free surface on the Oxy section was extracted to show the water sloshing in the tank most of the
dangeroustank.
Therefore,
section
under was in
seismic order
calculated to here
excitation. saveThistime
mainly and
result calculation
based on the
usually costs, the
thewater
2D simulation.
indicated most movement
The insection
free surface
dangerous the most
of dangerous
shape andtank.
the the
section
time was
history
Therefore, calculated
inof elevation
order to savehere mainly
between
time and based on the
thecalculation 2D
2D and 3Dcosts, simulation.
simulation
the water The free
are movement surface
further shown in the shape
in most and
Figures 8the
andtime
dangerous 9.
history
They of elevation
indicated that thebetween
2D the
results 2D
were and 3D simulation
acceptable. are
Therefore, further
the shown
motion in Figures
response
section was calculated here mainly based on the 2D simulation. The free surface shape and the time of 8 and
water 9.
and They
the
conclusions drawn from
history of elevation it should
between the 2D be and
meaningful.
3D simulation are further shown in Figures 8 and 9. TheyWater 2020, 12, x FOR PEER REVIEW 8 of 14
Water 2020, 12, x FOR PEER REVIEW 8 of 14
indicated that the 2D results were acceptable. Therefore, the motion response of water and the
Water 2020, 12,
indicated
conclusions 2098
that the 2D
drawn results
from were
it should meaningful.Therefore, the motion response of water and8 ofthe
beacceptable. 14
conclusions drawn from it should be meaningful.
Figure 9. Free surface response at selected moments.
Figure 9. Free surface response at selected moments.
Figure 9. Free surface response at selected moments.
3.2. Free
3.2. Free Surface
Surface Motion
Motion in in the
the High
High Filling
Filling Depth
Depth Water
Water Tank
Tankunder
underSeismic
SeismicExcitation
Excitation
3.2. Free Surface Motion in the High Filling Depth Water Tank under Seismic Excitation
From Figure
From Figure6,6,the
the largest
largest longitudinal
longitudinal andand vertical
vertical PGDs PGDs appeared
appeared in thein the10first
first s of 10
the sground
of the
ground From Figure
motions. 6,
Duringthe largest
this longitudinal
stage, the shapes and
of vertical
the free PGDs
surface appeared
at the
motions. During this stage, the shapes of the free surface at the 2 s, 4 s, 6 s, 8 s, and 10 s and2 s, 4in
s, the
6 s, first
8 s, 10
and s
10 of the
s and
the
ground
the Fast
Fast motions.
FourierFourier During
Transfer
Transfer (FFT)this stage,
(FFT) the shapes
of twoofof
amplitudes
amplitudes the ground
two
ground free surface
motions at shown
motions
are theare
2 s,shown
4 s,
in 6 s,in8 10.
Figure s, and
FigureThe1010.s and
The
results
the Fast
results Fourier
showed Transfer
that the (FFT)
response amplitudes
of the of
water two ground
reached the motions
maximum are
showed that the response of the water reached the maximum displacement of the tank. Although the shown
displacement in Figure
of 10.
the The
tank.
results
Although
maximum showed
the that the instantaneous
maximum
instantaneous response
displacement of the
of water
displacement
the reachedof the
SF earthquake theSFmaximum
earthquake
was larger (seedisplacement
was
Tablelarger offree
(see
1), the the
Tabletank.
1),
liquid
Although
the free the
liquid maximum
motion instantaneous
caused by it was displacement
not significantly of the
more SF earthquake
violent than
motion caused by it was not significantly more violent than that of the IV earthquake. The maximum was
that oflarger
the IV (see Table
earthquake. 1),
the
Thefree
free liquid
maximum
surface motion
elevations caused
free surface
of the twoby itearthquakes
was notofsignificantly
elevations the more
two earthquakes
reached 11.20 mviolent thanm,that
reached
and 11.19 of the
11.20 m IV
respectively. andearthquake.
The11.19
ratio m,
of
The maximum
respectively. Thefree
ratiosurface
of the elevations
free surface of the
elevation twoto earthquakes
the original reached
water level
the free surface elevation to the original water level was only 3.7% for the IV earthquake and 3.6% for 11.20
was m
onlyand 3.7%11.19
for m,
the
respectively.
IV earthquake
the The
SF earthquake.andratio
3.6%offor
There the
is free
the
no SFsurface elevation
earthquake.
significant There
difference to isthe
here. nooriginal water
significant level was
difference only 3.7% for the
here.
IV earthquake and 3.6% for the SF earthquake. There is no significant difference here.
Figure 10. Free surface response at selected moments, (a) the free surface response under the Imperial
Valley 1940 earthquake, (b) the free surface response under the San Fernando 1971 earthquake.Imperial Valley 1940 earthquake, (b) the free surface response under the San Fernando 1971
earthquake.
