BARC/2015/E/001 BARC/2015/E/001 LIQUID SLOSHING IN GRAVITY DRIVEN WATER POOL OF ADVANCED HEAVY WATER REACTOR: POOL LIQUID UNDER DESIGN SEISMIC LOAD AND SLOSH CONTROL STUDIES by M. Eswaran and G.R. Reddy Structural and Seismic Engineering Section, Reactor Safety Division
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BARC/2015/E/001B
AR
C/2015/E
/001
LIQUID SLOSHING IN GRAVITY DRIVEN WATER POOL OFADVANCED HEAVY WATER REACTOR:
POOL LIQUID UNDER DESIGN SEISMIC LOAD AND SLOSH CONTROL STUDIES
byM. Eswaran and G.R. Reddy
Structural and Seismic Engineering Section,Reactor Safety Division
BARC/2015/E/001
GOVERNMENT OF INDIAATOMIC ENERGY COMMISSION
BHABHA ATOMIC RESEARCH CENTREMUMBAI, INDIA
2015
BA
RC
/201
5/E
/001
LIQUID SLOSHING IN GRAVITY DRIVEN WATER POOL OFADVANCED HEAVY WATER REACTOR:
POOL LIQUID UNDER DESIGN SEISMIC LOAD AND SLOSH CONTROL STUDIES
byM. Eswaran and G.R. Reddy
Structural and Seismic Engineering Section,Reactor Safety Division
BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT(as per IS : 9400 - 1980)
01 Security classification : Unclassified
02 Distribution : External
03 Report status : New
04 Series : BARC External
05 Report type : Technical Report
06 Report No. : BARC/2015/E/001
07 Part No. or Volume No. :
08 Contract No. :
10 Title and subtitle : Liquid sloshing in gravity driven water pool of Advanced Heavy Water Reactor - Pool liquid under design seismic load and slosh control studies
11 Collation : 84 p., 67 figs., 13 tabs.
13 Project No. :
20 Personal author(s) : M. Eswaran; G.R. Reddy
21 Affiliation of author(s) : Structural and Seismic Engineering Section, Reactor Safety Division, Bhabha Atomic Research Centre, Mumbai
22 Corporate author(s) : Bhabha Atomic Research Centre, Mumbai - 400 085
23 Originating unit : Reactor Safety Division, Bhabha Atomic Research Centre, Mumbai
24 Sponsor(s) Name : Department of Atomic Energy
Type : Government
Contd...
BARC/2015/E/001
BARC/2015/E/001
30 Date of submission : January 2015
31 Publication/Issue date : February 2015
40 Publisher/Distributor : Head, Scientific Information Resource Division, Bhabha Atomic Research Centre, Mumbai
Abstract : Sloshing phenomenon is well understood in regular cylindrical and rectangular liquidtanks subjected to earthquake. However, seismic behaviour of water in complex geometry suchas a sectored annular tank, e.g., Gravity Driven Water Pool (GDWP) which is located in AdvancedHeavy Water Reactor (AHWR) need to be investigated in detail in the view of safety significance.Initially, for validation of Computational Fluid Dynamics (CFD) procedure, square and foursectored square tanks are taken. Slosh height and liquid pressure are calculated over time throughtheoretical and experimental procedures. Results from theoretical and experimental approachesare compared with CFD results and found to be in agreement. The present work has two mainobjectives. The first one is to investigate the sloshing behaviour in an un-baffled and baffledthree dimensional single sector of GDWP of AHWR under sinusoidal excitation. Other one is tostudy the sloshing in GDWP water using simulated seismic load along the three orthogonaldirections. This simulated seismic load is generated from design basis floor response spectrumdata (FRS) of AHWR building. For this, the annular tank is modelled along with water andnumerical simulation is carried out. The sinusoidal and earthquake excitations are applied asacceleration force along with gravity. For the earthquake case, acceleration-time history isgenerated compatible to the design FRS of AHWR building. The free surface is captured byVolume of Fluid (VOF) technique and the fluid domain is solved by finite volume method whilethe structural domain is solved by finite element approach. Un-baffled and baffled tankconfigurations are compared to show the reduction in wave height under excitation. Theinteraction between the fluid and pool wall deformation is simulated using a partitioned fluid–structure coupling. In the earthquake case, a user subroutine function is developed to convertFRS in to time history of acceleration in three directions. Wavelet analysis is performed to findthe frequency variation of sloshing with respect to time. Results such as sloshing heights andhydrodynamic pressure considering with and without structure interaction effects have beenpresented
ii
ABSTRACT
Sloshing phenomenon is well understood in regular cylindrical and rectangular liquid
tanks subjected to earthquake. However, seismic behaviour of water in complex geometry
such as a sectored annular tank, e.g., Gravity Driven Water Pool (GDWP) which is located
in Advanced Heavy Water Reactor (AHWR) need to be investigated in detail in the view
of safety significance. Initially, for validation of Computational Fluid Dynamics (CFD)
procedure, square and four sectored square tanks are taken. Slosh height and liquid
pressure are calculated over time through theoretical and experimental procedures. Results
from theoretical and experimental approaches are compared with CFD results and found
to be in agreement.
