Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank
The analytic technique and numerical experiments are employed to show that the orbital elliptic translational ex citations of a square-base container can, depending on the ratio of the semiaxes of the elliptic orbit, lead, when the forcing frequency is close to the lowest natural sloshing frequenc...
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| Cite this: | Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank / O.E. Lagodzinskyi, A.N. Timokha // Доповіді Національної академії наук України. — 2021. — № 6. — С. 45-51. — Бібліогр.: 8 назв. — англ. |
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Lagodzinskyi, O.E. Timokha, A.N. 2022-07-17T14:27:05Z 2022-07-17T14:27:05Z 2021 Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank / O.E. Lagodzinskyi, A.N. Timokha // Доповіді Національної академії наук України. — 2021. — № 6. — С. 45-51. — Бібліогр.: 8 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2021.06.045 https://nasplib.isofts.kiev.ua/handle/123456789/184816 532.595 The analytic technique and numerical experiments are employed to show that the orbital elliptic translational ex citations of a square-base container can, depending on the ratio of the semiaxes of the elliptic orbit, lead, when the forcing frequency is close to the lowest natural sloshing frequency, to both the counter- and co-directed (relative to the orbital forcing direction) stable swirling-type steady-state resonant waves. For a non-zero damping in the hydrodynamic wavy system, the passage to circular orbits makes the stable counter-directed swirling impossible. Застосовується аналітична техніка та чисельні експерименти для того, аби показати, що орбітальні еліптичні поступальні збурення баку квадратного перерізу можуть призвести в залежності від співвідношен ня напіввісей еліптичної орбіти до як проти- так і співнаправленої (відносно напрямку збурення баку) стійкої усталеної кругової хвилі. Частоти збурення близькі до першої власної частоти коливання рідини. Для ненульового демпфування в гідродинамічній системі перехід до кругових орбіт робить неможливими протинаправлені кругові хвилі. The authors acknowledge the financial support of the National Research Foundation of Ukraine (Project number 2020.02/0089). The second author also acknowledges a partial support of Centre of Autonomous Marine Operations and Systems (AMOS) whose main sponsor is the Norwegian Research Council (Project number 223254-AMOS). en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Механіка Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank Проти- та співнаправлені кругові хвилі за орбітальних збуреннях баку квадратного перерізу Article published earlier |
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Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank |
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Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank Lagodzinskyi, O.E. Timokha, A.N. Механіка |
| title_short |
Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank |
| title_full |
Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank |
| title_fullStr |
Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank |
| title_full_unstemmed |
Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank |
| title_sort |
counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank |
| author |
Lagodzinskyi, O.E. Timokha, A.N. |
| author_facet |
Lagodzinskyi, O.E. Timokha, A.N. |
| topic |
Механіка |
| topic_facet |
Механіка |
| publishDate |
2021 |
| language |
English |
| container_title |
Доповіді НАН України |
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Видавничий дім "Академперіодика" НАН України |
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Article |
| title_alt |
Проти- та співнаправлені кругові хвилі за орбітальних збуреннях баку квадратного перерізу |
| description |
The analytic technique and numerical experiments are employed to show that the orbital elliptic translational ex citations
of a square-base container can, depending on the ratio of the semiaxes of the elliptic orbit, lead, when the
forcing frequency is close to the lowest natural sloshing frequency, to both the counter- and co-directed (relative
to the orbital forcing direction) stable swirling-type steady-state resonant waves. For a non-zero damping in the
hydrodynamic wavy system, the passage to circular orbits makes the stable counter-directed swirling impossible.
Застосовується аналітична техніка та чисельні експерименти для того, аби показати, що орбітальні еліптичні поступальні збурення баку квадратного перерізу можуть призвести в залежності від співвідношен ня напіввісей еліптичної орбіти до як проти- так і співнаправленої (відносно напрямку збурення баку) стійкої усталеної кругової хвилі. Частоти збурення близькі до першої власної частоти коливання рідини.
