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Journal of Modern Technology & Engineering
Vol.3, No.1, 2018, pp.15-52
15
FREE AND FORCED VIBRATIONS OF LIQUID STORAGE TANKS
WITH BAFFLES
Elena Strelnikova1,2*, Vasyl Gnitko1, Denys Krutchenko1, Yury Naumemko1
1 A.N. Podgorny Institute for Mechanical Engineering Problems, Kharkiv, Ukraine
2 V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
Abstract. In this paper we consider vibrations of shells of revolution partially filled with a liquid. The
liquid is supposed to be an ideal and incompressible one and its flow introduced by the vibrations of the
shell is irrotational. The problem of the fluid-structure interaction is solved using single-domain and
multi-domain reduced boundary element methods. The rigid and elastic baffled tanks with different
annular orifices are considered. The dependencies of frequencies via orifice radius at different values of
the filling level are obtained numerically for vibrations of the fluid-filled tanks with and without baffles.
Keywords: fluid-structure interaction, baffles, liquid sloshing, free vibrations, boundary element
method, singular integral equations.
Corresponding Author: Prof. Elena Strelnikova, Department of General Research in Power
Engineering of A.N. Podgorny Institute for Mechanical Engineering Problems, 2/10 Pozharsky St.,
Kharkiv, 61046, Ukraine, e-mail: elena15@gmx.com
Manuscript received: 10 August 2017
1. Introduction
Practicing engineers face many issues and challenges on design and seismic
simulation of liquid storage tanks. The liquid storage tanks are important components of
lifeline and industrial facilities. Ground-supported cylindrical tanks are used to store a
variety of liquids water for drinking and firefighting, crude oil, wine, liquefied natural
gas, etc. Failure of tanks, following destructive earthquakes, may lead to environmental
hazard, loss of valuable contents, and disruption of fire-fighting effort. Inadequately
designed or detailed tanks have suffered extensive damage in past earthquakes that has
resulted in disastrous effects (see Jung et al., 2006; Malhotra, 1997; Ru-De, 1993;
Sanchez-Sanchez et al., 2004).
Liquid sloshing near free surfaces can damage roofs and upper shells of storage
tanks. High stresses in the vicinity of poorly detailed base anchors can rupture the tank
wall. Base shears can overcome friction causing the tank to slide.
Early simulations of the liquid sloshing problem relied upon constructing
mechanical analogies that comprise pendulums or spring mass elements whose
parameters are designed to simulate the resultant dynamic pressure loads imparted on a
tank during sloshing are presented in (Degtyarev et al., 2015; Degtyarev et al., 2016).
Housner (Degtyarev et al., 2015) obtained classical solutions for impulsive and
convective parameters of ground supported rectangular and circular tanks (under
horizontal accelerations). The work was further extended to analyze elevated water
tanks (Degtyarev et al., 2016). The tanks walls were considered rigid. It should be noted
that engineering procedures for seismic analysis and design of storage tanks are often
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
16
based on the Housner multicomponent spring-mass analogy (Degtyarev et al., 2015).
The analogy allows the complex dynamic behaviour of the tank and its contents to be
considered in a simplified form. The principal modes of response include a short period
impulsive mode, with a period of around 0.5 seconds or less, and a number of longer
period convective (sloshing) modes with periods up to several seconds. For most tanks,
it is the impulsive mode that dominates the loading on the tank wall. The frequency of
the first convective mode is usually much less than the impulsive one, and the higher
order convective modes can be ignored. Later Veletsos and Yang (Veletsos & Yang,
1976) considered the effect of flexibility of cylindrical tanks.
Comprehensive reviews of the phenomenon of sloshing, including analytical
predictions and experimental observations were done in the work of Abramson
(Abramson, 1966) and Ibrahim (Ibrahim, 2005). Ibrahim also suggested in (Ibrahim,
2005) that exact solutions for the linear liquid sloshing are limited to regular tank
geometries with straight walls, such as rectangular and upright-cylindrical containers.
Note that fluid-free-surface natural frequencies and mode shapes for two- and three-
dimensional rectangular tanks have been obtained by Abramson (Abramson, 1966) and
Ibrahim (Ibrahim, 2005) using the method of separation of variables. Owing to
difficulties associated with this classical method for analysis of linear slosh in most
practical tank geometries (e.g., horizontal cylinders, spherical tanks), several other
methods have been developed for linear slosh analysis.
Since analytic solutions do not exist for tanks and reservoirs with complicated
geometrical shapes, in addition to the analytical methods, numerical methods have been
employed for solutions of linear boundary value problems of liquid sloshing. The
dynamic analysis of shell structures is often performed by use of finite element
programs (Jung et al., 2006).
Ru-De (Ru-De, 1993) presented a finite element analysis of linear liquid slosh in
an upright cylindrical tank under a lateral excitation. Arafa in (Arafa, 2006) developed a
finite element formulation to investigate the sloshing of liquids in partially filled rigid
rectangular tanks undergoing base excitation. Hydro-elastic oscillations of rectangular
plates, resting on Pasternak foundation and interacting with an ideal incompressible
liquid with a free surface, are studied in (Kutlu et al., 2012).
But such 3-D nonlinear finite element analysis, including the contained fluid as
well as the foundation soil, and elasticity of the shell walls is complex and extremely
time consuming. Several simplified theoretical investigations were also conducted, and
other numerical methods were elaborated. Some of these studies have been used as a
basis for current design standards.
Faltinsen and Timokha (Faltinsen & Timokha, 2012) developed a linear
multimodal method to study the two-dimensional liquid slosh in a horizontal cylindrical
tank. Based on the linear multimodal approach, the free-surface elevation and velocity
potential were expressed by series of the natural sloshing modes. This reduced the
associated linear boundary value problem to a set of ordinary differential equations.
McIver (McIver, 1989) also solved the potential flow equation for the free liquid
sloshing in two-dimensional cylindrical and spherical containers using the conformal
mapping technique and reported the natural frequencies in terms of liquid filling level.
Ergin and Ugurlu (Ergin & Ugurlu, 2004), investigated the effects of different
boundary conditions on the response behaviour of thin circular cylindrical shell
structures fully in contact with flowing fluid using finite and boundary element
methods.
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
17
The adequate definition of the fluid free-surface needs to be tracked using
alternate methods such as the volume-of-fluid method This method was developed by
Kim and Lee (Kim & Lee, 2003) and Kim et al. (Kim et al., 2003) on the basis of
fractional volumes of liquid in a cell, which can be used to identify the position of the
free-surface.
The viscous effects on sloshing frequencies were studied in (Bauer & Chiba,
2007)
To damp the liquid motion and prevent instability a lot of slosh-suppression
devices have been proposed. Such devices are used to reduce structural loads induced
by the sloshing liquid, to control liquid position within a tank, or to serve as deflectors.
These devices include rigid or elastic ring baffles of various sizes and orientation,
rectangular plates submerged into a fluid-filled tank, different plates partly covering the
free surface. The selection and design of suppression systems require quantitative
knowledge of the slosh characteristics.
In practice, the effect of baffles usually can be seen after the baffle has been
installed. But often this experimental work is too expensive. So developing
computational methods for qualified numerical simulation is a very topical issue. One of
the pioneering papers in the area was written by Miles (Miles, 1958).
The linear sloshing in a circular cylindrical tank with rigid baffles has been
studied by many authors in the context of spacecraft applications. Experimental and
numerical results were reported in Watson (Watson & Evans, 1991).