The present work has two main objectives. The first one is to investigate the sloshing
behaviour in an un-baffled and baffled three dimensional single sector of GDWP of
AHWR under sinusoidal excitation. Other one is to study the sloshing in GDWP water
using simulated seismic load along the three orthogonal directions. This simulated seismic
load is generated from design basis floor response spectrum data (FRS) of AHWR
building. For this, the annular tank is modelled along with water and numerical simulation
is carried out. The sinusoidal and earthquake excitations are applied as acceleration force
along with gravity. For the earthquake case, acceleration-time history is generated
compatible to the design FRS of AHWR building. The free surface is captured by Volume
of Fluid (VOF) technique and the fluid domain is solved by finite volume method while
the structural domain is solved by finite element approach. Un-baffled and baffled tank
configurations are compared to show the reduction in wave height under excitation. The
interaction between the fluid and pool wall deformation is simulated using a partitioned
fluid–structure coupling. In the earthquake case, a user subroutine function is developed to
convert FRS in to time history of acceleration in three directions. Wavelet analysis is
performed to find the frequency variation of sloshing with respect to time. Results such as
sloshing heights and hydrodynamic pressure considering with and without structure
interaction effects have been presented.
iii
CONTENTS
Contents
ABSTRACT ........................................................................................................................................ ii
CONTENTS ....................................................................................................................................... iii
LIST OF FIGURE .............................................................................................................................. v
Fluid in each sector has different dynamic characteristics, however some of the sectors
are symmetric. As shown in sector arrangement Fig. 4.4, the each sector has different
slosh frequencies. The slosh frequencies for sectors, 1 and 5 are same. Also, similarly
sectors 2, 4, 6 and 8 are same and remaining sectors 3 and 7 are same slosh frequencies.
These frequencies are found using ERM approach and listed in Chapter 5 along with
frequencies from CFD approach for comparison.
Fig. 4-3 Sectional view of GDWP (updated
dimensions) Fig. 4-4Plan of GDWP
4.2 Equivalent Rectangular Method (ERM): Since the design code for GDWP (8- sectored annular water pool with outer spherical
wall) geometry is not available directly in seismic design code, as instructed [31], the one
sector s taken alone and ERM analysis is made. The rectangular domain is constructed as
shown in Fig. 4.5. While converting from GDWP fluid domain to equivalent rectangular
domain the following assumptions are taken.
(i) Liquid volume should be equal in GDWP sector and rectangular model.
(ii) Ratio between liquid height (h) and length of the tank (L) is taken almost equal to
the GDWP sector dimensions.
Hydrodynamic forces exerted by liquid on tank wall shall be considered for analysis in
addition to hydrostatic forces. These hydrodynamic forces are evaluated with the help of
spring mass model of rectangular tanks as shown in Fig. 4.6.
Excitation direction
1
23
4
5
67
8
15
Fig. 4-5Equivalent rectangular model Fig. 4-6Spring –mass model
4.3 Spring-Mass Damper Model
For the purposes of incorporating the dynamic effects of sloshing in the pools, it is
convenient to replace the liquid conceptually by an equivalent linear mechanical system.
The equations of motion of oscillating masses and rigid masses are included more easily in
the analysis than are the equations of fluid dynamics. Fig. 4.6 illustrates generalized
spring-masses model for the rectangular tank and the symbols used in the analysis. The
width of the tank are 2L and height of the liquid is denoted as h. The center of the mass of
the liquid is represented as C.G, while, the locations (Hn) of the masses are references to
the C.G. The tank is excited by a small time-varying linear acceleration . Rigidly
attached mass is denoted as m0, while the convective (slosh) masses are showed as m1
through mn .The deflection of the mass is represented as xn which is relative to the tank
walls as a result of the tank motion.
The mathematical equations can be derived from static and dynamic properties of spring-
mass model. These equations and derivations can be found in ACI, Housner (1963) and
Dodge (2000) for simple geometries like rectangular and cylindrical tanks. According to
static properties, the sum of all the masses must be equal to the liquid mass and center of
mass of the model must be same elevation as the liquid. These can be derived as follows,
m0+ m1+ m2+…..+ mn = mliq (4.1)
Z
H = 8m
W=11 mL= 11.5 m
x
yRectangular model
GDWP/ sector
(11.5 x11 x 8) = 1012 m3
1000 m3
16
. m0H0+ m1H1+ m2H2+ ….+ mnHn = 0 at C.G (4.2)
Equation of motion can be derived by inserting the acceleration terms and applying
static properties into the force equation. The equations of motion for each of the spring-
masses is expressed as
0 (4.3)
From the above equations forces acting on the rectangular tank can be estimated. The
slosh height is estimated using the well derived equations as listed here (Eqs. 4.1 to 4.8).
The convective and impulsive masses are,
.)12(
/)12tanh(833 hn
LhnLmm liqn −−
=π
π (4.4)
.)12(
/)12tanh(81
330 ⎥⎦
⎤⎢⎣
⎡−−
−=hn
LhnLmm liq ππ
(4.5)
Height of the convective masses and natural frequency,
.)12(
/)12tanh(8)12(
sinh
12/)12tanh(
)12(2 33 hnLhnL
Lhn
Lhnn
LhH n −−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
−−=
ππ
ππ
π (4.6)
⎟⎠⎞
⎜⎝⎛=
LH
Lg
58.1tanh58.1ω (4.7)
The first mode slosh height,
1
)58.1coth(527.0
2
max,1
−==
Lg
LhLd
h
n
θω
where, n =1, (4.8)
Based on the above formula, the slosh height and other parameters have been calculated
and tabulated for sector 1 of fully curved outer wall GDWP and partially curved outer wall
GDWP in Tables 4.1 and 4.2 respectively.However the detailed computations for both
models are shown in Tables 4.3 and 4.4.