The free surface shapes are shown in Figure 10, indicating that the location of the largest node
of water vibration is different. The nodes in IV are located around 1/3D and 2/3D, and the nodes in
Water 2020, 12, 2098 9 of 14
SF are located at the side wall. They corresponded to the first and third modes sloshing in the water
tank, respectively [39]. Based on the Fast Fourier Transfer (FFT), the seismic excitations were
transferredTheto free
thesurface shapes domain,
frequency are shownand in Figure 10, indicating
the first that thefrequencies
three natural location of the (ωi,largest
i = 1,node
2, 3)ofwere
marked. The results showed that the primary frequencies of SF and IV were close to ωSF
water vibration is different. The nodes in IV are located around 1/3D and 2/3D, and the nodes in are
1 and ω3,
located at the side wall. They corresponded to the first and third modes sloshing in the water tank,
respectively. Therefore, the corresponding sloshing modes were excited. On the other hand, the odd
respectively [39]. Based on the Fast Fourier Transfer (FFT), the seismic excitations were transferred to
resonance modes of sloshing are (n = 1, 3, …) usually activated by the anti-symmetric motion (sway
the frequency domain, and the first three natural frequencies (ωi, i = 1, 2, 3) were marked. The results
in the horizontal
showed that the direction) of the container,
primary frequencies of SF and IVand wereeven
close the
to ω1resonance modes (nTherefore,
and ω3 , respectively. = 2, 4, …)
corresponded to the symmetric motion (heavy in the vertical direction).
the corresponding sloshing modes were excited. On the other hand, the odd resonance modes The FFT results in Figure
of 10
indicated thatare
sloshing the(nresonance
= 1, 3, . . . )sloshing is more likely
usually activated by the to occur under motion
anti-symmetric the IV earthquake.
(sway in the horizontal
The envelopes
direction) of the maximum
of the container, and even thefreeresonance
surface modes
revealed(n =that . . . )effects
2, 4,the of water
corresponded to on
the the tank walls
symmetric
were similar (Figure 11), although the excitations of the ground motion were irregular and
motion (heavy in the vertical direction). The FFT results in Figure 10 indicated that the resonance
sloshing is
asymmetric. more likely to
Meanwhile, theoccur under the
maximum IV surface
free earthquake.elevation appeared near the side wall, and the
The envelopes of the maximum free surface
minimum point was located at the center of the still water level. revealed that the The effects of water level
high-water on theattank
the walls
tank wall
were similar (Figure 11), although the excitations of the ground motion
of the water tank is also usually the cause of the destruction and overturning of the tank. The were irregular and asymmetric.
Meanwhile, the maximum free surface elevation appeared near the side wall, and the minimum point
maximum rate of free surface elevation was 6.58% for the IV earthquake and 8.20% for the SF
was located at the center of the still water level. The high-water level at the tank wall of the water
earthquake. These results mean that earthquakes with large instantaneous displacements, such as
tank is also usually the cause of the destruction and overturning of the tank. The maximum rate of
the SF earthquake,
free may lead
surface elevation to more
was 6.58% for dangerous situations.
the IV earthquake This for
and 8.20% wasthe inconsistent
SF earthquake. with the conclusion
These results
reflected
meanin theearthquakes
that response of withthelarge
water motion during
instantaneous the first 10
displacements, suchs when
as the aSFlarge groundmay
earthquake, displacement
lead to
occurred. Therefore,situations.
more dangerous the time This history
wasof the free surface
inconsistent with theelevation
conclusionatreflected
the right sideresponse
in the wall is of shown
the in
Figure 12 motion
water to helpduring
understand
the firstthe
10 sabove
when aphenomenon. Figure 12 shows
large ground displacement thatTherefore,
occurred. the sloshing was not
the time
fullyhistory
excitedofinthe thefree
early stageelevation
surface of the earthquake. The wall
at the right side maximum
is shown free in surface
Figure 12 response did not appear
to help understand
thestage
at the abovewhenphenomenon.
the maximumFigure 12ground
shows that the sloshing occurred,
displacement was not fully andexcited in the early
the sloshing stage of
phenomenon
the earthquake. The maximum free surface response did not appear
continued and kept growing, despite the large ground displacement having passed. The elevation at the stage when the maximum
groundafter
decreased displacement
10 s in theoccurred, and the sloshing
SF earthquake event.phenomenon
The water kept continued
sloshing and stably
kept growing,
in the IV despite the
earthquake
large ground displacement having passed. The elevation decreased after 10 s in the SF earthquake
event until a new seismic pulse peak appeared, and the elevation increased again. The above results
event. The water kept sloshing stably in the IV earthquake event until a new seismic pulse peak
indicated that the sloshing excited by the seismic motion was hysteretic compared with the ground
appeared, and the elevation increased again. The above results indicated that the sloshing excited by
motion. Therefore, the water storage tank still faced the threat of collapse, although the violent
the seismic motion was hysteretic compared with the ground motion. Therefore, the water storage
ground
tankmotion
still facedpassed.
the threat of collapse, although the violent ground motion passed.