Hasheminejad and Mohammadi (Hasheminejad & Mohammadi, 2011) employed
the conformal mapping technique to study the effect of surface-touching horizontal side
baffles, bottom mounted vertical baffle and surface piercing vertical baffle in cylindrical
containers under lateral excitations. The study showed that a long pair of surface-
touching horizontal side baffles have considerable effect on the natural sloshing
frequencies while the bottom mounted vertical baffle was not recommended as an
effective anti-sloshing device. A surface-piercing vertical baffle, however, was found to
be efficient for controlling the liquid sloshing under high fill levels. The same
conclusion was also drawn by Hasheminejad and Aghabeigi (Hasheminejad &
Aghabeigi, 2009, 2011, 2012) where elliptical tanks with the same baffle configurations
were considered.
Cho et al. (Cho & Lee, 2004; Cho et al., 2005) and Arafa (Arafa, 2006) developed
a finite element formulation for linear liquid slosh in two-dimensional baffled
rectangular tanks. Also finite element method was applied in (Arafa, 2006; Cho & Lee,
2004; Cho et al., 2005) to examine numerically the damping effects of disc-type elastic
baffle on the dynamic characteristics of cylindrical fuel-storage tank boosting with uniform
vertical acceleration.
Askari et al. (Askari et al., 2011) developed an analytical method to investigate
the effects of a rigid internal body on bulging and sloshing frequencies and modes of a
cylindrical container partially filled with a fluid using the Rayleigh quotient, Ritz
expansion, and Galerkin method. Askari and Daneshmand (Askari & Daneshmand,
2009) proposed finite element method using Galerkin method to analysis coupled
vibrations of a partially fluid-filled cylindrical container with a cylindrical internal
body. The effects of baffles on the natural sloshing frequencies were also investigated
by Gedikli and Erguven (Gedikli & Erguven, 2003) using a variational boundary
element method (BEM). Gedikli and Erguven (Gedikli & Erguven, 1999) also reported
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
18
seismic responses for an upright cylindrical tank with a ring baffle using BEM and
superposition of modes. The major contribution of these works was the significant
reduction of computational cost compared to other numerical methods such as the finite
element method. Gedikli and Erguven in (Gedikli & Erguven, 1999, 2003) used BEM to
investigate the sloshing problem using Hamilton method and evaluated the influence of
baffle size and location on natural frequencies of upright rigid cylindrical tank.
So then boundary integral equations have been widely used for the solution of a
variety of problems in engineering. This approach has certain advantages. In the basic
equations the functions and their derivatives will be defined on the domain boundaries
only. That allows reducing the problem dimension. This method gives new qualitative
possibilities in modeling dynamic coupled problems.
Though BEM formulations have provided robust solutions to engineering
problems, the resulting discretized systems are typically dense and non-symmetric, thus
entailing increased computational cost especially when compared to domain
discretization methods. It was the reason that multi-domain methods, or domain
decomposition methods based exclusively on boundary elements have also appeared for
both interior and exterior boundary value problems (Brebbia, 1984; Crotty, 1982; Rigby
and Aliabadi, 1995). The main idea of multi-domain is in dividing the original domain
into smaller ones (sub-domains or macro-elements). In each domain the BEM
formalism is employed. Fictitious (interface) boundaries are involved to delimit the
domains when necessary and will be described in terms of pressure and velocity similar
to solid boundaries. Continuity equations are written on these fictitious boundaries.
Then the BEM algebraic equations are established for each sub-domain; and the global
system of equations is formed by assembling results of all sub-domains in terms of the
equilibrium and consistence conditions over common interface nodes. The Blocked
Equation Solvers (Crotty, 1982; Rigby & Aliabadi, 1995) are proposed to obtain the
solutions of these sparse systems of algebraic equations.
The multi-domain BEM is especially effective at numerical simulation of tanks
with baffles.
When liquids slosh in closed containers, one can observe the multiple
configurations (modes) in which the surface may evolve. Commonly, the different
modes can be defined by their wave number (number of waves in the circumferential
direction) and by their mode number n.
Although baffles are commonly used as the effective means of suppressing the
sloshing magnitudes, the only few studies have assessed the role of baffle design
factors. The size and location effects of a baffle orifice on the sloshing has been
reported in only two studies devoted rectangular Popov et al (Popov, 1993) and generic
Guorong et al. (Guorong & Rakheja, 2009) cross-section tanks.
It should be noted that anti-slosh properties of baffle designs have been
investigated through laboratory experiments by using small size tanks of different
geometry Lloyd et al. (Lloyd et al., 2002).
The overview of the research on the topic (Bermudez & Rodrigues, 1999;
Guorong & Rakheja, 2009; Jung et al., 2006; Lloyd et al., 2002) demonstrates that the
dynamic response of liquid-containing structures can be significantly influenced by
vibrations of their elastic walls in interaction with the sloshing liquid.
In (Degtyarev et al., 2015, 2016; Gnitko et al., 2016, 2017; Strelnikova et al.,
2016; Ventsel et al., 2010) the authors developed an approach based on using the
coupled finite and boundary element method to the problem of natural vibrations of the
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
19
fluid-filled elastic shells of revolution, as well as to the problem of natural liquid
vibrations in the rigid vessels. But in (Degtyarev et al., 2016; Gnitko et al., 2016;
Strelnikova et al., 2016) the only rigid vessels were under consideration. In (Ventsel et
al., 2010) the fluid-structure interaction was considered without including the sloshing
effects, and in (Degtyarev et al., 2015; Gnitko et al., 2017) the effects of sloshing and
elasticity of walls were considered separately.
This paper is summarizing the authors’ efforts in the area. We consider here free
and forced liquid vibrations in cylindrical, conical and spherical tanks with and without
baffles, carry out the numerical simulation of elasticity effects, and mutual influence of
sloshing and elasticity of tank walls on the frequencies.
2. Problem statement
Consider a coupled problem of dynamic behavior of an elastic shell of revolution
partially filled with a liquid under a short-time impulsive load. Also free and forced
vibrations of such shells are under consideration.
Suppose that the fluid-filled elastic shell of revolution of an arbitrary meridian has
internal baffles installed to damp the liquid sloshing. The shell is of uniform thickness
h, and height L, made of homogeneous, isotropic material with elasticity modulus E,
Poisson's ratio and mass density s. The shell structure and its sketch are shown in
Fig. 1.
Figure 1. Shell structure with an internal baffle and its sketch.
Denote the wetted part of the shell surface through and the liquid free surface as
S0. The liquid volume is divided here into two domains 1 and 2 by the surface
SbafSint , where Sbaf is a baffle surface, Sint is an interface surface (Biswal et al., 1984),
see Fig.1. The shell surface consists of four parts,
bafbot21 SSSS ww
. Here
S1 and S2 are lateral surfaces of first and second fluid domains, respectively, and Sbot is a
surface of the tank bottom.
Let U
321 ,, UUU
denote the vector-function of shell displacements. Consider
at first stage the free vibrations of the shell without a liquid (the empty shell). For the
problems of free vibrations we assume that the time dependent shell displacements are
given by
321 ,,);exp( uuuti uuU
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
20
Here is the vibration frequency; the time factor
)exp( i
will be omitted further
on. After the separation of the time factor, the vibrations of the shell without a liquid are
described by the system of three partial differential equations
3,2,1,
3
1
2
juuL j
iiij
,
where
ij
L
are linear differential operators of the Kirchhoff - Love shell theory Levitin et
al. (Levitin & Vassiliev, 1996).