17
Table 4-1: Slosh height estimation through seismic codes for sector 1 of fully curved outer
wall GDWP
Length direction Width direction unit
Length of pool 2L 11.5 11 m
Width of pool W 11 11.5 m
Height of pool 10 10 m
Height of water 8 8 m
Density of water 1000 1000 kg/m3
Mass of water 1.01E+06 1.01E+06 kg
Volume of water 1012 1012 m3
Convective acceleration 0.13 0.145 g
Impulsive acceleration 1.11 1.88 g
Convective acceleration 1.275 1.422 m/sec2
Displacement (A) 0.485 0.515 m
Convective
frequency
ω 1.622 1.662 rad/sec
f 0.258 0.265 Hz
Slosh height
TID-7024 0.778 0.856 m
Slosh height
ACI 350.3 (2001) 0.748 0.798 m
18
Table 4-2: Slosh height estimation through seismic codes for sector 1 of partially curved
outer wall GDWP
Length direction Width direction unit
Length of pool 2L 13.5 12.5 m
Width of pool W 12.5 13.5 m
Height of pool 8 8 m
Height of water 6 6 m
Density of water 1000 1000 kg/m3
Mass of water 1.01E+06 1.01E+06 kg
Volume of water 1012.5 1012.5 m3
Convective acceleration 0.13 0.145 g
Impulsive acceleration 1.11 1.88 g
Convective acceleration 1.275 1.422 m/sec2
Displacement (A) 0.627 0.632 m
Convective
frequency
ω 1.427 1.501 rad/sec
f 0.227 0.239 Hz
Slosh height
TID-7024 0.893 0.953 m
Slosh height
ACI 350.3 (2001) 0.878 0.906 m
4.4 Result and Discussions
The fully curved wall GDWP slosh height and other calculations are shown in Table 4.2,
while slosh calculations are shown for partially curved wall GDWP in Table 4.3.The
approved model of GDWP’s outer wall is fully curved (Fig. 4.5). During internal review
meeting, a partially curved wall GDWP model (Fig. 4.6) is proposed (at AHWR Review
Meeting 201, BARC). To address the both cases, two cases are studied and tabulated in
Tables 4.2 and 4.3.
19
Table 4-3: X and Y direction values of sector 1 of fully curved outer wall GDWP.
Sl. No
Term
Longitudinal
(x)direction
Lateral (y) direction
Unit Comments
1 Impulsive mass (mi) 6.88E+05 7.06E+05 kg Sum of Impulse and convective mass is slightly higher than total mass of fluid. 2 Convective mass (mc) 3.75E+05 3.60E+05 kg
3
Height of the impulsive mass above the bottom of the tank wall (hi) (without considering base pressure)
3.00 3.00 m See the Fig. 4.7
4
Height of the convective mass above the bottom of the tank wall (hc)(without considering base pressure)
5.09 5.15 m See the Fig. 4.6
5
Height of the impulsive mass above the bottom of the tank wall (hi*)(with considering base pressure)
4.88 4.73 m See the Fig. 4.7
6
Height of the convective mass above the bottom of the tank wall (hc*)(with considering base pressure)
5.91 5.87 m See the Fig. 4.6
7 Wall deflection (d) due to load
0.0192 0.0180 m Considered as fixed at three edges and free at top
8 Impulsive frequency (fi)
3.600 3.715 hz ACI 350.3 (2001)
9 Convective frequency (fc)
0.243 0.255 hz ACI 350.3 (2001)
10
Seismic co-efficient (Ah)
FRS at 137 m height of AHWR building. Impulsive (Ah)i 1.11g 1.88g
Convective (Ah)c 0.13g 0.145g
20
11
Total shear force (V) at bottom of the wall
13765.85 23402.02 KN Lateral base shear 23.2 % of total seismic weight in x direction while same in 24 % in y direction
Impulsive Vi 13757.55 23396.42 KN
Convective Vc 477.86 512.11 KN
12
Total bending moment at bottom of the wall (M)
50.72 85.79 MN SRSS rule as followed in all international code except Eurocode 8 (1988).
Impulsive Mi 50.66 131.71 MN Convective Mc 2.43 3.52 MN
13 Over turning moment at bottom of base slab. (M*)
78.59 131.76 MN-m Housner(1963)
14 Impulsive time period (Ti)
0.28 0.27 sec ACI 350.3 (2001) and NZS 3106 (1986)
15 Convective time period (Tc)
4.11 3.92 sec ACI 350.3 (2001) Housner (1963)
16 Slosh height (hs) 0.748 0.798 m Free board is 1 m (Importance factor 1 )
17
Impulsive pressure on wall (y=0) (Piw)
63.89 106.15 KN/m2 ACI 350.3 Housner(1963) See the Fig. 4.7
Impulsive pressure on top of base (y=0) (Pib)
30.74 51.83 KN/m2
18
Convective pressure on wall (y=0) (Pcw)
1.34 1.29 KN/m2 ACI 350.3 Housner(1963) See the Fig. 4.7
Convective pressure on wall (y=h) (Pcwt) 6.11 6.52 KN/m2
Convective pressure on top of base (y=0) (Pcb)
1.34 1.29 KN/m2
19
Pressure due to wall inertia (Pww)
13.61 23.05 KN/m2 ACI 350.3 ; Housner(1963) Pressure due to
vertical excitation (Pv) 41.20 41.20 KN/m2
20 Maximum hydrodynamic pressure (P)
87.78 135.62 KN/m2 Hydro static pressure is 78.8KN/m2.
21
Table 4-4: X and Y direction values of sector 1 of partially curved outer wall GDWP.
Sl. No
Term
Longitudinal
(x)direction
Lateral (y) direction
Unit Comments
1 Impulsive mass (mi) 4.99E+05 5.32E+05 kg
Sum of Impulse and convective mass is slightly higher than total mass of fluid.