Figure 11. The maximum free surface envelope under two different ground motions.
Figure 11. The maximum free surface envelope under two different ground motions.Water 2020, 12, 2098 10 of 14
Water 2020, 12, x FOR PEER REVIEW 10 of 14
Figure 12. Free surface elevations at right side wall.
3.3. Seismic Design of the Water Storage Tank and the Anti-Sloshing Ability of the Damping Baffle
12. Free surface elevations at right side wall.
Even a short-term seismicFigure pulse
Figure 12.will
Free cause continuous
surface elevations sloshing
at right side wall. in the water storage tank and
bring a sloshing load, which delayed more than the ground motion.
3.3. Seismic Design of the Water Storage Tank and the Anti-Sloshing Ability of the For
Dampingthe Baffle
safety design of the
3.3. Seismic Design of the Water Storage Tank and the Anti-Sloshing Ability of the Damping Baffle
water tank in Evenearthquake-prone
a short-term seismicareas, a well-studied
pulse will cause continuous vertical
sloshingdamping baffle
in the water (seetank
storage Figure
and 13) was
used to test Even
bringits a short-term
suppression
a sloshing seismic pulse
on sloshing,
load, which delayed more will
such cause
thanasthecontinuous
[15,20,40,41]. sloshing
ground motion.The in the
For height water
the safety storage
ofdesign
the baffle tanka and
= 2/3d. The
of the water
mesh around the baffle was refined. Figure 14 gives the free surface elevations at the righttheside wall
bring
tank ina sloshing load,
earthquake-prone which
areas, delayed
a more
well-studied than the
vertical ground
damping motion.
baffle (seeFor the
Figure safety
13) wasdesign
used of
to test
water
its tank in earthquake-prone
suppression on sloshing, such as areas, a well-studied
[15,20,40,41]. vertical
The height damping
of the baffle abaffle
= 2/3d.(see
TheFigure
mesh 13) was
around
in the cases of tanks with and on without the damping baffle.The The results showed that the vertical
the baffle was refined. Figure 14 gives the free surface elevations at the right side wall in the casesThe
used to test its suppression sloshing, such as [15,20,40,41]. height of the baffle a = 2/3d. of
dampingtanks
baffle
mesh with had
aroundandthea baffle
better
without was
the effect
damping onbaffle.
refined. the free
Figure surface
14 gives
The the
results elevations
free
showed surface
that the at thedamping
elevations
vertical atside wall,
the right under
side
baffle wall
had a the SF
earthquake.
in theeffect
better casesonofthe
tanks
freewith andelevations
surface without at thethe
damping
side wall,baffle.
underThe the results showed that the vertical
SF earthquake.
damping baffle had a better effect on the free surface elevations at the side wall, under the SF
earthquake.
Figure 13. Location of the damping baffle in the water tank and mesh geometry.
13. Location of the damping baffle in the water tank and mesh geometry.
Figure Figure
13. Location of the damping baffle in the water tank and mesh geometry.
On the other hand, the sloshing pressure on the two measuring points with and without the
damping baffle was compared. The two measuring points were located near the bottom and the
free surface of the water tank, which were 1 m and 10 m away from the bottom, respectively. Then,
the pressure variation at these two points is shown in Figure 15, and the sloshing pressures were
normalized by p/ρgh-1 to express the fluctuation of sloshing pressure. Here, the ρgh was the hydrostatic
pressure at the selected points. The results showed that the pressure variation ratio near the water
surface was larger than that in the bottom area. The damping baffle had a significant suppression effect
on the sloshing pressure near the free liquid surface, but this only occurred after the violent ground
motion passed. The pressure and free surface elevation had similar trends due to the seismic excitation.
With the damping baffle, the maximum response of the sloshing surface and the pressure at the
selected points P1 and P2 were suppressed by 16.8%, 5.0%, and 3.5% under the IV earthquake and
37.2%, 0.5%, and 4.2% for the SF earthquake. Then, the conclusion can be drawn that the suppression
of sloshing pressure by the vertical damping baffle was limited under the nonlinear seismic excitation,
although it can suppress the water motion well. That is probably because the vertical baffle suppressed
the violent sloshing through its function of resonant frequency tuning. Since the nonlinear seismic
excitation and the structural differences in size and the filling level may cause the different pressure
response and the water motion, the results of the baffle configuration in the previous sloshing studyWater 2020, 12, 2098 11 of 14
cannot be applied directly to the seismic design of the tank. Therefore, analysis of the water motion
Water 2020, 12, x FOR PEER REVIEW 11 of 14
and pressure within a water storage tank under an earthquake load is still needed.