Для ненульового демпфування в гідродинамічній системі перехід до кругових орбіт робить неможливими
протинаправлені кругові хвилі.
|
| issn |
1025-6415 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/184816 |
| citation_txt |
Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank / O.E. Lagodzinskyi, A.N. Timokha // Доповіді Національної академії наук України. — 2021. — № 6. — С. 45-51. — Бібліогр.: 8 назв. — англ. |
| work_keys_str_mv |
AT lagodzinskyioe counterandcodirectedswirlingtypewavesduetoorbitalexcitationsofasquarebasetank AT timokhaan counterandcodirectedswirlingtypewavesduetoorbitalexcitationsofasquarebasetank AT lagodzinskyioe protitaspívnapravleníkrugovíhvilízaorbítalʹnihzburennâhbakukvadratnogopererízu AT timokhaan protitaspívnapravleníkrugovíhvilízaorbítalʹnihzburennâhbakukvadratnogopererízu |
| first_indexed |
2025-11-25T23:31:20Z |
| last_indexed |
2025-11-25T23:31:20Z |
| _version_ |
1850581963881054208 |
| fulltext |
45ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2021. № 6: 45—51
Ц и т у в а н н я: Lagodzinskyi O.E., Timokha A.N. Counter- and co-directed swirling-type waves due to orbital
excitations of a square-base tank. Допов. Нац. акад. наук Укр. 2021. № 6. С. 45—51.
https://doi.org/10.15407/dopovidi2021.06.045
https://doi.org/10.15407/dopovidi2021.06.045
UDC 532.595
O.E. Lagodzinskyi 1,
A.N. Timokha 1,2, https://orcid.org/0000-0002-6750-4727
1 Institute of Mathematics of the NAS of Ukraine, Kyiv
2 Centre of Excellence “Autonomous Marine Operations and Systems”, Department of Marine Technology,
Norwegian University of Science and Technology, Trondheim, Norway
E-mails: lagodzinskyi@gmail.com; tim@imath.kiev.ua, atimokha@gmail.com
Counter- and co-directed swirling-type waves due
to orbital excitations of a square-base tank
Presented by Academician of the NAS of Ukraine A.N. Timokha
The analytic technique and numerical experiments are employed to show that the orbital elliptic translational ex ci-
tations of a square-base container can, depending on the ratio of the semiaxes of the elliptic orbit, lead, when the
forcing frequency is close to the lowest natural sloshing frequency, to both the counter- and co-directed (relative
to the orbital forcing direction) stable swirling-type steady-state resonant waves. For a non-zero damping in the
hydrodynamic wavy system, the passage to circular orbits makes the stable counter-directed swirling impossible.
Keywords: sloshing, swirling, stability, orbital forcing
МЕХАНІКА
MECHANICS
In the recent paper [1], the authors have finalized a series of systematic theoretical studies on the
steady-state resonant sloshing in a square-base tank, which performs a periodic (cyclic) motion
with five degrees of freedom (sway, pitch, surge, roll, and yaw, when no vertical excitations are
allowed) with the forcing frequency close to the lowest natural sloshing frequency. The Nari-
manov--Moiseev-type nonlinear modal system [2, Chapter 9] is employed. These studies estab-
lished an asymptotic equivalence between periodic solutions of the nonlinear modal system (these
solutions are associated with steady-state resonant surface waves) for an arbitrary cyclic non-
parametric sway-pitch-surge-roll-and-yaw excitation and those ones coming from the modal sys-
tem, when the tank performs an elliptic horizontal orbital motion. The equivalence implies that,
for any periodic non-parametric (non-heave) tank excitation, one can match a horizontal elliptic
tank orbit, which causes, to within the highest-order asymptotic quantities in the corresponding
periodic solutions of the modal system, the same steady-state surface waves. Furthermore, the
authors were concentrating on effective frequency domains of the almost standing and swirling
steady-state wave modes for translational, diagonal, and oblique positions of the matched elliptic
46 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2021. № 6
O.E. Lagodzinskyi, A.N. Timokha
orbits. These steady-state wave results were compared with those for an upright circular cylindri-
cal tank [3—6] for which the longitudinal horizontal forcing [3] leads to standing, swirling and
irregular wave modes, circular horizontal tank orbit [4, 5] yields the co-directed (to the orbit di-
rection) swirling, and the elliptic tank excitation may cause both counter- and co-directed swirl-
ing-type waves [6]. The square-base cross-section makes the mentioned `classification’ of the
steady-state resonant wave modes much more complicated, so that, e.g., stable nearly-standing
waves become possible even for circular orbits, while, as we remember, this excitation type of
circular-base tank only yields the co-directed swirling. A very special question appears on wheth-
er the counter-directed stable swirling wave mode exists, when the excitation orbit of a square-
base tank approaches the circular shape. The present paper addresses the question after conduct-
ing a parametric analytic-and-numerical analysis following the paper [1].