2 Convective mass (mc) 5.33E+05 5.06E+05 kg
3
Height of the impulsive mass above the bottom of the tank wall (hi) (without considering base pressure)
2.25 2.25 m See the Fig. 4.7
4
Height of the convective mass above the bottom of the tank wall (hc)(without considering base pressure)
3.41 3.47 m See the Fig. 4.8
5
Height of the impulsive mass above the bottom of the tank wall (hi*)(with considering base pressure)
5.34 4.96 m See the Fig. 4.7
6
Height of the convective mass above the bottom of the tank wall (hc*)(with
considering base pressure)
5.67 5.31 m See the Fig. 4.8
7 Wall deflection (d) due to
load 0.0068 0.0060 m Considered as fixed at
three edges and free at top
8 Impulsive frequency (fi) 6.063 6.433 hz ACI 350.3 (2001)
9 Convective frequency (fc) 0.224 0.239 hz ACI 350.3 (2001)
10
Seismic co-efficient (Ah) FRS at 137 m height of AHWR building.
Impulsive (Ah)i 1.11g 2.16g
Convective (Ah)c 0.13g 0.145g
11
Total shear force (V) at bottom of the wall
11224.95 22294.37 KN
Lateral base shear 23.2 % of total seismic weight in x direction while same in 24 % in y direction
Impulsive Vi 11204.35 22282.76 KN
Convective Vc 679.80 719.36 KN
22
12
Total bending moment at bottom of the wall (M)
32.51 63.96 MN SRSS rule as followed in all international code except Eurocode 8(1988).
Impulsive Mi 32.42 116.76 MN
Convective Mc 2.32 4.54 MN
13 Over turning moment at
bottom of base slab. (M*) 60.57 116.85 MN-m Housner (1963)
14 Impulsive time period (Ti) 0.16 0.16 sec ACI 350.3 (2001) and NZS 3106 (1986)
15 Convective time period
(Tc) 4.46 4.18 sec
ACI 350.3 (2001) Housner (1963)
16 Slosh height (hs) 0.878 0.906 m Free board is 1 m (Importance factor 2 )
17
Impulsive pressure on wall (y=0) (Piw)
54.33 104.29 KN/m2
ACI 350.3 Housner (1963) See the Fig. 4.7
Impulsive pressure on top of base (y=0) (Pib)
20.73 41.95 KN/m2
18
Convective pressure on wall (y=0) (Pcw)
3.32 3.10 KN/m2
ACI 350.3 Housner (1963) See the Fig. 4.7
Convective pressure on wall (y=h) (Pcwt) 7.17 7.41 KN/m2
Fig. 5.4 shows the numerical results for the liquid heights at the left, right and center point
in a three-dimensional GDWP subject to harmonic motions under 0.1 m amplitude for
different frequencies.
27
Table 5-3: Numerical cases taken for investigation.
Sl. No Excitation
Sectors consider
ed in GDWP
tank
Baffle Excitation Amplitude
(m)
Excitation
direction
Excitation Frequency
(Hz)*
Condition
1
Sinusoidal excitation
1 (Fully curved outer wall
GDWP)
No baffle
0.01
x
1ω
Fixed wall
2 0.02 1ω
3 0.06 1ω
4
0.1
1ω 5 0.5 1ω
6 0.8 1ω
7 1.2 1ω
8 0.1 1ω Flexible wall
9 Annul
ar 0.1
1ω Flexible wall
10 5 1ω
11 10 1ω
12 Cap-Plate
0.1 1ω Flexible
wall 13 5 1ω
14 10 1ω
15 Sinusoidal excitation
3 (partially curved outer wall
GDWP)
No baffle
0.03 xyz 1ω Flexible wall
16 Earthquake excitation
3 (partially curved outer wall
GDWP)
No baffle
Max. floor response acceleration
xyz Design FRS data
Flexible wall
*First mode natural frequency of liquid is 1ω =0.312 Hz.
When the excitation frequencies are close to the natural frequency as shown in Figs. 5.4
(a) and (c), the beat phenomena are noticeable (Eswaran et al., 2009). It can be observed
from Fig. 5.4 (d) that when the excitation frequency is far-off from the natural frequency,
i.e., 1.92 rad/sec, the liquid heights are very small and frequency is equal to excitation
28
frequency. When the frequency is almost near to the first mode natural frequency, i.e. Fig.
5.4 (b), the amplitude grows monotonically with time. There is a slight difference in liquid
elevation between right and left corner of the water pool.
Fig.5-4 Time history of free surface elevation at 0.1 m excitation amplitude and
for different excitation frequencies.
0 2 4 6 8 10 12 14 16 18 20 22 24-2
-1
0
1
2
Right Left Center
Time (Sec)
-2
-1
0
1
2
Right Left Center
Free
sur
face
Ele
vatio
n (m
)
-2
-1
0
1
2
Right-2
-1
0
1
2
ωx/ω1 = 0.5
ωx/ω1 = 0.8
ωx/ω1 = 0.99
ωx/ω1 = 1.2
(d)
(c)
(b)
(a)
Right Left Center
29
6 EXPERIMENTAL AND ANALYTICAL VALIDATION OF CFD
SIMULATIONS
In this chapter, slosh height is computed through experimental and analytical methods
and compared with CFD procedure for validation.
6.1 Case Study 1: Analytical Validation
In this section, a 2D model partially filled tank has been taken and the liquid elevation has
been captured under sinusoidal excitation by numerical as well as analytical relation. The
2-D rigid tank which is 570 mm long and 300 mm high is excited with )sin( tA ω as shown
in Fig. 6.1. The water depth is 150 mm and excited amplitude is 5 mm. The lowest natural
frequency 1ω for this case is 6.0578 rad/sec. The natural frequency is calculated from
Equation 6.2. Liquid free surface elevation has been calculated from the following third
order analytical relations (Faltinsen et al., 2000) and compared with the present numerical
simulation results for frequency ratio 0.583.
