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Procedia Engineering 144 (2016) 1187 – 1194
12th International Conference on Vibration Problems, ICOVP 2015
Pressure Based Eulerian Approach for Investigation of
Sloshing in Rectangular Water Tank
Kalyan Kumar Mandala*, Damodar Maitya
a
Department of Civil Engineering, Indian Institute Technology, Kharagpur, India, 721302
Abstract
The present paper deals with the finite element analysis of the water tanks in Eulerian approach. In the governing equations of
fluid in a container, pressure is considered to be the independent nodal variables. The sloshing behaviour of fluid due to harmonic
excitation for different tank lengths is investigated. The free surface of the fluid gets higher elevation when the length of the tank
is equal to the height of the water in the tank. The hydrodynamic pressure along the wall increases with the decrease of length of
the tank. The fluid with in the tank has tendency to rotate for comparatively lower tank length which enhances the hydrodynamic
pressure on tank walls.
© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
© 2016 The Authors. Published by Elsevier Ltd.
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-reviewunder
under
responsibility
of organizing
the organizing
committee
of ICOVP
Peer-review
responsibility
of the
committee
of ICOVP
2015 2015
Keywords: Lagrangian approach, Eulerian approach, Irrotational motion, Inviscid.
1. Introduction
The sloshing behavior is of important concern in the design of liquid containers. The clear understanding of
sloshing characteristics is essential for the determination of the required freeboard to prevent overflow of the
contaminated cooling water. The sloshing motion can affect the stability of the free-standing spent fuels during
earthquakes and it may vary considerably depending on the size of the tank. In general, the two classical
descriptions of motion, Lagrangian and Eulerian, are used in the problem of fluid-structure interaction involving
finite deformation.
* Corresponding author. Tel.: +91(033)24572709
E-mail address: kkma_iitkgp@yahoo.co.in
1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the organizing committee of ICOVP 2015
doi:10.1016/j.proeng.2016.05.098
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Kalyan Kumar Mandal and Damodar Maity / Procedia Engineering 144 (2016) 1187 – 1194
However, Lagrangian approach is generally not suitable when the fluid undergoes large displacements since the
mesh will become highly distorted and the Eulerian description is desirable when the fluid experiences large
distortions.
Out of the different Eulerian approaches, displacement potential and velocity potential based approaches are very
common in modelling of fluid in tank [1-3]. However, requirement of computational time becomes higher as
number of unknown parameters gets increased. In these cases, the analysis procedure also needs large computer
storage. On the other hand, the pressure formulation has certain advantages in the computational aspect, as the
number of unknown per node is only one. In the present study, a pressure based Eulerian approach is used to model
the water. Water in the tank is assumed to be compressible and is discretized by eight node isoperimetric elements.
A computer code in MATLAB environment has been developed to obtain hydrodynamic pressure and sloshed
displacement of water in tanks. The sloshing characteristic of fluid is investigated for different frequencies of
harmonic excitation. The sloshed behavior is also studied for different lengths of the tank.
2. Theoretical formulation
Assuming water to be linearly compressible and inviscid with small amplitude irrotational motion, the
hydrodynamic pressure distribution due external excitation is given as
1
’2 p( x, y,t )
p( x, y,t )
(1)
C2
Where, C is the acoustic wave velocity in the water. The pressure distribution in the fluid domain is obtained by
solving eq. (1) with the following boundary conditions. The geometry of the tank is shown in Fig. 1
Y
Surface I
Surface II
H
Fluid
Surface IV
2.5 m
Point A
Surface III
X
L
Fig.1. Geometry of tank-water system
wp
1
p+ =0
g
wy
At surface I
(2)
wp
(0, y, t) = U f aeiZ t
wn
is the horizontal component of the ground acceleration and
(3)
At surface II and IV
Where,
ae
iZ t
At surface III
wp
x,0,t
wn
0.0
U f is the mass density.