A finite element method was applied in (Ravnik et al., 2016; Ventsel et al., 2010)
to evaluate the natural frequencies
k
and modes
k
u
,
Nk ,1
of the shell of revolution
without a liquid. After forming the global stiffness
L
and mass M matrices, the
following equation of motion for the shell containing fluid was obtained in (Ravnik et
al., 2016; Ventsel et al., 2010):
QnUMLU d
p
, (1)
where n is an external unit normal to the shell wetted surface, a term pd n gives the fluid
dynamical pressure upon the shell, normal to its surface,
tQQ
is a vector of
external load.
To model the fluid motion, a mathematical model has been developed based on
the following hypotheses: the fluid is incompressible, the motion of the fluid is
irrotational and inviscid, only small vibrations (linear theory) need to be considered. So
a scalar velocity potential Φ(x,y,z,t) whose gradient represents the fluid velocity can be
introduced.
The fluid pressure
tzyxpp ,,,
acting on the wetted shell surface is obtained
from the linearized Bernoulli’s equation for a potential flow, Lamb (Lamb, 1993).
t
pgzppgz
t
pldlsl
;;
0
,
where g is the gravity acceleration, z is the vertical coordinate of a point in the liquid, l
is the liquid density, ps and pd are static and dynamic components of the fluid pressure,
p0 is for atmospheric pressure.
Assuming the flow to be inviscid and irrotational, the incompressible fluid motion
in the 3D tank is described by the Laplace equation for the velocity potential
0
2
. (2)
To determine this potential a mixed boundary value problem for the Laplace
equation is formulated in the double domain 12 (Fig. 2). The non-penetration
condition on the wetted tank surfaces is following [56]:
t
w
n
,
nU,w
. (3)
Let function
yxt ,,
be the free surface elevation. The kinematics and dynamic
boundary conditions on S0 can be expressed as follows (Gnitko et al., 2016):
0; 0
0
0
S
Spp
tn
. (4)
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
21
To apply the multi-domain approach we divide the fluid domain into two sub-
domains 1 and 2, shown in Fig. 2. Here we introduce the artificial interface surface
Sint. Let
bafbot11 SSS
and
baf22 SS
are the shell surfaces contacting with a
liquid in sub-domains 1 and 2. Then boundaries of sub-domains 1 and 2 are
int11 S
and
02 S 2
.
a) b)
Figure 2. Fluid sub-domains
Denote by 1, 2, 0 the potential values in nodes of 1, 2 and S0, respectively.
The fluxes on 1, 2 are known from the no-penetration boundary condition as
21,ww
,
and on the free surface the unknown flux is denoted as
0
q
. The potential and flux
values on the interface surface Sint will be unknown functions ji and
j
q
,
2,1,
int jS j
, and we have (Brebbia et.al., 1984)
2112 ;qq
ii
(5)
Equations (1), (2) are solved simultaneously using the shell fixation conditions
relative to U, boundary conditions (3)-(5), relative to and the following expressions
for the dynamical component of the liquid pressure on elastic walls:
t
pld
),( nP
.
To define modes of free harmonic shell vibrations coupled with liquid sloshing,
we represent displacements of the fluid-filled tank as U=ufexp(it). Here and uf are
natural frequencies and vibration modes of the fluid-filled shell structure.
3. The mode superposition method for coupled dynamic problems
Consider the vibration modes of the fluid-filled tank in a form
N
kkk
c
1uU
, (6)
where
tcc kk
are unknown coefficients, and uk are eigenmodes of the empty tank.
In other words, the mode of vibration of the fluid-filled tank is determined as a linear
combination of eigenmodes of the empty shell structure. Note that the following
relationships are fulfilled (Ventsel et.al., 2010):
kjjkkkk )),((,)()( 2uuMuMuL
. (7)
Hence
kjkjk 2
)),(( uuL
, (8)
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
22
where k is the k-th frequency of the empty tank vibrations. Equations (7), (8) show
that the abovementioned vibration modes have to be orthonormalized with respect to the
mass matrix.
Consider the potential Φ as a sum of two potentials
21
, as it was done
by Degtyarev et al in (Degtyarev et al.,2015).
The series for potential Φ1 can be written as
N
kkk tc
111
.
Here time-dependant coefficients ck(t) are defined in equation (6). To determine
functions 1k the following boundary value problems are formulated:
0
1 k
,
k
kw
n
1
,
0
0
1 S
k
,
nu ,
kk
w
,
Nk ,1
(9)
The solution of boundary value problems (9) was done by Ventsel et al in
(Venstel et al., 2010). Thus the dynamic analysis of elastic shells of revolution with a
liquid, neglecting the gravity force, is formulated in terms of the functions U and 1.
The above functions satisfy the system of differential equations (1), (2) the no-
penetration condition and the lack of the pressure on a free surface, as well as the
conditions of the shell fixation. The solutions of the boundary value problems (9) can be
represented in the symbolic form as
kk iuH1
, where
k
uH
is the inverse
operator of the hydrodynamic problem (Ventsel et.al., 2010).
Suppose that
tiCtc kk exp
, where is an own frequency of the shell with a
fluid. Based on the equations (1), (2), (9) we obtain
N
kjkkljkjkjkCC 1
22 ,uuH
. (10)
The above equation represents a generalized eigenvalue problem. Solving this
problem yields the natural frequencies of the vibrations of the elastic shell conveying
fluid, but without the gravity effects.
When the potential 2 is known, the low frequency sloshing modes will be
obtained. To determine the potential 2 we have a problem of fluid vibrations in a rigid
shell including gravity effects.
Use the expansion
M
kkk td
122
, where dk.(t) are unknown coefficients,
functions 2k are natural modes of the liquid sloshing in the rigid tank. To obtain these
modes the following boundary value problems are considered:
0
2 k
;
;0
2
nk
t
g
tS
k
S
k
0
0
22 ;0 n
,
Nk ,1
(11)
The zero eigenvalue obviously exists for problem (11), but we exclude it with the
help of the following orthogonality condition:
0
0
2
0
dS
S
k
n
Differentiate the third equation in relationship (11) with respect to t and substitute
there the expression for
t
from the forth one of (11). Suppose
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
23
zyxezyxt k
ti
kk,,,,, 22
and obtain the next conditions on the free surface for
each mode 2k with the sloshing frequency k:
Mk
gk
kk ,1,
2
2
2
n
. (12)
It leads to the following eigenvalue problems
0
2 k
;
;0
2
nk
,
2
2
2k
kk g
n
MkdS
Sk,1,0
02
0
. (13)
Solving these problems yields the sloshing frequencies k. and modes 2k.
So to solve the free vibration problem for an elastic shell of revolution coupled
with liquid sloshing it is necessary to determine three systems of basic functions: modes
of liquid in rigid shell under force of gravity; own modes of empty shell; modes of
fluid-filled elastic shell without including the force of gravity.
Thus, the problem under consideration involves the following steps.
First, it is necessary to obtain the sloshing frequencies and modes
k2
using rigid
wall assumption.
Second, we obtain the natural frequencies
k
and modes
k
u
of the empty tank
with elastic walls. It would be noted that the Kirchhoff-Love shell theory is employed
here because of considering the thin shells, but for defining basic functions
k
u
one can
involve another shell theory.
Then we define the free vibration frequencies and modes
k1
of the elastic tank
without considering effects of sloshing.
