Natural sloshing frequencies and modes in an upright circular container with poles
Natural sloshing modes are approximated, semi-analytically, for an uprightcircular cylindrical tank equipped with an array of vertical poles (pipes)as it happens, e.g., in shafts of ocean platforms.
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| author | Timokha, A. N. Timokha, A. N. |
| author_facet | Timokha, A. N. Timokha, A. N. |
| author_institution_txt_mv | [
{
"author": "A. N. Timokha",
"institution": "Institute of Mathematics of NAS of Ukraine, Kiev, Ukraine; Centre of Excellence AMOS, Norwegian University of Science and Technology, Trondheim, Norway"
}
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| author_sort | Timokha, A. N. |
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| datestamp_date | 2018-01-23T12:01:13Z |
| description | Natural sloshing modes are approximated, semi-analytically, for an uprightcircular cylindrical tank equipped with an array of vertical poles (pipes)as it happens, e.g., in shafts of ocean platforms. |
| first_indexed | 2026-08-04T01:01:30Z |
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Збiрник праць Iнституту математики НАН України 2015, т. 12, № 5, 275–283
УДК 532.595
Natural sloshing frequencies and
modes in an upright circular container
with poles∗
A.N. Timokha
Institute of Mathematics of NAS of Ukraine, Kiev, Ukraine;
Centre of Excellence AMOS, Norwegian University of Science and
Technology, Trondheim, Norway
Natural sloshing modes are approximated, semi-analytically, for an upright
circular cylindrical tank equipped with an array of vertical poles (pipes)
as it happens, e.g., in shafts of ocean platforms.
Отримано чисельно–аналiтичнi наближення власних форм коливання
рiдини у вертикальному цилiндричному контейнерi, в якому встанов-
лено вертикальнi стовпи (труби), як то буває, наприклад, в шахтах
океанських платформ.
1. Introduction
Interior space of tanks contains, normally, devices and structures [2, ch. 1]
which play different roles, that includes, suppression of sloshing and stiff-
ening the construction. Vertical pipes (poles) are installed, e.g., in shafts
of ocean platforms [1]/cooling tanks of nuclear reactors [7]. The poles
modify the natural sloshing modes and frequencies as well as cause an
extra damping due to flow separation [2, ch. 6]. The both effects are
poorly investigated. This inhibits progress of analytical approaches to
nonlinear sloshing problems for those tanks.
The present paper constructs an analytically approximate solution
of the spectral boundary problem on the natural sloshing modes for an
∗ The work was partly supported by the Grant № 0112U001015. The author also
acknowledges the financial support of the Centre of Autonomous Marine Operations
and Systems (AMOS) whose main sponsor is the Norwegian Research Council (Project
number 223254–AMOS).
c© A.N. Timokha, 2015
276 A.N. Timokha
upright circular cylindrical tank with an array of vertical poles. The Tr-
efftz variational method is adopted that facilitates efficient computations
and parameter studies versus position, number and radii of the poles.
Numerical examples are mainly associated with computations in [1].
D
x=x
y=y
y
xn
0
0
n
0
k
k
k
Γ0
kΓ
D
Fig. 1. The left panel: Geometric notations to the spectral boundary prob-
lem (1) in a circular cheese–like domain D associated with the cross–sectional
area of an upright circular cylindrical tank equipped with vertical poles. The
domain is confined by the external (circular) boundary Γ0 and has, in addi-
tion, circular holes Γk, k = 1, . . . ,M . The outer normal n to D is the same as
the set of nk on Γk, k = 0, . . . ,M . The solution method requires introducing
the local Cartesian Okxkyk (O0x0y0 = Oxy) and polar (rk, θk), k = 0, . . . ,M
coordinates. The right panel: A schematic cross-sectional plan of the Draugen
platform at the sea level (dimensions are reported in [1]).
2. Statement of the problem
The spectral (eigenvalue) boundary problem on natural sloshing modes
and frequencies in an upright cylindrical tank admits separation of the
vertical coordinate z and, thereby, reduces to the boundary eigenvalue
problem in the cross–sectional area D (see, an extensive discussion, e.g.,
in [2, ch. 4] and [1]). When choosing the Oxy plane coinciding with
the flat mean free surface and h is the mean liquid depth, separating z
implies the natural sloshing modes Φ(x, y, z;λ) = ϕ(x, y;λ) cosh(λ(z +
h))/ cosh(λh) where ϕ(x, y;λ) (λ > 0, the trivial solution λ = 0,Φ =
ϕ = const does not satisfies the volume conservation condition and should
Sloshing frequencies and modes in container with poles 277
be excluded) comes from the well–known spectral boundary problem
−∇2ϕ = λϕ in D; ∂nϕ = 0 on ∂D; λ > 0 (1)
for the Laplace operator ∆ = ∇2. The natural sloshing frequencies are
then computed by σ =
√
gλ tanh(λh) where g is the gravity acceleration.