A square-base container moves translatorily along a horizontal elliptic orbit, as it is sche-
matically shown in Fig. 1. For the prescribed elliptic semiaxes values xe and ye and the angle
between the major semiaxis and the horizontal coordinate axis, the two non-dimensional ge-
neralized coordinates describing the horizontal translational orbital tank motion can be defined
to within the time-lag substitution t t + const as follows:
1
2
( ) ( ) [ cos ]cos [ sin ]sin ;
( ) ( ) [ sin ]cos +[ cos ]sin .
x x y
y x y
t t e t e t
t t e t e t
(1)
Because of the symmetry planes Oxz and Oyz for the square base, one can, without loss of ge-
nerality, assume 0 / 4 and associate the major semiaxis with xe . Mathematically, the
two generalized coordinates (1) determine either counterclockwise or clockwise orbit or imply
a reciprocating tank excitation. For 0 / 4 , the sign of x ye e discriminates the counter-
clockwise ( > 0)x ye e or clockwise ( < 0)x ye e forcing, or the reciprocation
Fig. 1. A top view on a schematic elliptic orbital
trajectory, which is characterized by the non-di men-
sional sizes of semiaxes xe and ye , as well as the
angle between the major semi- and horizontal-
coordinate axes. Because of the symmetry of two co-
ordinate-planes, one can, without loss of generality,
assume 0 / 4 . The cyclic tank motions along
the elliptic orbit occur either counterclockwise or
clockwise so that, when the two translational gene-
ralized coordinates in (1) determine an elliptic or bit
with a non-zero 0xe , 0 y xe e , the forcing di-
rection is defined from (2). The resonant swirling
wave may be counter- or co-directed with respect to
the orbital forcing direction.
47ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2021. № 6
Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank
( 0)x ye e . (2)
The forcing amplitude should be an asymptotically small value, which means that
( ) ~ ( ) 1 1,2i t O , i .
The Narimanov--Moiseev asymptotic approximation of the free-surface resonant waves
suggests the following ansatz:
1/3 2/3
(1) (2) (1) (2) (1) (2)
1 1 2 2 11 1 2 2 1 1
( ) ( )
(1) (2) (1) (2) (1)
3 3 21 123 3 2 1 1
( , , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
O O
z f x y t a t f x b t f y a t f x b t f y c t f x f y
a t f x b t f y c t f x f y c t f
(2)
2
( )
) ( ) [lin.]+o( ),
O
x f y
(3)
where (1) (2)[ ( ) ( )]i jf x f y are the natural sloshing modes in the non-dimensional tank (scaled by
the tank width), the term [lin.] implies the linear wave component, and
(1)( ) cos( ( +1/2))if x i x ; (2)( ) cos( ( +1/2)) 0.if y i y , i (4)
Based on t he modal approximation (3), (4), the Narimanov--Moiseev-type modal system of
nonlinear differential equations was derived [2, Chapter 9], which couples the sloshing-related
generalized coordinates, 1 1 2 2 1 3 3 21 12( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )a t , b t , a t , b t , c t , a t , b t , c t , c t . This system is
studied in [1] for the arbitrary orbital tank forcing (1). The steady-state asymptotic (periodic)
solution of the modal system contains the dominant asymptotic component
1/3
1 1( ) cos + sin + ( ); ( ) cos + sin + ( ) ~ ~ ~ ~ ( )a t a t a t O b t b t b t O , a b a b O , (5)
where is the forcing frequency, and the non-dimensional amplitude parameters a, a, b, b are
real solutions of the following system of nonlinear algebraic (secular) equations:
2 2 2 2
1 2 3 2 3 1
2 2 2 2
1 2 3 2 3 1
2 2 2 2
1 2 3 2 3 1
2 2 2 2
1 2 3 2 3 1
[ + ( + )+ + ]+ [( ) ] cos ,
[ + ( + )+ + ]+ [( ) ] sin ,
[ + ( + )+ + ]+ [( ) ] cos ,
[ + ( + )+ + ]+ [( ) + ] sin
x
y
y
x
a m a a m b m b a m m bb+ P e
a m a a m b m b a m m bb P e
b m b b m a m a b m m aa P e
b m b b m a m a b m m aa P e
,
(6)
in which the so-called Moiseev detuning parameter
2 2 2/3
11 / ( )O (7)
depends on the forcing frequency (the smallness of on the ( )O -scale reflects the resonance
condition, i.e. it says that the forcing frequency should be close to the lowest natural sloshing
frequency 1 ), the relatively small damping coefficient 2/3
0,1 1,02 2 ( )O is associated
with the logarithmic decrement of the lowest natural sloshing modes (1)
1[ ( )]f x and (2)
1[ ( )]f y ,
but 1 2m , m and 3m , as well as 1P , are functions of the non-dimensional liquid depth h . Explicit
48 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2021. № 6
O.E. Lagodzinskyi, A.N. Timokha
formulas for 1P , 1 2 3,m , m m and can be found in [1], but [2, Fig. 9.7] graphically illustrates
the values of 1 2m , m and 3m as functions of h , specifically,
2 1 3 1 30 0 0m < m < , < m , m +m > as 0.3368h > …. (8)
In the most general case, computing the non-dimensional amplitude parameters a, a, b, b
from the secularity equations (6) for each given frequency [solution of (6) may not be unique]
and substituting the result into (5) and (3) determine the lowest-order approximation of reso-
nant steady-state surface-wave patterns (modes), which can be either stable or unstable. An ana-
lytic stability criterion is derived and proven in [7, Eq. (4.10)].