Finally, for the sum of potentials
21
the following expression can be
written
M
kkk
N
kkk tdtc 12
11
. (14)
The unknown function takes the following form:
M
k
k
k
N
k
k
kn
td
n
tc 1
2
1
1
. (15)
So, the total potential satisfies the Laplace equation and non penetration
boundary condition
0
;
t
w
S
1
n
due to validity of relations (11),(13). Noted that also satisfies the condition
tn S
0
as a result of representation (15).
Satisfying the condition
0
0
s
gz
t
on the free surface, one can obtain the next equality
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
24
0
1
2
1
1
12
11
M
k
k
k
N
k
k
k
M
kkk
N
kkk z
d
z
cgdc
.
When functions
k1
and
k2
are defined, we substitute them in eqns (1),(4) and
obtain the system of ordinary differential equations as it was done in (Gnitko et al.,
2017).
N
k
M
kkkkkl
N
kkk
N
kkk tdtctcMtcL 1 1 21
11 )()()()(
uu
;. (16)
0
1 1 2
2
1
12
N
k
M
kkkk
k
k
M
kkk d
n
cgd
.
The first equation here is valid on the wetted surface of the shell and the second
one – on the free surface of liquid.
Considering the result of dot product of first equation in (16) by uj and second one
by
j2
, taking also into account relationships (7),(8) and orthogonality of natural modes
of fluid vibrations in rigid vessel, we come to the next set of N+M second order
differential equations to determine unknown coefficients
tdtc kk ,
:
N
k
M
kjkkjkkljjj wtdwtctctc 1 1 21
2,)(,)()()(
(17)
0, 2
2
1
1
tdg
n
tcgtd jjj
k
N
kkj
To define coupled modes of harmonic vibrations we represent the time-dependant
unknown coefficients as
ti
kk
ti
kk eDtdeCtc ;
, (18)
where is an own frequency, and
kk DC ,
are unknown constants.
Taking into account equations (18), one can obtain that equations (17) can be
expressed as
NjwDwCCC N
k
M
kjkkjkkljjj ,1,0,,
1 1 2
2
1
222
(19)
.,1,0, 2
1
1
22 Ml
n
CgDD l
k
m
kklll
Introducing the following matrixes and vectors
;
...
;
... 2
1
2
1
nm D
D
D
D
C
C
C
C
2
2
1
00
......0
00.
n
H
;
2
2
1
00
......0
00.
m
H
;
NjkwppP jkkjkj ,1,;,; 1
;
kjjkjk wbbB ,; 2
;
MjNk
n
aaA j
k
jkjk ,1;,1;,; 2
1
,
we come to the next eigenvalue problem
0
222 BDPCCHEC ll
;
0
2 DHgACED
.
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
25
Let also introduce for simplicity vectors and matrix of doubled dimension
MN
D
C
X
;
E
BPE
H022
;
HgA
H
G0
.
It brings us to the following eigenvalue problem
0
2 XHG
. (20)
So free vibration analysis of an elastic shell coupled with liquid sloshing is
reduced to the solution of generalized eigenvalue problem (20) where both elasticity and
gravity effects are taken into account (Degtyarev et al., 2015). It would be noted that
hereinbefore we did not assume that the shell considered is a shell of revolution only.
The effective numerical procedure for solution of this eigenvalue problems using the
single and multi-domain boundary element methods (BEM) has been developed in
(Gnitko et al., 2016; Ravnik et al., 2016).
4. Reducing to the system of one-dimensional integral equations
To define functions
k1
and
k2
we use the boundary element method in its direct
formulation (Brebbia et al., 1984). Dropping indices 1k and 2k one can obtain the main
integral equation in the following form
dS
PP
dS
PP
qP
SS 00
011
2
n
. (21)
Here
0
SS
, points P and P0 belong to the surface S. The value
0
PP
represents Cartesian distance between the points P and P0. In doing so, the function
defined on the wetted tank surface presents the pressure, and the function q defined on
the free surface S0, is the flux,
n /q
.
The basic procedure is to start with the standard boundary integral equation for
potential (21), replace Cartesian coordinates (x, y, z) with cylindrical ones (r,
, z), and
integrate with respect to , taking into account that
00
2
0
2
0
2
0cos2 rrzzrrPP
,
where points P and P0 have the following coordinates
0000 ,,;,, zrPzrP
.
Furthermore we represent unknown functions as Fourier series by the
circumferential coordinate
,...2,1;2,1;2,1;cos,,,;cos,,, kjizrzrzrwzrw iiii
jkjkkk
, (22)
where is a given integer (the number of nodal diameters). In this case, the solution is
independent of the angular coordinate , and the three-dimensional problem is reduced
to a two-dimensional one in the radial coordinate r and the axial coordinate z.
Let be a generator of the surface . Using (21), (22) we have obtained the
following system of singular integral equations for unknown functions and q in
problem (9):
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
26
010
0000 ;,,,2 PdzrPPzwdPPqdzrzzQzz R
;
.;,,, 0010
000 SPdzrPPzwdPPqdzrzzQz R
(23)
Here
;
2
14
,0
2
0
2
0
2
0
zr nk
ba zz
nkk
ba zzrr
r
ba
zzQ EFE
;
4
,0k
ba
PP
F
2/
0
222 sin12cos411 dkkE
;
2/
022 sin1
2cos
1k
d
kF
;
;2; 0
2
0
2
0
2 bzza
ba b
k
2
2
.
Letting
0
in expressions (23), we obtain the standard elliptic first and second
kind integrals.
The system of singular integral equations for mixed boundary value problems (11)
has been obtained in (Gnitko et al., 2016).
To define potentials 2 we introduce as in (Gnitko et al., 2016) next integral
operators:
1
0
111 ),( 1
2
1
dS
PPrn
AS
;
;
1
0
000
SdS
r
B
0
000 1
SdS
rz
C
;
1
0
11 1
1
dS
PPn
DS
;
0
000 1
SdS
r
F
.
Then the boundary value problem (13) takes the form
00
2
1/ CBgA
;
10 SP
;
0
2
01 /2 FgED
;
00 SP
.
After excluding function 1 from these relations, we obtain the eigenvalue
problem and its solution gives natural modes and frequencies of liquid sloshing in the
rigid tank
gFBDAECDA /;0)()( 2
0
1
0
1
.
It should be noted that there are two types of kernels in the integral operators
introduced above, namely
.;
1
,;
1
,0
00
PdS
PP
SBdS
PP
SA
SS n
(24)
At integration with respect to one can conclude that the internal integrals in (24)
are complete elliptic integrals of first and second kinds. As the first kind elliptic
integrals are non-singular, one can successfully use standard Gaussian quadratures for
their numerical evaluation. For second kind elliptic integrals we have applied here the
approach based on the characteristic property of the arithmetic geometric mean AGM
(a,b) (see Cox David, 1984). The above-mentioned characteristic property consists in
following:
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
27
baAGM
ba
d,2
sincos
2
02222
.
To define AGM(a,b) there exist the simple Gaussian algorithm, described below
;...;
2
;....;
2
;; 11001
00
100 nnn
nn
nbab
ba
abab
ba
abbaa
.limlim,n
n
n
nbabaAGM
(25)
It is a very effective method to evaluate the elliptic integrals of the second kind.
Convergence
8
10
nn ba
is achieved after 6 iterations (namely,
6n
in (25)).
So we have the effective numerical procedures for evaluation of inner integrals,
but integral equations (23) involve external integrals of logarithmic singularities and
thus the numerical treatment of these integrals will also have to take into account the
presence of this integrable singularity. Here integrands are distributed strongly non-
uniformly over the element and standard integration quadratures fail in accuracy. So we
treat these integrals numerically by special Gauss quadratures (Brebbia et al., 1984)
and applying technique proposed in (Naumenko & Strelnikova, 2002).