(4)
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Kalyan Kumar Mandal and Damodar Maity / Procedia Engineering 144 (2016) 1187 – 1194
2.1. Finite element formulation for fluid domain
³N
rj
:
1
ª 2
«’ ¦ N ri pi c 2
¬
¦N
ri
º
pi » d :
¼
(5)
0
Where, Nrj is the interpolation function for the reservoir and Ω is the region under consideration. Using Green's
theorem eq. (5) may be transformed to
wN rj wN ri º
ª wN rj wN ri
³«
pi pi » d:
¦
¦
w
x
w
x
w
y
w
y
¬
¼
:
1
wN
rj
N rj ¦ N ri d: pi ³ N rj ¦
d* pi 0
³
w
n
c2 :
*
(6)
in which i varies from 1 to total number of nodes and Γ represents the boundaries of the fluid domain. The last term
of the above equation may be written as
^B` ³ N rj wp d*
wn
(7)
*
The whole system of equation (6) may be written in a matrix form as
ª¬ E º¼ ^P` ª¬G º¼ ^P`
^F `
(8)
Where,
ª¬ E º¼
1
C2
ª¬G º¼
ªw
¦ ³ « wx > N @
¦ ³>N @ >N @d:
T
r
r
(9)
:
T
:
r
¬
> F @ ¦ ³ > Nr @
T
*
º
w
w
T w
> Nr @ > Nr @ > Nr @»d :
wx
wy
wy
¼
wp
d*
wn
^FI ` ^FII ` ^FIII ` ^FIV `
(10)
(11)
Here the subscript I to IV stand for different surfaces as shown in Fig. 1. For surface wave, the eq. (2) may be
written in finite element form as
1
(12)
^FI ` ª¬ R f º¼ ^ p`
g
In which,
T
Rf
Nr
Nr d
(13)
f
At the fluid-structure interface (surface II and surface IV) if {a} is the vector of nodal accelerations of generalized
coordinates, {Ffs} may be expressed as
^FII `
U ª¬ R fs º¼ ^a`
(14)
^FIV `
U ª¬ R fs º¼ ^a`
(15)
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Kalyan Kumar Mandal and Damodar Maity / Procedia Engineering 144 (2016) 1187 – 1194
In which,
ª¬ R fs º¼
¦ ³ > N @ >T @> N @ d *
T
r
d
* fs
(16)
Where, [T] is the transformation matrix at fluid structure interface and Nd is the shape function of dam.
At surface III
^FIII `
0
(17)
After substitution all terms the eq. (6) becomes
> E @^P` >G@^P` ^F`
(18)
^Fr `
(19)
Where,
U ª¬ R fs º¼ ^a`
For any given acceleration at the fluid-structure interface, the eq. (18) is solved to obtain the hydrodynamic
pressure within the fluid.
2.2. Computation of velocity and displacement of fluid
The accelerations of the fluid particles can be calculated after computing the hydrodynamic pressure in reservoir.
The velocity of the fluid particle may be calculated from the known values of acceleration at any instant of time
using Gill’s time integration scheme [4] which is a step-by-step integration procedure based on Runge-Kutta method
[5]. This procedure is advantageous over other available methods as (i) it needs less storage registers; (ii) it controls
the growth of rounding errors and is usually stable and (iii) it is computationally economical. At any instant of time
t, velocity will be
Vt
Vt 't 'tVt
(20)
Velocity vectors in the reservoir may be plotted based on velocities computed at the Gauss points of each
individual point. Similarly, the displacement of fluid particles in the reservoir, U at every time instant can also
computed as
Ut
Ut 't 'tVt
(21)
This section describes the normal structure of a manuscript. For formatting reasons sometimes it is desired to
break a line without starting a new paragraph. Line breaks can be inserted in MS Word by simultaneously hitting
Shift-Enter. Table 1 summarized the font attributes for different parts of the manuscript. Normal text should be
justified to both the left and right margins.
3. Results and Discussion
3.1. Validation of the Proposed Algorithm
In this section proposed algorithm is validated by comparing fundamental time period from present study and a
bench marked problem solved Virella et al. [6]. The geometric and material properties of the water with in the tank
are considered as follows: height of water in the tank (H)= 6.10 m, length of tank = 30.5 m, density of water = 983
kg/m3, aquatic wave velocity = 1451 m/s. Here, the fluid is discretized by 4 ×8 (i.e., Nh= 4 and Nv = 8). The first
Kalyan Kumar Mandal and Damodar Maity / Procedia Engineering 144 (2016) 1187 – 1194
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three time periods of the fluid in the tank are listed and compared in Table 1. The tabulated results show the
accuracy of the present method.