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−−
+
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−
−−
++
+= )2cos(
)cos()4(
3
2
1)2cos(*
)4(2
33
8
1
8
1
)cos()cos(),(
222
22
22422
2
22
22
224
4
224
4
224
2
xk
tkgt
kgkgkg
gaxktAtx n
nnnn
nnnnn
nnn
nn
n
nn
n
nn
nnn
ωωωω
ωωωω
ωωωω
ωω
ωω
ωωζ (6.1)
where, the linear sloshing frequencies
)tanh( snnn hkgk=ω and )2tanh(22 snnn hkkg=ω . (6.2)
The initial conditions are )cos(),(0
xkanx xt=
=ζ and 0),(
0=
=tzxφ , where A is the
amplitude of the initial wave profile, bnk n /π= is the wave profile for n = 0,1,2... , hs is
still water level and x is the horizontal distance from the left wall. This analytical result is
30
compared with the present numerical approach. From Fig. 6.2, it is observed that the free
surface elevation of analytical and present numerical coincides with each other.
Fig.6-1 The sketch of the 2-D rigid rectangular
tank. Fig.6-2 Comparisons of free surface elevation.
Fig.6-3 Details of the experimental setup.
6.2 Case Study 2: Experimental Validation
This case study shows the comparison of experimental and numerical results. For this
purpose, a model square tank with sectored arrangement was built and experiments were
conducted. The experiments were performed on a shake table (1.2 m x 1.0 m) coupled
with a servo-controlled hydraulic actuator of 250 KN capacity. The test setup is specially
designed for the sloshing experiments as shown in Fig. 6.3. The perspective view of the
setup is shown in Fig. 6.4 with the actuator coupling arrangement.
570
Probe
20
All dimensions are in mm
Water
Air
300xy
Time (Sec)
Free
surfa
ceel
evat
ion
(m)
0 2 4 6 8 10-0.015
-0.01
-0.005
0
0.005
0.01
0.015
AnalyticalPresent Numerical
Platform
Liquid A sin (ω t)
DAS
10020
0
500250
1000
A
arrang
all dir
install
Water
(a) S
Fig.6
10
20
30
40
50
60
70
80
Pres
sure
(Pa)
square tan
gement as s
rections. Pr
led at 100 m
r fill level in
Fig.6-4
Square tank
6-6 Compari
0 1 20
00
00
00
00
00
00
00
00
nk has 1 m
shown in Fi
ressure var
mm (positio
n the tank is
4 Experimen
at excitatio
1ω .
ison of expe
3 4 5Time (Sec
m length an
ig. 6.5 for th
riations are
on 1) and 2
s maintained
ntal setup.
on frequency
erimental an
(
6 7 8
Numerical Experimenta
c)
31
nd 0.5 m h
he experim
e sensed by
200 mm (po
d as 250 mm
y 0.57 (
nd CFD pre
(position 2)
9 10
al
Pres
sure
(Pa)
height alon
ments in orde
y two flush
osition 2) f
m.
Fig.6-5 Is
(b) Sectored
fr
essure data a
).
0 1 20
100
200
300
400
500
600
700
800
Pres
sure
(Pa)
ng with rem
er to allow
h type pre
from the bo
sometric vie
square ta
d square tan
requency 0.
at 200 mm f
2 3 4 5
Time (
movable 4
liquid moti
ssure trans
ottom of the
ew of 4-sect
ank
nk with exci
99 1ω .
from tank b
6 7 8
Sec)
Numerical Experimen
Baffle
Wate
-sector
ions in
sducers
e tank.
tored
tation
ottom
9 10
ntal
er
32
Fig.6-7Actuator with shake table Fig.6-8 Water oscillation during the
excitation
Fig.6-9Water spill out snapshots during base excitation in an 4-sectored rectangular tank
33
For this case the liquid natural frequency has been calculated as 0.71 Hz through
analytical relation and sine sweep experiments. The comparison of experimental and
numerical time history of pressure at position 2 for square tank and 4- sectored square tank
cases are shown in Fig. 6.6 (a) and (b) respectively. For these cases, the excitation
frequency ratio is taken as 0.57 and 0.99 of the first mode square tank. The comparisons of
numerical with experiment results are shown and found the CFD results are good in
agreement with the experiment. Figs. 6.7 through 6.9 show the snap shots of the
experimental setup to display the tank and actuator arrangement and water spill outs.
In this chapter, experimental and analytical studies are also performed to validate present
numerical results. For experimental validation, a simple square and four-sectored squared
tanks was taken. The pressure variations were captured at different locations under the
surge motions of the tank and found the CFD results are good in agreement with the
experiment.
34
7 CFD SIMULATION OF LIQUID SLOSHING CONTROL IN
GDWP UNDER SINUSOIDAL EXCITATION
In this chapter, the detailed sloshing studies are carried out for studying the effect of
amplitude on liquid sloshing and structural analysis to compute the wall displacement and
induced stress. Un-baffled and baffled tank configurations are compared to show the
reduction in wave height under sinusoidal excitation. For this, annular baffle and cap-plate
baffles are taken for analysis. The slosh height is compared between un-baffled and
baffled configurations under design acceleration.