Figure 14. The suppression effect of the vertical damping baffle on sloshing water, (a) the free surface
elevation under the Imperial Valley 1940 earthquake, (b) the free surface elevation under the San
Fernando 1971 earthquake.
On the other hand, the sloshing pressure on the two measuring points with and without the
damping baffle was compared. The two measuring points were located near the bottom and the free
surface of the water tank, which were 1 m and 10 m away from the bottom, respectively. Then, the
pressure variation at these two points is shown in Figure 15, and the sloshing pressures were
normalized by p/ρgh-1 to express the fluctuation of sloshing pressure. Here, the ρgh was the
hydrostatic pressure at the selected points. The results showed that the pressure variation ratio near
the water surface was larger than that in the bottom area. The damping baffle had a significant
Figure 14.
suppression Theon
effect suppression effect
the sloshing of the vertical
pressure near thedamping baffle
free liquid on sloshing
surface, water,
but this only(a) the free after
occurred
Figure 14. The suppression effect of the vertical damping baffle on sloshing water, (a) the free surface
the violent ground motion passed. The pressure and free surface elevation had similar trendsthe
surface elevation under the Imperial Valley 1940 earthquake, (b) the free surface elevation under due to
elevation under the Imperial Valley 1940 earthquake, (b) the free surface elevation under the San
San Fernando
the seismic 1971 earthquake.
excitation.
Fernando 1971 earthquake.
On the other hand, the sloshing pressure on the two measuring points with and without the
damping baffle was compared. The two measuring points were located near the bottom and the free
surface of the water tank, which were 1 m and 10 m away from the bottom, respectively. Then, the
pressure variation at these two points is shown in Figure 15, and the sloshing pressures were
normalized by p/ρgh-1 to express the fluctuation of sloshing pressure. Here, the ρgh was the
hydrostatic pressure at the selected points. The results showed that the pressure variation ratio near
the water surface was larger than that in the bottom area. The damping baffle had a significant
suppression effect on the sloshing pressure near the free liquid surface, but this only occurred after
the violent ground motion passed. The pressure and free surface elevation had similar trends due to
the seismic excitation.
Figure 15.
Figure The fluctuation
15. The fluctuation of
of sloshing
sloshing pressure
pressure at
at selected
selected points.
points.
4. Conclusions
The water motion in a real-scale water storage tank under two well-studied ground motions
were investigated based on the OpenFOAM in this study. The experimental tests were conducted for
validation of the present numerical model. The larger tank size, the high filling level, and the nonlinear
seismic excitation were considered in this research. Other than the previous studies on sloshing in
Figure 15. The fluctuation of sloshing pressure at selected points.Water 2020, 12, 2098 12 of 14
a rectangular water tank, the results under the current conditions were slightly different in terms of
baffle configuration design, pressure response, and water sloshing.
(1) A comparison between 2D and 3D results of the water sloshing in a cylindrical tank was
conducted. The 2D analysis could be used to predict the water sloshing response in a cylindrical tank.
The Oxy section gave the maximum free surface elevation of the sloshing, and the most dangerous
condition can be covered.
(2) Due to the huge size of the water tank and the large mass of the filled water, the free surface
motion was small, even under the strong ground motion. The maximum free surface elevation did not
exceed 10%.
(3) Similar to the response under the harmonic excitation, the free surface motion still satisfied the
sloshing modal characteristic under seismic excitation. The maximum free surface elevation appeared
at the side wall, and there were two nodes when the seismic frequency was close to the third-order
natural frequency of the water tank.
(4) In addition, the water motion was hysteretic compared with the ground motion. The maximum
free surface elevation appeared after the seismic wave passed instead of the seismic wave was passing.
Meanwhile, the pressure response was different, which kept decreasing after the strong seismic
wave passed.
(5) The effect of the well-used vertical damping baffle on the suppression of sloshing pressure was
limited when the strong ground motion was passing. Further study about the seismic design of the
water storage tank is still needed.
Author Contributions: Writing—original draft, H.J.; Writing—review and editing, R.S. and Y.L. All authors have
read and agreed to the published version of the manuscript.
Funding: This research was funded by the Natural Science Foundation of Zhejiang, grant number LQ19E090005,
the Natural Science Foundation of Ningbo, grant number 2019A610164 and National Natural Science Foundation
of China, grant number 51605431.
Conflicts of Interest: The authors declare no conflict of interest.
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