The solution of system (23) is independent of the angular coordinate , and the
three-dimensional problem is reduced to a two-dimensional one in the radial coordinate
r and the axial coordinate z. Using dependence
rzz
, we finally reduce the system of
singular integral equations to a one-dimensional one.
So 3-D problem of determining the pressure and free surface elevation is reduced
to solution of the one-dimensional system of singular integral equations.
5. Multi-domain approach
To estimate the liquid vibrations in the presence of the baffle, we use the multi-
domain method (boundary super-elements). In doing so, we introduce an "artificial"
interface surface Sint (Brebbia et al., 1984; Gnitko et al., 2016; Gnitko et al., 2017),
divide the region filled with the liquid into two parts
21;
, bounded by surfaces Sbot,
S1, Sbaf, Sint and S2, Sbaf, Sint, S0 and shown in Fig. 2. Let
bafbot11 SSS
and
baf22 SS
are the surfaces of the shell contacting with a liquid in sub-domains 1
and 2. Then boundaries of sub-domains 1 and 2 are
int11 S
and
02 S 2
.
Denote by 1, 2, 0 the potential values in nodes of 1, 2 and S0, respectively, and by
w1 , w2 the values of function
nU,w
. The fluxes on 1, 2 are known from the no-
penetration boundary condition as
21,ww
and on the free surface the unknown flux is
denoted as
0
q
. The potential and flux values on the interface surface Sint will be
unknown functions ji and
j
q
,
2,1,
int jS j
, and we have the following
compatibility conditions
2112 ;qq
ii
. (26)
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
28
Consider the boundary value problem for determining the potential 1.
Introducing
1
~
S
=
1
,
2
~
S
=Sint,
3
~
S
=
2
,
4
~
S
=S0 allows us to obtain matrixes
4,1,,
~
,
~
;
~
,
~ jiSSBBSSAA jiijjiij
. By using the multi-domain approach to
determine the potential 1 the next system of integral equations in the operator form
was obtained in (Gnitko et al., 2017):
;;
;;
int0122121122121
10112111112111
SPqBwBAA
PqBwBAA
i
i
;; 20034132233233132 PqBqBwBAA i
(27)
.;
;;
00044142243243142
int0024122223223122
SPqBqBwBAA
SPqBqBwBAA
i
i
It would be noted that compatibility conditions (26) are taking into account in
system (27). As a result of solving equations (27), we have
Qw
,
2
1,
2
1
2
1;;
ji
ij
i
i
i
iQw Qw
,
where
ij
Q
are obtained in [26]. So for each 1k the pressure on the surface will be
defined by formulae
baf
2
1
1
12bot11 ;;; SPPPtcpSSSPPtcp kkklk
ikklk
.
The boundary value problem for determining the potential 2 with multi-domain
BEM (MBEM) was solved by Gnitko et al in (Gnitko et al., 2016).
Hereinafter the results of numerical simulation are described. In Section 6 the
convergence of proposed method is shown. Sections 7-10 are devoted to liquid sloshing
in rigid shells, Sections 11-12 present results of fluid-structure interaction including
both sloshing and elasticity effects.
6. Comparing with analytical solution and convergence
As it was mentioned above, for vibration analysis of elastic shells it is necessary
to determine three systems of basic functions. One of them is represented by free
vibrations modes of the liquid in the rigid shell under the force of gravity. So the first
stage of our research is connected with the liquid vibrations in the rigid shells. We
consider here rigid spherical, cylindrical and conical shells with and without baffles.
The initial 3D problem is reduced to solution of the one–dimensional system of singular
integral equations in the form (23). In Fig. 3 the drafts of shells are shown with
discretized geometry based on BEM.
In Fig. 3 the following designation are introduced: N0 is the number of boundary
elements along the free surface radius; Nw is the number of boundary elements along the
shell wall; Ninf is the number of boundary elements along the interface surface; Nbaf is
the number of boundary elements along the baffle; and Nbot is the number of boundary
elements along the shell bottom.
To validate the proposed method the rigid cylindrical is considered. The first set
of calculations is therefore to determine the requisite number of boundary elements for a
precise determination of the natural frequencies.
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
29
Figure 3. Drafts of shells and discretization
We consider liquid sloshing in the rigid cylindrical shell. For testifying the
proposed numerical algorithm we use the analytical solution (Abramson, 2000) that can
be expressed in the next form:
,..2,1,tanh
2
k
R
H
Rg k
kk
;
H
R
z
R
r
R
Jkkk
k1
2coshcosh
. (28)
Here R is the shell radius, H is its height, values
k
are roots of the equation
0
xJ
, where
xJ
is Bessel function of the first kind, k, 2k are frequencies and
modes of liquid sloshing in the rigid cylindrical shell. The numerical solution is
obtained by using the BEM as it was described beforehand.
Consider the rigid circular cylindrical shell with a flat bottom, without baffles, and
having the following parameters: the radius and height are R = 1 m, and H=1m. Table 1
below provides the numerical values of the natural frequencies of liquid sloshing for
nodal diameters =0 and =1 obtained by proposed numerical method for different
numbers N0, Nw, and Nbot and analytical values received by formula (28). Here we
choose equal numbers N0 =Nw = Nbot because radii of the free surface and bottom, and
the height of the wetted part of the are equals to 1m. So we consider the following sizes
of one-dimensional boundary elements according to numbers N0=Nw=Nbot: 0.04m; 0.02
m, and 0.01m.
Table 1. Slosh frequency parameters
g
n/
2
of the fluid-filled rigid cylindrical shell
BEM
n=1
n=2
n=3
n=4
n=5
0
N0
Nw
Nbot
25
25
25
3.8289
7.0163
10.1761
13.3152
6.47089
50
50
50
3.8285
7.0159
10.1735
13.3243
6.47066
100
100
100
3.8281
7.0156
10.1732
13.3233
6.47060
Analytical solution
3.8281
7.0156
10.1734
13.3236
6.47063
1
N0
Nw
Nbot
n=1
n=2
n=3
n=4
n=5
25
25
25
1.6590
5.3301
8.5385
11.7071
14.8684
50
50
50
1.6579
5.3297
8.5372
11.7082
14.8655
100
100
100
1.6573
5.3293
8.5366
11.7066
14.8635
Analytical solution
1.6573
5.3293
8.5363
11.7060
14.8635
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
30
The results of Table 1 testify convergence of proposed BEM. In should be noted
that the accuracy
4
10
has been achieved here for N0 =Nw = Nbot = 100. So further
we consider boundary elements with length near 1% of the characteristic size.
In Fig. 4 the distributions of first three sloshing modes for = 0 on the free
surface are shown. The solid lines denote modes obtained by analytical expression (28)
at z = H. The lines pointed with circles and squares denote numerical solutions at N0
=Nw = Nbot = 100.
Figure 4. Numerically and analytically obtained modes
Fig. 4 also demonstrates good agreement between numerical and analytical data.
7. Cylindrical shells with and without baffles
The study of free vibration characteristics of the rigid cylindrical shell interacting
with the liquid is presented here. It is supposed that =0,1 in equation (22), i.e. we
consider both axisymmetric and non- axisymmetric modes.
Consider the circular cylindrical shell with a flat bottom and having the following
parameters: radius is R = 1 m, the thickness is h = 0.01 m, the length L = 2 m. The fluid
filling level is denoted by H. The baffle is considered as a circle flat plate with a central
hole (the ring baffle), fig. 5. The vertical coordinate of the baffle position (the baffle
height) is denoted as H1 (H1 < H). The interface surface radius is denoted as Rint and we
also have
21 HHH
. So the baffle radius is
intbaf RRR
.