The stable steady-state wave solutions of the Narimanov-Moiseev-type modal equations
describe swirling (rotary), standing, or/and nearly-standing waves. When all steady-state so-
lutions for the fixed forcing frequency are unstable, irregular (chaotic) surface wave patterns
are theoretically expected. In the lowest-order approximation (using only the 1/3( )O -order
terms in (3)), one can discriminate the following stable wave types as functions of the ampli-
tude parameter +ab ab :
2/3( ) 0O -counterclockwise swirling; 2/3( ) 0O < - clockwise swirling
( )O - nearly standing wave; ( )o - standing wave. (9)
When taking 1P , 1 2 3,m , m m and from [1] and/or [2] for a fixed non-dimensional liquid
depth h , as well as the elliptic forcing parameters, which include xe , ye , and , and varying
by (7) in (6), the solution of the secular system (6) with respect to the non-dimensional amp-
litude parameters a, a, b, b makes it possible to draw the wave-amplitude response curves in
the 1( / ), A, B -space, where the resulting 1/3( )O -order non-dimensional amplitudes in the
horizontal directions (along Ox and Oy ) are
22 2 2+ +A a a , B b b . (10)
Implementing the stability criterion [7, Eq. (4.10)] identifies for every point on the response
curves whether it corresponds to stable or unstable solution (steady-state wave).
The limiting (longitudinal and diagonal) forcing cases with 0 and / 4 are in-
ves tigated in [2, Chapter 9] in an analytic way. For these cases, it was proved in [1] that both
coun ter- and co-directed stable swirlings exist, when 1/ 0y xe e (the reciprocating tank mo-
tion), but counter-directed resonant steady-state swirling disappears as 1 1 (the circular
tank orbit). Specifically, the developed analytic technique is not applicable to oblique positions
of the elliptic tank orbit, i.e., when 0 / 4< < . That is why, the oblique elliptic forcing is only
exemplified in [1] with / 6 and /12 , and other input parameters associated with ex-
periments in [8].
Utilizing a modified semianalytic algorithm from [1], we conducted an exhaustive nume rical
analysis of response curves in the 1( / ), A, B -space for 0 / 4< < , 0.4 0.8h (the res-
pon se curves practically do not change with larger liquid depths), 10.001 0.01 0 <1xe , < ,
and 0.001 0.01 . The input values belong/cover the most realistic and physically admissible
domains. Even though the response curves may significantly change with the chosen input values,
49ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2021. № 6
Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank
globally, all of them confirm that the counter-directed swirling mode becomes impossible (di-
sappears), as 1 1 .
Normally, the counter-directed swirling exists, as 1 <0.2 . This fact is illustrated in Fig. 2
for 0.6, 0.00727, 0.0256xh e , and 2 9 / , which correspond to the experimental in-
put data in [8]. Case (a) implies the reciprocating oblique tank motion ( 1( 0.0) but panel (b)
depicts the response curves for the counterclockwise elliptic oblique forcing with 1 0.2 . In
these both cases, the oblique forcing is nearly diagonal. The solid lines correspond to stable solu-
tions (sloshing), whereas the dashed lines imply the instability. One can see the two continuous
branches, 0 1 0( )l rP TD U U U D VS WP and 1 0 0V S P (loop-type) on which the stable swirling is pre-
Fig. 2. The wave amplitude response curves in the 1( / ), A, B -space computed and drawn for 0.6h ,
0.00727, 0.0256xe , and 2 / 9 . Panel (a) corresponds to the case 1 0.0 , but (b) is drawn with
1 0.2 (the counterclockwise elliptic orbital forcing with the nearly diagonal position of the elliptic orbit).