Domain D and its boundary ∂D = ∪Mk=0Γk are shown in fig. 1 (the left
panel introduces, schematically, geometric notations, but the right panel
illustrates the actual geometry for the Draugen platform shaft). Circles
Γk have the radii Rk and their centres are at (x0k, y0k) in the global
Cartesian coordinates Oxy. Henceforth, (1) is scaled by R0 that suggests
h := h/R0, R0 = 1, Rk := Rk/R0, x0k := x0k/R0, y0k := y0k/R0, k =
0, . . . ,M and so on.
Remark 2.5. The spectral problem (1) has exact analytical solu-
tions for M = 0 (no holes, a purely circular cylindrical tank) and M = 1
when the origins coincide, O0 = O1 (an annular cylindrical tank). These
solutions are documented, in some detail, in [2, 4, 6]. Eigenvalues λ are
then computed from a transcendental equation involving the Bessel func-
tions. This is even though (1) is a linear spectral problem which reduces,
naturally, to a linear algebraic problem when using traditional numerical
methods. The latter methods do not provide the analytical solutions and,
generally speaking, demonstrate a rather low convergence [1, 5, 7] to the
exact analytical λ.
Remark 2.6. A rough approximation of the natural sloshing frequen-
cies (and, therefore, λ) can also be found utilising the strip theory and
the Reynolds quotient formulation of the original spectral sloshing prob-
lem ([1] and [2, ch. 4]). This is, generally, unable to construct the natural
sloshing modes (required in the analytical sloshing), assumes, explicitly,
that the poles are located far from each other (the proximity effect is
neglected) and their radii Ri, i ≥ 1 are small relative to R0.
3. Solution method
The present paper uses a variational Trefftz method which suggests a
λ-depending representation of ϕ(x, y;λ) and trial functions δϕ(x, y;λ) so
that getting λ > 0 and ϕ(x, y;λ) follows from the variational equation [3]
∫
∂D
∂nϕ δϕdΓ ≡
M∑
k=0
∫
Γk
(∇ϕ · nk) δϕ dΓ = 0. (2)
278 A.N. Timokha
The eigenfunction ϕ is furthermore posed as
ϕ(x, y;λ) =
M∑
k=0
2Nk∑
i=0
ak,i φk,i(rk, θk;λ), Nk → ∞, (3)
where λ and {ak,i} are the unknowns, but the functional basis {φk,i}
consists of
{φ0,i(x, y;λ), i = 0, . . . , 2N0} = {J0(λr0), J1(λr0) cos θ0, . . . ,
JN0(λr0) cosN0θ0, J1(λr0) sin θ0, . . . , JN0(λr0) sinN0θ0}, (4)
{φk,i(x, y;λ), i = 0, . . . , 2Nk} = {Y0(λrk), Y1(λrk) cos θk, . . . ,
YNk
(λrk) cosNkθk,
Y1(λrk) sin θk, . . . , YNk
(λrk) sinNkθk}, k = 1, . . . ,M, (5)
defined in the polar coordinates (rk, θk) but considered, altogether, as
functions of x, y and λ; Jl(·) and Yl(·) are the Bessel functions of the
first and second kinds, respectively, and λ > 0. The functional subset (4)
approximates a “regular” component (no singularities inside the circles
Γk, k = 1, . . . ,M), but (5) fits the local singular behaviour of ϕ(x, y;λ)
inside the circular holes. The functional set (4), (5) is complete since the
Dirichlet and Neumann traces of {φl,i} on Γl = {x = xl0 +Rl cos θl, y =
xl0 + Rl sin θl; 0 ≤ θl < 2π} are, within to multipliers, the 2π–periodic
Fourier basis.