The numerical solution is obtained by using the BEM as it is described
beforehand. In present numerical simulation we used 100 boundary elements along the
bottom (Nb), 120 elements along wetted cylindrical parts (Nw), and 100 elements along
the radius of free surface (N0). At the interface and baffle surfaces we used different
numbers of elements depending on radius of the baffle. In numerical simulations we
consider different values both for Rint and H1. We used for comparison and validation
the analytical solution (Ibrahim, 2005) that can be expressed by formulae (28).
To validate our multi-domain BEM approach we also have calculated the natural
sloshing frequencies at H1=0.5m, H1=0.9m, and with Rint=0.7m, H=1.0m. The
comparison of results obtained with proposed MBEM and the analytically oriented
approach presented by I. Gavrilyuk et al. in (Gavrilyuk et al., 2008) is shown in Table 2.
X
Y
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
-0.45
-0.375
-0.3
-0.225
-0.15
-0.075
0.075
0.15
0.225
0.3
0.375
0.45
0.525
0.6
0.675
0.75
0.825
0.9
0.975
0
Created with a trial version of Advanced Grapher - http:// www.alentum.com/ agrapher/
1
2
3
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
31
Figure 5. Cylindrical shell with internal baffle and its sketch
Table 2. Comparison of numerical results for
g/
22
, =0.
baffle position
method
n=1
n=2
n=3
n=4
H1=0.5
MBEM
3.756
7.012
10.176
13.328
(Gavrilyuk et al., 2008)
3.759
7.010
10.173
13.324
H1=0.9
MBEM
2.278
6.200
9.609
12.810
(Gavrilyuk et al., 2008)
2.286
6.197
9.608
12.808
These results have demonstrated a good agreement and validated the proposed
multi-domain approach. In all tables we have compared the frequency parameters
g/
22
of the problems described beforehand.
The three first modes of liquid vibrations for =0 are shown on Fig.6. Here we
consider Rint=0.2m and the height of baffle installation H1=0.9m.
Here numbers 1,2,3 correspond to the first, second and third sloshing modes. The
combination of Rint=0.2m and H1=0.9m brings to frequencies’ maximal decreasing.
From Figure 6 one can conclude that modes of vibrations of baffled and un-baffled
tanks are similar, and numerical values do not differ significantly.
Consider =1. In this case values
k
are roots of the equation (see the handbook
of I.S. Gradshteyn and I.M Ryzhik, (Gradshteyn & Ryzhik, 2000))
xJxJxJ 201 2
. (29)
Table 3 hereinafter provides the numerical values of the frequencies parameters of
liquid sloshing for nodal diameters =0 and H=1.0m. The numerical results obtained
with proposed MBEM are compared with those received using formulae (20), (21).
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
32
Figure 6. Modes of vibrations of un-baffled and baffled tanks
Table 3. Comparison of analytical and numerical results, =1
Modes
n=1
n=2
n=3
n=4
n=5
MBEM
1.750
5.332
8.538
11.709
14.870
Analytical solution,
(Gradshteyn &
Ryzhik, 2000)
1.750
5.331
8.536
11.706
14.864
We also have calculated the natural sloshing frequencies for =1 at H1=0.5m, H1=
0.9m, and with Rint =0.7m. The comparison of results obtained with proposed MBEM
and the analytically oriented approach presented by I. Gavrilyuk et al (Gavrilyuk et al.,
2008) has been demonstrated in Table 4.
Table 4. Comparison of analytical and numerical results, =1
Position
method
n=1
n=2
n=3
n=4
H1=0.5
MBEM
1.3663
5.2941
8.5359
11.7097
(Gavrilyuk et al., 2008)
1.3662
5.2940
8.5357
11.7092
H1=0.9
MBEM
0.7078
4.5066
8.1947
11.5556
(Gavrilyuk et al., 2008)
0.7079
4.5068
8.1945
11.5550
The three first modes of liquid vibrations are shown on Fig. 7. Here we consider
the ring baffle with Rint=0.2m and the height of baffle installation H1=0.9m.
Curves with numbers 1, 2, 3 correspond to the first, second, and third sloshing
modes. These results demonstrate that modes of vibrations of baffled and un-baffled
tanks at =1 differ more significantly than those at =0.
Results presented here may serve as the basis for designing liquid containers
subjected to external excitations whose frequencies may be close to the lowest natural
frequency of the free surface.
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
33
Figure 7. Modes of vibrations of un-baffled and baffled tanks, =1
8. Conical shell with and without baffles
Conical shells in interaction with a fluid have received a little attention in
scientific literature in spite of the usage of thin walled conical shells is of much
importance in a number of different branches of engineering. In aerospace engineering
such structures are used for aircraft and satellites. In ocean engineering, they are used
for submarines, torpedoes, water-borne ballistic missiles and off-shore drilling rigs,
while in civil engineering conical shells are used as containment vessels in elevated
water tanks. The difficulty of using the analytical methods arises due to the fact that
walls are not parallel to the axis of symmetry.
Boundary element method retains its advantages in this case.
We consider both V-shape and Ʌ-shape conical tanks with radius R1=1.m, and
=/6, Fig. 8. Note, that for V-shape tank R1 is the free surface radius, whereas for Ʌ-
shape tank R1 is radius of the bottom, and for V-shape tank R2 is radius of bottom,
where as for Ʌ-shape tank R2 is the free surface radius.
If R1, R2 and are known quantities, than the corresponding value of H can be
easy found as
cot
21 RRH
.
Figure 8. Baffled conical shells of Ʌ- and V shapes
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
34
In Table 5 the results of numerical simulation of the un-baffled tanks frequencies
are presented for =0, 1, 2 and different values of R2. Our numerical simulation was
dedicated to the frequencies
g
kk /
22
for =0, 1 ,2 and k =1 because these are the
lowest natural frequencies that give the essential contribution to the hydrodynamic load.
Table 5. Frequency parameter
g
kk /
22
of V – shape and Ʌ- – shape conical tanks
V – shape
Ʌ – shape
R2
0.2
0.4
0.6
0.8
0.9
0.2
0.4
0.6
0.8
0.9
=0, k =1
(Gavrilyuk
et al.,
2008)
3.386
3.386
3.382
3.139
2.187
24.153
10.014
6.665
4.550
2.683
MBEM
3.389
3.390
3.391
3.192
2.200
20.027
10.034
6.669
4.545
2.678
=1, k =1
(Gavrilyuk
et al.,
2008)
1.304
1.302
1.254
0.934
0.542
11.332
5.629
3.515
1.661
0.726
MBEM
1.305
1.307
1.259
0.954
0.574
11.303
5.626
3.481
1.651
0.732
=2, k =1
(Gavrilyuk
et al.,
2008)
2.263
2.263
2.255
2.015
1.361
17.760
8.967
5.941
3.724
1.923
MBEM
2.265
2.270
2.269
2.048
1.394
17.939
8.965
5.941
3.726
1.951
The comparison of results obtained by proposed method with data of I. Gavrilyuk
et al (Gavrilyuk et al., 2008) is presented here. The results are in good agreement except
the data for Ʌ- shape tank with for =0 and R2=0.2m. But it was noted in (Gavrilyuk et
al., 2008) that in this case the low convergence was achieved using the proposed there
analytical method. Next, we have carried out the numerical simulation of the natural
frequencies of liquid sloshing for tanks with baffles. Both V-shape and Ʌ-shape baffled
tanks are under consideration. We consider tanks of height H=H1+H2=1.0m with
different baffle positions H1. We use R1 = 1.0m and R2=0.5.m for both type of tanks (see
Fig. 8).