Table 1 First three natural time periods of the fluid in the tank
Natural Time period in sec
Mode
number
Present Study
Virella et al. [6]
1
8.46
8.38
2
3. 94
3.70
3
2.89
2.78
3.2. Responses of water in tanks
In order to investigate the sloshing characteristic of water in rectangular tanks, water tank with following
properties are considered: depth (H) = 1.6 m, acoustic speed (C) = 1440 m/sec, mass density (ρf) = 1000 kg/m3. The
analysis is carried out for different tank length (L), i.e, 1.6 m and 2.4 m and fluid domain is discretized by 8 × 8 (i.e.,
Nh = 8 and Nv = 8) and 12 × 8 (i.e., Nh = 12 and Nv = 8) respectively. The tank walls are assumed to be rigid. The
number of time steps per cycle of excitation for solution of tank by the Newmark’s, Nt is considered as 32 at which
the results are found to converge sufficiently. Thus, the time step ( Δt) for the analysis of water tank is taken as T/32.
The responses of fluid in tanks such as sloshed displacement, velocity within the fluid, hydrodynamic pressure are
determined due to the harmonic excitation (TC/Hf=100) with amplitude of 1.0g. Maximum hydrodynamic pressure
for different tank lengths is presented in Table 2. From Table 2, it is noted that the hydrodynamic pressure is
increased with the decrease of tank length. The time history of sloshed displacements for different length of the tank
at the middle of the free surface are compared in Figure 2. This comparison depicts that more free board is required
for comparatively lower tank because the free surface for lower tank length gets higher elevation due to external
excitation (Figure 3). It is also observed from Figure 3 that the length L1.5H is higher than the length LH. This may the
main reasons for getting higher hydrodynamic pressure at lower tank length (Figures 4-5). The velocity vectors in
the water domain are plotted in Fig. 6. The vertical axis represents the height of the water and the horizontal axis
represents the length between two vertical walls of the tank. Figs. 6 (a) clearly show that the fluid generates a
tendency to rotate more in case of lower tank length and this rotating tendency increases the hydrodynamic pressure.
However, for L=1.5H, rotation water is only at the upper part of the tank.
Table 2 Pressure coefficient at point A for tank with different lengths
L/H
Pressure coefficient (p/(aρH)
1
2.759
1.5
1.653
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Kalyan Kumar Mandal and Damodar Maity / Procedia Engineering 144 (2016) 1187 – 1194
Fig. 2. Comparison of sloshed displacement at middle point of free surface
LH
L1.5H
Fig. 3. Comparison of sloshed displacement of free surface
Fig. 4. Hydrodynamic pressure along tank wall
Kalyan Kumar Mandal and Damodar Maity / Procedia Engineering 144 (2016) 1187 – 1194
1193
Fig. 5. Variation of Hydrodynamic pressure at point A
(a)
(b)
Fig. 6. Comparison of velocity plot (a) L=1.0H (b) L=1.5H
4. Conclusions
A pressure based Eulerian approach in analysis of water tank with different lengths is presented. The sloshed
displacement decreases with the increase of length of tank. The fluid in lower tank length has a tendency to rotate
more. This higher sloshed displacement and higher rotating tendency are responsible for higher hydrodynamic
pressure on the tank wall for comparatively lower tank length.
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Kalyan Kumar Mandal and Damodar Maity / Procedia Engineering 144 (2016) 1187 – 1194
References
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Engineering Structures , 28,704-715.
[2] Tung CC (1979), Hydrodynamic forces on submerged vertical circular cylindrical tanks underground excitation. Appl. Ocean Res, 1, 75-78.
[3] Liu WK (1981), Finite element procedures for fluid-structure interactions and applications to liquid storage tanks. Nucl. Eng. Des., 65, 221238.
[4] Gill S (1951), A process for the step-by-step integration of differential equations in an automatic digital computing machine. Proceedings of
the Cambridge Philosophical Society,47, 96-108.
[5] Ralston A, Wilf HS (1965), Mathematical Models for Digital Computers. Wiley1965,New York.
[6] Virella JC, Prato CA, Godoy LA (2008). Linear and nonlinear 2D finite element analysis of sloshing modes and pressures in rectangular
tanks subject to horizontal harmonic motions. Journal of Sound and Vibration, 312, 442–460.