Substituting (3) into (2) within δϕ = φn,j gives the
[∑M
k=0(2Nk + 1)
]
×
[∑M
k=0(2Nk + 1)
]
system of linear homogeneous algebraic equations
M∑
k=0
2Nk∑
i=0
[
M∑
l=0
Rl
∫ 2π
0
[(∇φk,i|Γl
· nl)φn,j ]Γl
dθl
]
︸ ︷︷ ︸
B(n,j)(k,i)(λ)
ak,i = 0 (6)
with respect to {ak,i} and the symmetric matrix (B(n,j)(k,i) = B(k,i)(n,j)
due to the second Green identity) must have the zero-determinant
det{B(n,j)(k,i)(λ)} = 0 (7)
which is, in fact, a transcendental equation with respect to λ in terms of
the Bessel functions.
Sloshing frequencies and modes in container with poles 279
R
11
11
λ 11
λ
x
y
1
1R
0.5
sin
cos
1
λ
1
0 0.1 0.2 0.3 0.4 0.5
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
2
Fig. 2. Two lowest eigenvalues λsin
11 and λcos
11 splitting from λ11 =
1.841 . . . ; J ′
1(λ11) = 0 for D without holes (the corresponding exact analytical
eigenfunctions are then J1(λr) sin θ and J1(λr) cos θ) when a single eccentric
pole of the nondimensional radius 0 < R1 < 0.5 is installed with the x = 0.5-
offset. The computed values of λsin
11 and λcos
11 versus R1 are marked by the
solid lines. The dashed lines follow from the asymptotic (R1 ≪ 1) formula [2,
eq. (4.185)].
Remark 3.7. As discussed in [3, p. 179], roots of (7) approximate
eigenvalues λ of both Neumann’s (1) and Dirichlet’s
−∇2ϕ = λϕ in D; ϕ = 0 on ∂D (8)
spectral boundary problems. Readers can justify this fact (after a rather
tedious derivation) for circular and annular tanks when analytical solu-
tions exist. A posteriori numerical procedure is therefore needed to select
which of computed λ correspond to (1).
Remark 3.8. The spectral problem (1) possesses degenerate eigen-
values (the second-order multiplicity) for circular, annular, and any other
cross-section that is invariant relative to the π/2-rotation. For that case,
the determinant of (7) does not change the sign at the corresponding λ
that means a difficulty in a numerical detection of its zeros. One can
then focus on finding the minima of the determinant modulus [1]. Al-
ternatively, the weight coefficients in (2) may be recombined to provide,
a priori, the required eigenfunction symmetry excluding, thereby, the
multiplicity.
280 A.N. Timokha
0.8
−0.4
−0.8
0
0.4
0.8
0.4
−0.4
−0.8
0
−0.8
−0.4
0
0.4
0.8
−0.4 0 0.4 0.8
−0.4
0
0.8
0.4
−0.8 0.8
0.4
0
−0.4
−0.8
−0.8
−0.4
0
0.4
0.8
−0.8
−0.8 −0.4 0 0.4
0.8
Fig. 3. Two eigenfunctions ϕ(x, y;λcos
11 ), the first row, and ϕ(x, y;λsin
11 ), the
second row, for the case in fig. 2 with R1 = 0.4. Three-dimensional and topo-
graphic views.
Remark 3.9. Computing integrands in (6) requires to know, ex-
plicitly, ∇φk,i|Γl
and φk,i|Γl
as functions of θl for all 0 < k < M
(n0 = (cos θ0, sin θ0) and nl = −(cos θl, sin θl)). Originally, φk,i and
∇φk,i = (∂xφk,i, ∂yφk,i),
∂xφk,i = ∂xk
φk,i = cos θk ∂rkφk,i − r−1
k sin θk ∂θkφk,i,
∂yφk,i = ∂ykφk,i = sin θk ∂rkφk,i + r−1
k cos θk ∂θkφk,i,
(9)
are functions of the polar coordinates (rk, θk) where rk =
√
x2k + y2k, θk =
atan2(yk, xk) (atan2 is the acrtangent function of two arguments well–
known in the computer languages) in terms of the local Cartesian co-
ordinate system Okxkyk. However, substituting xk = x0l − x0k +
Rl cos θl, yk = y0l− y0k+Rl sin θl makes them exclusively depending on
0 ≤ θl < 2π.
Sloshing frequencies and modes in container with poles 281
4. Numerical examples
16
95
914
650
38
00
659
18
50
25
00
1000
4260
7500
0 0.3 R0 0.3 R0 0.3 R00.3 R00.25 R 0
0.
3
R
0
0.
3
R
0
0.
3
R
0
0.
3
R
0
0.
3
R
0
0.