In Table 6 the results of numerical simulation are presented for =0, 1 and
different baffle positions, described by the height H1. Here we consider four eigenvalues
for =0,1. Radius of the conical shell at the baffle position is denoted as Rb, and the free
surface radius is Rint (Fig. 8). First, we have obtained the natural frequencies of V-shape
and Ʌ-shape conical tanks without baffles. It corresponds to values H1= H2 =0.5m,
Rint/Rb=1. The values of H1 and H2 can be arbitrary chosen, but H1+H2=1.0m. Then we
have put baffles at the different positions H1=0.5m and H1=0.8m and considered the
different sizes of baffles, namely Rint/Rb=0.5 and Rint/Rb=0.2.
The results obtained show different behaviour of decreasing frequencies for V-
shape and Ʌ-shape conical tanks. For Ʌ-shape tanks the baffle positions and their sizes
are not affected essentially on the values of frequencies. For V- shape tanks the effects
of baffle characteristics is more considerable.
It would be noted also that the first harmonic frequencies are lower than
axisymmetric ones both for V-shape and Ʌ-shape conical tanks.
The analytical numerical treatment for conical containers may require choosing a
coordinate system where most of boundary conditions may be exactly satisfied. The
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
35
proposed BEM does not require any transformations of initial equations and involving
the special coordinate system.
Table 6. Natural frequencies of V- shape and Ʌ-shape conical tanks with baffles
n
1
2
3
4
1
2
3
4
H1
H2
Rint/Rb
V-shape
Ʌ-shape
=0
0.5
0.5
1
3.466
6.681
9.845
12.99
7.985
14.37
20.70
27.01
0.5
0.5
0.5
3.408
6.668
9.843
12.99
7.968
14.37
20.69
27.01
0.5
0.5
0.2
3.405
6.635
9.843
12.99
7.960
14.37
20.69
27.01
0.8
0.2
0.5
2.527
6.387
9.724
12.92
7.344
14.25
20.66
26.99
0.8
0.2
0.2
2.443
6.059
9.565
12.88
7.113
14.20
20.65
26.99
=1
0.5
0.5
1
1.416
4.997
8.206
11.37
4.424
11.09
17.46
23.79
0.5
0.5
0.5
1.228
4.974
8.197
11.37
4.192
11.06
17.46
23.79
0.5
0.5
0.2
1.172
4.943
8.196
11.37
4.037
11.06
17.45
23.79
0.8
0.2
0.5
0.815
4.742
8.003
11.20
3.128
10.78
17.42
23.77
0.8
0.2
0.2
0.630
4.191
7.849
11.23
2.529
10.66
17.36
23.75
9. Partially filled rigid spherical baffled and un-baffled shells
Spherical tanks partially filled with liquid are difficult to analyze the free-liquid
natural frequencies and mode shapes using analytical methods. The difficulty arises due
to the fact that walls are not straight. Liquid spattering and sloshing in spherical tanks
was studied in papers (Faltinsen & Timokha, 2012; Kulczycki et al., 2016). A
characteristic feature of spherical tanks is the change in radius of a free surface with
changing in a filling level. There exist known analytical solutions for almost completely
filled tanks with small radii of the free surface, the so-called "ice fishing problems"
formulation. The effect of baffles on sloshing frequencies was studied by Biswal et al.
(Biswal et al., 1984).
In this paper we consider the problem of fluid vibrations in the rigid spherical
shells with and without baffles. To reduce the sloshing in the shell, an internal baffle is
installed, Fig. 9.
Figure 9. Spherical fuel tank with internal baffle
We denote here the wetted surface of the shell by Sw=
21 SS
, S1 and S2 are used
when the baffled shell is under consideration.
Consider the spherical shell of radius R = 1 m, partially filled with the ideal
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
36
incompressible fluid, with the filling level H. Numerical analysis is carried out for
99.1/2.0 RH
and various
3,0
.
Both single (SBEM) and multi-domain (MBEM) boundary element methods are
applied here. The boundary elements with constant approximation of unknowns inside
elements are used. In SBEM there are 200 elements along the spherical surface (Nw) and
150 elements along the free surface. In MBEM we divide the computational domain
into two parts by the artificial interface surface at
Hh 5.0
int
using 100 boundary
elements in each sub-domain along the spherical surface and 150 elements along the
free surface. We use practically the same mesh to find a numerical approximation of
low eigenvalues for the so called “ice-fishing problem”. In this problem, formally, we
should consider an infinitely wide and deep ocean covered with ice, with a small round
fishing hole. Sloshing in such “containers” was studied by McIver (McIver, 1989). We
approximate this infinite case using the spherical tank with the very small round hole on
its top. It allows us to compare our numerical results with those obtained in papers
(Kulczycki et al., 2016; McIver, 1989).
In Tables 7-8 we compare our results obtained by using SBEM and MBEM with
those obtained in (Kulczycki et al., 2016) – (McIver, 1989) for axisymmetric (=0) and
non-axisymmetric (=1) modes. Four first frequencies (
4,1m
) are evaluated for each
. Here we consider different filling levels h1. The value h1/R1=1.99 corresponds to the
ice-fishing problem. Obtained results and results of Faltinsen et al (Faltinsen &
Timokha, 2012) and Kulczycki et al. (Kulczycki et al., 2016) are very close.
Different levels of fluid filling are considered, including H/R = 1.99, that
corresponds to «ice-fishing problem», (McIver, 1989).
Table 7. Axisymmetric slosh frequencies parameters
g
n/
2
of the fluid-filled spherical shell.
m
method
Filling level H, m
H =0.2
H =0.6
H =1.0
H =1.8
H =1.99
1
(Kulczycki et al., 2016)
3.8261
3.6501
3.7451
6.7641
29.0500
(Cho & Lee, 2004)
3.8261
3.6501
3.7451
6.7641
29.2151
MBEM
3.4034
3.5455
3.7294
6.6098
30.7081
SBEM
3.8314
3.6510
3.7456
6.7665
29.1811
2
(Kulczycki et al., 2016)
9.2561
7.2659
6.9763
12.1139
51.8122
(McIver, 1989)
9.2561
7.2659
6.9763
12.1139
52.0467
MBEM
9.2636
7.2893
6.9796
12.0008
52.9393
SBEM
9.2686
7.2684
6.9780
12.1205
52.0255
3
(Kulczycki et al., 2016)
14.7556
10.7443
10.1474
17.3960
74.2909
(Cho & Lee, 2004)
14.7556
10.7443
10.1474
17.3960
74.5537
MBEM
14.9214
10.7483
10.1496
17.3136
75.3139
SBEM
14.7763
10.7502
10.1512
17.4086
74.5547
4
(Kulczycki et al., 2016)
20.1187
14.1964
13.3041
22.6579
96.6207
(Cho & Lee, 2004)
20.1187
14.1964
13.3041
22.6570
96.9560
MBEM
20.2066
14.2023
13.3083
22.5962
97.7771
SBEM
20.1498
14.2056
13.3110
22.6777
96.9021
The results of calculations with SBEM and MBEM are close, in some cases
SBEM gives more accuracy compared with MBEM, but the matrix size in SBEM is
twice larger compared with MBEM. If un-baffled tanks are at low filling levels, it is
preferable to use SBEM.