7.1 Effect of Amplitude of GDWP
Fig. 7.2 depicts the effect of excitation amplitude (A) under its first mode natural
frequency. For this purpose, the non-dimensional free surface elevation is captured at the
right corner of the water pool for 20 seconds. If the excitation amplitude is increased, the
fluid response becomes large. Fig. 7.2 is plotted with the assumption of with and without
fluid-structure interaction conditions. In the case of without FSI, the boundaries are
considered as rigid wall. It is also found in the Fig. 7.3 that FSI consideration has slight
more free surface elevation than the without FSI. It is caused due to the interaction of the
fluid domain with structure produces the relative pressure component. However, there is a
large gap between the first mode frequency of the structure and fluid portions. Deviations
are not high as the excitation frequency is low and faraway from the structural first mode
frequency. It is also observed that the amplification of the fluid motion is relatively larger
at lower amplitude while at the higher amplitude; the amplification of free surface
elevation is less than the lower amplitude case. Fig. 7.1 shows the power spectral density
of liquid elevation wave at 0.1 m excitation amplitude. Closer to natural frequency, a
single dominant frequency is absorbed. The phase-plane diagram is plotted in Fig. 7.3,
shows that non-linearity exists in the flow.
35
Fig.7-1Power spectral density for amplitude 0.1 m and 1ω =0.312 Hz.
Fig.7-2Time history of non-dimensional free surface elevation at 1ω =0.312 Hz for different excitation amplitudes
Fig.7-3Phase-plane diagram for amplitude 0.1 m and 1ω =0.312 Hz.
Pressure waves are captured in different locations of the water pool and the locations
(i.e., A through E) are depicted in Fig. 7.4. Positions A through C are 1 m below from the
liquid free surface and positions D and E are in 5 m and 8 m from the free surface
respectively. The time histories of pressure at the different places of the water pool for un-
baffled water pool are plotted in Fig. 7.5 for 0.1 m amplitude and 1ω =0.312 Hz. It can be
seen from the Fig. 7.5 that when the water pool is excited, the impulse pressures occur
because of the relatively large amplitude of the external excitation. If liquid oscillation is
not controlled efficiently, sloshing of liquids in storage water pools may lead to large
dynamic stress to cause structural failure. On the other hand, if the baffle exists in the
water pool, the dynamic pressure will be minimal. The horizontal displacement histories
0 2 4 6 8 10 12 14 16 18 20-40
-20
0
20
40
Amplitute = 0.01 m
Time (Sec)
-40
-20
0
20
40
Amplitute = 0.02 m
ξ / A -40
-20
0
20
40
(d)
(c)
(b)
(a)
Amplitute = 0.06 m
-40
-20
0
20
40
Amplitute = 0.1 m (FSI) Amplitute = 0.1 m (without FSI)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.000
0.005
0.010
0.015
0.020
0.025
Frequency (Hz)
Pow
er
ζ/A
(∂ζ/
∂t)/(
A*ω
)
-15 -10 -5 0 5 10 15 20-1500
-1000
-500
0
500
1000
1500
36
of the container inner and outer walls are drawn in Figs. 7.6 and 7.7. The Fig. 7.1 shows
that the steady state values are reached from around 12 sec for 0.1 m excitation amplitude.
The displacement is captured at 12.04 sec. The displacement frequency is almost equal to
wave frequency. And it can be seen that the horizontal displacement is symmetric in both
side walls as shown in Fig. 7.6.
Fig.7-4Pressure point locations. Fig.7-5Time history of pressure at various locations for 0.1 m amplitude sec and 1ω=0.312 Hz.
Fig.7-6Wall horizontal displacement contour at time 1.417 sec
Fig.7-7Outer and inner walls x-displacement versus time
0 2 4 6 8 10 12 14 160
20
40
60
80
100
120 Position A Position B Position C Position D Position E
Pres
sure
(Kpa
)
Time (Sec)
0 5 10 15 20
-0.0010
-0.0005
0.0000
0.0005
0.0010 Displacement of outer wall Displacement of inner wall
Wal
l dis
plac
emen
t in
x di
rect
ion
(m)
Time (Sec)
37
7.2 Structural Analysis
Fig. 7.8 shows the VonMises stress on water pool wall at 12.4 sec. The maximum stress
created on the side wall is observed as 7 X 106 N/m2. Sloshing occurs primarily at the
liquid surface and oscillates, exerting forces on the tank structure walls. The liquid free
surface profile has a positive gradient when it moves towards right side. As soon as the
free surface elevation reaches its peak at the right wall, fluid vertical velocity will become
zero. Then, due to its own gravity and external applied forces move the liquid free surface
to down. Now, the direction of the fluid velocity switches from right to left and the
magnitudes of these velocities continue to increase until they reach their maximum. This
cycle will continue until the free surface stops its oscillations which may happen due to
the removal of external excitation to the system. Fig. 7.9 shows the velocity magnitude
from right to left at the time of 12.4 sec. During the surge motion of the water pool, a
single directional standing wave is moving upward and downward direction inside the
water pool.
Fig. 7-8VanMises stress contour at time 12.4 sec.
Fig. 7-9 Sectional view of velocity
magnitude at time 12.4 sec.
7.3 Effect of Baffles in GDWP
Tanks of asymmetric shapes and tanks with baffles, give rise to complications in fluid-
structure interaction, which is not amenable to analytical solution. The studies of liquid
sloshing in a tank with baffles are still very necessary (Eswaran and Saha, 2011; Xue et
al., 2012). Baffle is a passive device which reduces sloshing effects by dissipating kinetic
38
energy due to the production of vortices into the fluid. The linear sloshing in a circular
cylindrical tank with rigid baffles is being investigated by many authors in the context of
spacecraft and ocean applications. The shapes and positions need to be designed with the
use of either numerical model or experimental approaches. Nonetheless, the damping
mechanisms of baffle are still not fully understood. The effects of baffle on the free and
forced vibration of liquid containers were studied by Gedikli and Erguven (1999), Biswal
et al., (2006). To the author’s knowledge, there is a very limited set of analytically
oriented approaches to the sloshing problem in baffled tanks.