25
R
0
0.3 R
Fig. 4. The left panel: A simplified cross-sectional plane of the Draugen
platform shaft (see, the actual geometry in fig. 1) considered in [2, sect. 4.11.5]
suggesting that pipes of lower radii are neglected. Dimensions are in mm.
The right panel: A sample cross-sectional domain considered in [1] includes a
symmetric (with respect to the coordinate axes) array of equal poles of the
radius Ri, i = 1, . . . , 32.
A single eccentric pole with the 1/2–offset is considered with a fo-
cus on the two lowest eigenvalues λsin11 and λcos11 splitting from the two
equal lowest eigenvalues λ11 = 1.841 . . . ; J ′
1(λ11) = 0 which correspond
to eigenfunctions J1(λ11r) sin θ and J1(λ11r) cos θ. The numerical re-
sults are shown in fig. 2 by the solid lines versus the nondimensional
pole radius, 0 < R1 < 0.5. These are consistent with computations
in [1] (boundary element method); difference is invisible when superpos-
ing the graphs. In addition, the dashed lines are drawn determined by
the asymptotic formula [2, eq. (4.185)] which is derived by combining the
Rayleigh quotient formulation and the strip theory (the result depends
on h, the deep water approximation is used). Fig. 2 shows that the lowest
λsin11 monotonically decreases and using [2, eq. (4.185)] provides a good
approximation of this eigenvalue forR1 . 0.2. However, the second eigen-
value λcos11 is not accurately approximated by the asymptotic formula; the
function λcos11 (R1) is not monotonic but the formula does not fit that. The
corresponding eigenfunctions are shown in fig. 3 by the three-dimensional
282 A.N. Timokha
Table 1. The sample case in the right panel of fig. 4 suggesting 32 poles
of the equal radii Ri, i = 1, ..., 32. The poles are located symmetrically with
respect to the Cartesian axes so that the lowest eigenvalue implies two natural
sloshing modes. The computed lowest eigenvalue λ11 by the Trefftz method
(second column) is compared with the deep water prediction by the formula
[2, eq. (4.185)] (third column) and the boundary element (BE) computations
in [1] (columns 4-6) conducted with different numbers of facets per pipe (pole)
while 60 facets per the outer circle is used. The framed numbers correspond
to the exact solution detected by our method and the asymptotic formula [2,
eq. (4.185)].
present computations BEM [1], 60 facets/ outer circle
Ri/R0 Trefftz app. [2, eq. (4.185)] 6 fac./pipe 12 fac./pipe 24 fac./pipe
0.0000 1.8412... 1.8412... 1.847 1.847 1.847
0.0125 1.8359 1.8359 1.840 1.841 1.842
0.0250 1.8202 1.8201 1.818 1.823 1.825
0.0375 1.7946 1.7935 1.783 1.794 1.798
0.0500 1.7602 1.7542 1.736 1.755 1.761
0.0625 1.7178 1.7043 1.677 1.706 1.716
0.0750 1.6685 1.6253 1.607 1.649 1.663
and topographic views. The function ϕ(x, y;λcos11 ) is even by y and the
associated liquid flows mainly occur “along” the Ox-axis. The proxim-
ity effect (between the tank walls and the pole) may matter causing the
non-monotonic character of the λcos11 (R1) as R1 tends to 0.5. The lowest
natural sloshing mode is odd by y, ϕ(x, y;λsin11 ) = −ϕ(x,−y;λsin11 ).
An example from [2, sect. 4.11.5] is considered which appears after
excluding all pipes of lower diameters from the Draugen platform shaft.
This simplification is shown in the left panel of fig. 4 (compare it with
the actual cross–sectional plane in fig. 1). The highest natural sloshing
period for the clean (without pipes) shaft is T1 = 4.04881 s that corre-
sponds to the two degenerating natural sloshing modes, ϕ(x, y;λ11) =
J1(λ11r) cos θ and J1(λ11r) sin θ. This natural sloshing period T1 splits,
according to our computations, to T ∗
1 = 4.10807 s and T ∗∗
1 = 4.06965 s.
The highest period T ∗
1 was approximated in [2, sect. 4.11.5] by the asymp-
totic formula [2, eq. (4.185)]. The asymptotic result is 4.111 s.
Computations were done for the perforated domain in the right panel
of fig. 4. This sample geometry was introduced and analysed in [1]. Ta-
ble 1 compares our computations with [1] and the asymptotic formula [2,
eq. (4.185)]. Our numerical method (the asymptotic formula as well) fits
Sloshing frequencies and modes in container with poles 283
the exact solution for the clean tank (Ri/R0 = 0, framed numbers) but [1]
clearly overpredicts the corresponding exact eigenvalue, most probably,
due to a slow convergence of the boundary element method. The formula
[2, eq. (4.185)] remains rather accurate for the lower poles radii.