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
37
Table 8. Non-axisymmetric slosh frequencies parameters
g
n/
2
of the fluid-filled spherical shell
m
method
Filling level H, m
H =0.2
H =0.6
H =1.0
H =1.8
H =1.99
1
(Kulczycki et al.,
2016),
1.0723
1.2625
1.5601
3.9593
18.9838
(McIver, 1989)
1.0723
1.2625
1.5601
3.9593
19.1582
MBEM
1.1034
1.2777
1.5638
3.9606
19.1603
SBEM
1.0723
1.2626
1.5603
3.9508
19.1130
2
(Kulczycki et al.,
2016)
6.2008
5.3860
5.2755
9.4534
41.3491
(McIver, 1989)
6.2008
5.3860
5.2755
9.4534
41.7683
MBEM
6.1227
5.3534
5.2749
9.4582
41.5327
SBEM
6.2090
5.3697
5.2764
9.4538
41.5333
3
(Kulczycki et al.,
2016)
11.8821
8.9418
8.5044
14.7548
63.5354
(McIver, 1989)
11.8821
8.9418
8.5044
14.7548
64.0323
MBEM
11.9650
8.9529
8.5062
14.7648
63.9483
SBEM
11.8981
8.9429
8.5069
14.7574
63.8783
4
(Kulczycki et al.,
2016)
17.3581
12.4234
11.6835
20.0224
85.9166
(McIver, 1989)
17.3584
12.4234
11.6835
20.0224
86.3001
MBEM
17.4540
12.4276
11.6863
20.0394
86.2972
SBEM
17.3842
12.4291
11.6884
20.0278
86.2034
From results of Tables 7-8 one can observe sloshing frequencies behaviour with
increasing the fluid depth H. If radius R0 of the free surface increases, then the
frequencies decrease. In Table 9 the frequency parameters of axisymmetric sloshing are
compared for cylindrical and spherical shells via different ratios H/R0; where R0 is
radius for cylindrical shells. For defining the frequencies of cylindrical shells we use
formulae (28).
Table 9. Axisymmetric slosh frequencies parameters
g
n/
2
of the fluid-filled cylindrical and
spherical shells
H, m
0.2
0.6
1.0
1.8
1.99
R0
0.6
0.9165
1.0
0.6
0.1410
H/R0
0.3333
0.6546
1.0
3.0
14.106
cylinder
5.4649
3.8959
3.8281
6.3861
27.1752
sphere
3.8261
3.6501
3.7451
6.7641
29.2151
Frequency changing of cylindrical shells has similar non-monotonic behaviour.
Fig.10 below demonstrates the spatial wave patterns for =0, 1, n=1,2,3 at
H=1.8m; R=1m.
Considering our approximate natural sloshing modes one can observe how free
surface profiles change with the liquid depth.
These results are illustrated in Fig. 11 for the three lowest eigenvalues of the
mode = 0. Here numbers 1,2,3,4 correspond to the different filling levels, namely H
=1.0m; 0.2m; 1.8m; 1.9m, respectively.
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
38
Figure 10. Spatial wave patterns for =0,1
= 0, n=1
= 0, n=2
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
39
= 0, n=3
Figure 11. The radial wave profiles = 0, m=1,2,3, for different liquid depths H
We also consider non-axisymmetric modes, = 1, because the frequencies
corresponded to these modes are the lowest ones.
= 1, n=1
= 1, n=2
r
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
0.14
0.28
0.42
0.56
0.7
0.84
0.98
1.12
1.26
1.4
1.54
1.68
1.82
1.96
2.1
2.24
2.38
2.52
0
1
2
3
4
r
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
-1.98
-1.76
-1.54
-1.32
-1.1
-0.88
-0.66
-0.44
-0.22
0.22
0.44
0.66
0.88
1.1
1.32
1.54
1.76
1.98
2.2
2.42
0
1
2
3
4
JOURNAL OF MODERN TECHNOLOGY & ENGINEERING, V.3, N.1, 2018
40
= 1, n=3
Figure 12. The radial wave profiles = 1, m=1,2,3, for different liquid depths H
It would be noted that mode shapes presented in Fig. 12 have different magnitudes
in their picks, because these modes are not normalized. There dividing by R0 is applied
for comparison with results O.M. Faltinsen and A.N. Timokha (Faltinsen & Timokha,
2012) where namely such non-normalized modes are presented.
In the spherical tank with 0 < H/R< 0.5 the lowest mode presents a spatial wave
pattern that look like inclination of an almost flat free surface. Increasing the liquid
depth yields more complicated free surface profiles.
Next, the rigid spherical tank of radius R1=1m filled to the depth H=1.4m is
considered. The inner periphery of the tank contains a thin rigid-ring baffle. The baffle
position is hbaf=1m. The different annular orifices in the baffle are considered. Radii of
these orifices are radii Rint of the interface surfaces. The first three frequencies for mode
=1 are evaluated for radii Rint =1.0m, Rint=0.7m, and Rint=0.2m. Note that Rint =1.0m
correspond to the un-baffled tank. These frequencies are presented in Table 5.
Table 10. Vibrations of the tank with a baffle, frequency parameter
2/g
m
2/g
Rint=1.0 m
Rint=0.7m
Rint= 0.2 m
1
2.1232
2.0435
1.4234
2
5.9800
5.9723
5.8405
3
9.4789
9.4785
9.4567
Fig. 13 shows the first three forms of fluid vibrations in the spherical shell at =1
with baffles.
Figure 13. Modes of liquid vibrations in the baffled spherical shell.
r
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
0
1
2
3
4
E. STRELNIKOVA et al.: FREE AND FORCED VIBRATIONS OF LIQUID…
41
When the baffle is installed, the mode shape becomes almost flat. The presence of
the baffle has affected drastically only on the lower frequencies. Also one can see that
small baffles (when Rint is relatively large) do not affect the lower frequencies.
10. Impulse impacts on reservoirs
Consider the rigid cylindrical shell with the flat bottom partially filled with the
liquid. The tank parameters are following: radius is R = 1 m, thickness is h=0.01m,
length is L = 2 m, filling level is H =1.0m.
We determine pressure p upon shell walls from the linearized Cauchy-Lagrange's
integral by the following formula (Lamb, 1993):
xtapgzpstl
0
,
Here
tas
is a function characterizing external influence (a horizontal seism or an
impulse). The radial load is suddenly applied to cylindrical surface of the tank
as(t) = Q0a(t), where Q0 = 10 МPа – distributed pressure,
,
,0
,1
Tt
Tt
ta
T = 1.5 s.
Having defined the basic functions 2k, substitute them in expressions for velocity
potential
M
kkk
d
12
and for the free surface elevation
M
k
k
kn
td
1
2
(30)
Then substitute the received relations in the boundary condition on the free
surface
0
0s
st xtag
. (31)
As in cylindrical system of coordinates there is
cosrx
, we will be interested
only in the first harmonica, i.e. in a formula (22) we only consider =1. We come to
the following equation on the surface S0
0
1
2
12
rta
n
dgd s
M
k
k
k
M
kkk
.
Due to validity of relation (31) on the surface S0 the equality given above takes
the form
0
12
2
12
rtadd s
M
kkkk
M
kkk
. (32)
Accomplishing the dot product of equality (32) by
Ml
l,1
2
and having used
orthogonality of own modes (Gavrilyuk, 2006) we receive the system of ordinary
differential equations of the second order
MkrFFtadd kkkkks
k
kk ,1;,/,;0 222
2
. (33)