The solid lines mark the stable solutions, but the dashed lines imply the instability. There are two branches,
0 1 0( )l rP TD U U U D VS WP and 1 0 0V S P (loop-type), for the panels. The two subbranches 0V S and 1 0V S are asso-
ciated with the co- and counter-directed (counterclockwise and clockwise swirling waves), respectively. The
subbranches lP T and rW P change from the standing (nearly diagonal, `D’) to the nearly-standing (`S’) and,
further, to the counterclockwise swirling (`R’) as 1 increases. When the forcing frequencies belong to the
range TV , there is no stable steady-state sloshing, and, therefore, irregular (chaotic) wave motions are expected.
Fig. 3. The same as in Fig. 2 but with the ratio of semiaxes of the elliptic orbit 1 0.6 [panel (a)] and 1 1
(the circular counter-directed forcing [panel (b)]. For these ratios, the loop-type branch V1S0P0 on which the
stable counter-directed swirling was detected in Fig. 2, disappears. Specifically, the stable co-directed swirling
is associated with points of the subbranches lP T , 0V S , and rW P . The subbranch 0D U continues implying
the stable nearly standing wave mode.
50 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2021. № 6
O.E. Lagodzinskyi, A.N. Timokha
sented by the subbranches 0V S (the co-directed resonance swirling for the counterclockwise
forcing) and 1 0V S (the counter-directed [clockwise] swirling). The both subbranches appear in a
neighborhood of the primary resonance, i.e., around 1/ 1 . The subbranches lP T and rW P
change from the standing (nearly diagonal, `D’) to the nearly-standing (S) and, further, to
the counterclockwise swirling (R), as 1 increases. Comparing (a) and (b) demonstrates that
the zone of the counter-directed (clockwise) swirling increases with 1 .
For all the tested input parameters, the counter-directed (clockwise for the studied case)
swirling wave mode normally disappears, as 10.6 . This fact is illustrated in Fig. 3, which is
drawn with the same input parameters as in Fig. 2, but for 1 0.6 [panel (a)] and 1 1 (the
circular counterclockwise forcing [panel (b)]. The branching in panel (a) is characterized by the
unique continuous curve 0 1 0l rP TD U D V S W P where the stable co-directed (counterclockwise)
swirling is associated with lP T , 0V S , and rW P . The subbranch 0D U keeps implying the stable
nearly standing wave mode. When the ratio of semiaxes approaches 1, the subbranch 0D U splits
into the two loop-type branches. Fig. 3 (b) illustrates them for the given input parameters.
Conclusions. The exhaustive numerical analysis of response curves, which correspond to
the steady-state waves due to the elliptic orbital oblique forcing, shows for the finite non-di me-
n sional liquid depth 0.4h that the horizontal orbital elliptic forcing may cause both the co- and
c ounter-directed stable steady-state swirling wave regimes for smaller ratios of semiaxes of the
ellipse, but only the stable co-directed swirling exists, when the elliptic shape tends to the circle.
The authors acknowledge the financial support of the National Research Foundation of Uk-
raine (Project number 2020.02/0089). The second author also acknowledges a partial support of
Centre of Autonomous Marine Operations and Systems (AMOS) whose main sponsor is the Nor-
wegian Research Council (Project number 223254-AMOS).
REFERENCES
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https://doi.org/10.1017/jfm.2012.133
Received 08.08.2021
51ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2021. № 6
Counter- and co-directed swirling-type waves due to orbital excitations of a square-base tank
O.E. Лагодзинський 1
О.М. Тимоха 1,2, https://orcid.org/0000-0002-6750-4727
1 Інститут математики НАН України, Київ
2 Центр досконалості «Автономні морські операції та системи»,
Департамент морських технологій, Норвезький університет природничих та технічних наук,
Трондхейм, Норвегія
E-mails: lagodzinskyi@gmail.com; tim@imath.kiev.ua, atimokha@gmail.com
ПРОТИ- ТА СПІВНАПРАВЛЕНІ КРУГОВІ ХВИЛІ
ЗА ОРБІТАЛЬНИХ ЗБУРЕННЯХ БАКУ КВАДРАТНОГО ПЕРЕРІЗУ
Застосовується аналітична техніка та чисельні експерименти для того, аби показати, що орбітальні еліп-
тичні поступальні збурення баку квадратного перерізу можуть призвести в залежності від співвідно-
шен ня напіввісей еліптичної орбіти до як проти- так і співнаправленої (відносно напрямку збурення баку)
стійкої усталеної кругової хвилі. Частоти збурення близькі до першої власної частоти коливання рідини.
Для ненульового демпфування в гідродинамічній системі перехід до кругових орбіт робить неможливими
протинаправлені кругові хвилі.
Ключові слова: хлюпання рідини, кругова хвиля, стійкість, орбітальне збурення.
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