5. Conclusions
A semi-analytical approximation of the natural sloshing modes in an
upright circular cylindrical tank equipped with vertical poles can be ob-
tained by applying a variational Trefftz method to the corresponding
spectral boundary problem formulated in the cross–sectional plane. The
method employs the Bessel–type functional basis parametrically depend-
ing on the spectral parameter. The spectral parameter appears in a
nonlinear manner and the solvability condition yields a transcendental
equation with respect to the parameter. The equation turns to the well-
known form when the exact analytical solution exists. Numerical exam-
ples demonstrate the accuracy and effectiveness of the proposed method.
Comparisons are made with computations in [1] and the asymptotic for-
mula [2, eq. (4.185)] approximating the lowest natural sloshing frequen-
cies.
[1] Drake K. The effect of internal pipes on the fundamental frequency of liquid
sloshing in a circular tank // Applied Ocean Research.— 1999.— 21.—
P. 133–143.
[2] Faltinsen O.M., Timokha A.N. Sloshing.— Cambridge University Press,
2009.— 686 p.
[3] Faltinsen O.M., Timokha A.N. Analytically approximate natural sloshing
modes and frequencies in two-dimensional tanks // European Journal of
Mechanics. B/Fluids.— 2014.— 47.— P. 176–187.
[4] Ibrahim R. Liquid sloshing dynamics.— Cambridge University Press,
2005.— 900 p.
[5] Li T. F., Lin C. C., Luk C. Three-directional fluid pool seismic sloshing
analysis // ASME Journal of Pressure Vessel Technology.— 1981.— 103.—
P. 10–15.
[6] Lukovsky I.A. Nonlinear Dynamics: Mathematical Models for Rigid Bodies
with a Liquid.— De Gruyter, 2015.— 400 p.
[7] Takahara H., Hara K., Ishida T. Nonlinear liquid oscillation in a cylindrical
tank with an eccentric core barrel // Journal of Fluids and Structures.—
2012.— 35.— P. 120–132.
|
| id | oai:trim.imath.kiev.ua:article-43 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:01:30Z |
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| publisher | Інститут математики НАН України |
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| resource_txt_mv | trimimathkievua/5a/32cf57265eba878a62ceea26ea862f5a.pdf |
| spelling | oai:trim.imath.kiev.ua:article-432018-01-23T12:01:13Z Natural sloshing frequencies and modes in an upright circular container with poles Власні частоти і моди коливання рідини в вертикальному круговому циліндрами із вертикальними трубами Timokha, A. N. Timokha, A. N. Natural sloshing modes are approximated, semi-analytically, for an uprightcircular cylindrical tank equipped with an array of vertical poles (pipes)as it happens, e.g., in shafts of ocean platforms. Отримано чисельно–аналiтичнi наближення власних форм коливаннярiдини у вертикальному цилiндричному контейнерi, в якому встанов-лено вертикальнi стовпи (труби), як то буває, наприклад, в шахтахокеанських платформ. Інститут математики НАН України 2015-12-01 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/43 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 12 No. 5 (2015): Mathematical problems of mechanics and computational mathematics; 275-283 Сборник Трудов Института математики НАН Украины; Том 12 № 5 (2015): Математичні проблеми механіки та обчислювальної математики; 275-283 Збірник Праць Інституту математики НАН України; Том 12 № 5 (2015): Математичні проблеми механіки та обчислювальної математики; 275-283 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/43/14 Авторське право (c) 2015 Праці Інституту математики НАН України |
| spellingShingle | Timokha, A. N. Timokha, A. N. Natural sloshing frequencies and modes in an upright circular container with poles |
| title | Natural sloshing frequencies and modes in an upright circular container with poles |
| title_alt | Власні частоти і моди коливання рідини в вертикальному круговому циліндрами із вертикальними трубами |
| title_full | Natural sloshing frequencies and modes in an upright circular container with poles |
| title_fullStr | Natural sloshing frequencies and modes in an upright circular container with poles |
| title_full_unstemmed | Natural sloshing frequencies and modes in an upright circular container with poles |
| title_short | Natural sloshing frequencies and modes in an upright circular container with poles |
| title_sort | natural sloshing frequencies and modes in an upright circular container with poles |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/43 |
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