Here, two types of baffles are taken for analysis. First one is an annular baffle as
depicted in Fig. 7.10 (a) and (b). Few authors worked on this annular baffle for their own
geometries mainly two dimensional. This article is focused on annular baffle for a three
dimensional annular cylindrical water pool. Biswal et al. (2006) found that the baffle has
significant effect on the non-linear slosh amplitude of liquid when placed close to the free
surface of liquid. The effect is almost negligible when the baffle is moved very close to
the bottom of the tank. Past investigations also convey that the performance of the annular
baffle is better when it is near to the liquid free surface (Eswaran et al., 2009). Second one
is cap-plate baffle or shroud as shown in Fig. 7.10 (c) and (d) which is fixed at center of
the water pool. (More details about baffle for thermal stratification can be found in
Vijayan, 2010). Under reactor shutdown conditions, natural convection process starts due
to the strong heat source at the IC wall. Long-time effect of this natural convection
process leads to warm fluid layers floating on the top of gradually colder layers. This
results in a thermally stratified pool having steep temperature gradient along the vertical
plane. Over a period of time, the substantial part of this pool gets thermally stratified
except for the region close to the heat source where there is horizontal temperature
gradient as well (Gupta et al., 2009). Cap-plate model has been proposed to satisfy the
thermal stratification inside the water pool, since this water pool is mainly designed to
perform as a suppression pool to cool the steam and air mixture during LOCA in the
reactor vessel.
39
(a) Annular baffle. (b) Annular baffle in the water pool.
(c) Cap-plate baffle. (d) Cap-plate baffle (or shroud) in the water
pool.
Fig.7-10Baffles shape and its position in the water pool.
Liquid sloshing is violent near free surface and the liquid motion at the bottom of the
tank is almost zero. Here, one could assume that the mounting of the cap-plate does not
disturb the liquid sloshing as it is placed at bottom of the tank. The present work also
estimates and compares the annular and cap-plate baffle performance against the liquid
sloshing under the regular excitation. Fig. 7.11 illustrates that the comparison of liquid
elevations for un-baffle, annular baffle and cap-plate baffle cases. As expected, both the
baffle cases reduce the liquid oscillations as well. It is found that cap-plate baffle is more
effective in reducing the sloshing oscillations and sloshing pressure.
Volume = 7.3 m3
Thickness = 200 mmHeight = 250 mm
Volume = 36 m3
Thickness = 200 mmHeight = 5 m
40
Fig.7-11Comparison of liquid elevations with no baffle, annular baffle and cap-plate baffle cases at 1ω =0.312 Hz and amplitude 0.1 m.
Fig.7-12Effect of baffles at right corner of the water pool case at 15.05 sec and 1ω=0.312 Hz.
To elucidate the performance of baffles, at near right corner the liquid height is
captured and shown in Fig. 7.12. The liquid height is captured near the right corner of
water pool at 14.85 sec under the excitation frequency ( 1ω ) of 0.312 Hz. The liquid height
deviation for no baffle case is found at excitation amplitude between 0.01 m and 0.1 m is
0 5 10 15 20-2
-1
0
1
2
No baffle Annular baffle Cap-plate baffle
ζ (m
)
Time (Sec)
0.01 0.02 0.06 0.17.5
8.0
8.5
9.0
9.5
10.0
Line equivalent to design acceleration
Line of mean water level
Line of the top of the tank
Max
imum
Liq
uid
Hei
ght (
m)
Excitation Amplitude (m)
No baffle with FSI No baffle without FSI Annular Baffle with FSI Cap-plate Baffle with FSI
41
around 1.23 m. At the same time, this value for annular baffle and cap - plate baffle is
around 0.563 m and 0.182 m respectively. Moreover, it is found from the numerical
investigation that the liquid from the GDWP will spill out around 0.06 m excitation
amplitude ( ≈0.023 g acceleration) under liquid first mode frequency. The response
spectrum for the structure will give us design acceleration corresponding to first mode
frequency. Here, design acceleration is 0.16g at 0.312 Hz and corresponds to 0.028 m
equivalent harmonic amplitude. This line is shown as vertical in Fig. 7.12 to mark free
surface elevations for all cases. Baffle reduces the liquid slosh height 0.7 m to 0.3 m at
design acceleration as shown in Fig. 7.12. The Fig. 7.13 is drawn for qualitative
comparison between no baffle, annular and cap-plate baffle case. Here, snap shots of
liquid water pool (under regular excitation of 0.1 m amplitude) for different time step has
been shown.
7.4 Liquid Elevation in Higher Modes of GDWP
To study the effect of higher modes, both the annular and cap-plate baffled water pools
studied previously are analysed in this section. When the tanks are subjected to motions at
higher than first mode, the fluid in the tank will tend to undergo sloshing motions under
near to the same tank frequency. At the beginning of the disturbance, the fluid dynamic
pressure is dominated by the impulsive pressure. After few seconds, sloshing pressure or
convective pressure becomes the dominant component pressure. The small oscillations on
the pressure curve are the impulsive pressures. Figs. 7.14 and 7.15 show the free surface
elevation at 5 and 10 times of 1ω excitation frequency respectively for annular and cap-
plate baffle cases. Due to the strong impact forces at beginning, i.e., 0 - 2 seconds, liquid
rise is more, it reaches the steady state around 10 second on wards.
42
(a) Time at 0 sec (e) Time at 0 sec (i) Time at 0 sec
(b) Time at 14.6 sec (f) Time at 14.6 sec (j) Time at 14.6 sec
(c) Time at 15.6 sec (g) Time at 15.6 sec (k) Time at 15.6 sec
(d) Time at 16.6 sec (h) Time at 16.6 sec (l) Time at 16.6 sec
Fig.7-13Comparisons of free surface profile at different time instant for un-baffled and baffled water pool for 0.1 m amplitude sec and 1ω =0.312 Hz. (Figs. (a)-(d), (e)-(h), (i)-(l) show un-
baffled, annular baffled, cap-plate baffled water pools respectively).
43
Fig.7-14Comparison of free surface elevation of liquid at 0.1 m amplitude and 5
1ω excitation frequency.
Fig.7-15Comparison of free surface elevation of liquid at 0.1 m amplitude and
10 1ω excitation frequency.
0 2 4 6 8 10 12 14 16
-1.2
-0.8
-0.4
0.0
0.4
0.8
1.2
ζ (m
)
Time (Sec)
Annular Baffle Cap-plate baffle
0 2 4 6 8 10 12 14 16
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
ζ (m
)
Time (Sec)
Annular Baffle Cap-plate Baffle
44
8 CFD SIMULATION OF SLOSHING INGDWP UNDER SEISMIC
EXCITATION
In this chapter, a random wave is created using time histories compatible to design floor
response spectrum (FRS). For that, the floor response spectrum (FRS) at 137m height of
reactor building is used to generate the acceleration time history. A User-Defined Function
(UDF) subroutine has been developed to apply the random acceleration as a volume force.
The slosh height and forces on tank wall have been calculated for different excitation
directions. The combined effects in longitudinal and lateral directions are studied.
8.1 Introduction The sloshing studies are usually performed to ensure the safety of plants and to avoid
consequences of any seismic induced accidents.The mechanical mass–spring model
(Chapter 4) based on linear theory is generally employed by design standards to predict
the free surface displacement as well as other seismic design parameters of the liquid
storage tanks. However, nonlinear effects are always present and they occasionally
dominate the sloshing response. These nonlinear slosh effects arise mostly as a
consequence of large wave amplitudes (Chapter 7). Large amplitude waves may appear
when the great earthquakes are accompanied withpretty long period (3 to 10 sec)
components of seismic wave which coincide with the primary natural period of the
contained liquid (Goudarzia and Sabbagh-Yazdi, 2012). In this chapter, the floor response
spectrum (FRS) at 137m height of AHWR building is used to generate the acceleration
time history. This simulated earthquake accelearation data is used to study the sloshing
beheavior in GDWP.
The need to include nonlinearity in the hydrodynamics of the tank–liquid system arises
whenever high amplitude sloshing waves form on the liquid surface, leading to a nonlinear
influence of sloshing wave on the dynamic response of a tank. To estimate the non-linear
sloshing, the fluid momentum equations are solved as discussed in chapters 2 and 3. To
45
estimate the frequencies and slosh height in each sector of GDWP under seismic
excitation through CFD, three sectors are modelled among eight-sectored water pool as
depicted in Fig. 8.1. In each sector, three domains are modelled viz., water pool wall,
liquid and air domains. The sketch of the water pool are depicted as in Fig. 8.1.
8.2 Development of Random Waves
To study the response of GDWP under seismic load a time history is generated from FRS
along three orthogonal directions separately. FRS for AHWR building at 137 m is shown
in Fig. 8.2. The 5% broaden spectrum is generated from FRS. Acceleration time history is
generated using SIMQKE code [33] in three directions separately. These graphs are shown
in Fig. 8.3. A user subroutine function is developed to call the random acceleration data
and applied on the all fluid in terms of gravity force. Implicit pressure and implicit shear
stress conditions have been applied on the fluid solid interfaces. Air at top is at fixed
pressure condition (at atmosphere condition). The free surface elevation has been captured
every 0.005 sec.
8.3 Wavelet Analysis
In the past, the wavelet transform has been used to detect the frequencies at different
regions. In the field of fluid mechanics the wavelet analysis has been used to detect the
multi stable flow regions. In this problem, wavelet tool is used to find the sloshing
frequencies information. The Fourier transforms provide the spectral coefficients which
are independent of time i.e. they can give the amplitude-frequency information and donot
have any information about frequency with respect to time. Thus, it is useful only for a
stationary signal where the amplitude-frequency does not change with time. But, in real
life cases the signals are time dependent and also non-stationary. In such cases a scan
analysis using the Short Term Fourier Transform (STFT) is used but it has its limitations
like it can give information only about the amplitude and frequency, but not anything
about the time and frequency relation. The limitations of STFT are overcome by the
46
wavelet transform which gives a better idea about the time-frequency information about
the signals. The wavelet transform is a linear convolution of a given one dimensional
signal which is to be analysed and the mother wavelet (t). Mathematically a wavelet
transform is as shown below:
( ) dts
bttps
bsW ⎟⎠⎞
⎜⎝⎛ −
∫= *1),( ψ (8.1)
where, W(s, b) is the wavelet coefficient, the asterisk sign denotes the complex conjugate,
‘b’ is the translation parameter and ‘s’ is the scale parameter.There is a number of mother
wavelet which is used in practise but only some of the mother wavelets such as Mexican
hat wavelet, Gabor wavelet and Morlet wavelet are used in the field of fluid dynamics.
Wavelet has been used which is given by,
2/)/(4
12
)( γωωπψ tti oo eet −−= (8.2)
Where 2 2⁄ and is the number of wave in the wavelets. In practise the
value of varies from 5 to 12 and generally it is taken as 6. A frequency resolution of 12
is chosen when frequency of resolution of a signal is more important than time resolution.
(a) Plan (b) Isometric view
Fig. 8-1 Three sectors in GDWP
Sector 1
Sector 2
Sector3
Excitation direction
47
Table 8-1:Sectors frequencies in hertz computed by CFD simulations
Mode
number Sectors 1 and 5 Sectors 2,4, 6 and 8 Sectors 3 and 7