Learning the singledominant modal system on resonant sloshing in a rectangular tank
A machine learning technique is proposed to derive the damping rate functions in the co-called single-dominant modal system describing the resonant liquid sloshing in a rectangular tank performing harmonic longitudinal motions. Its implementation is demonstrated for the steady-state (periodic) res...
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irk-123456789-1879002023-02-03T01:26:44Z Learning the singledominant modal system on resonant sloshing in a rectangular tank Milyaev, A.O. Timokha, A.N. Механіка A machine learning technique is proposed to derive the damping rate functions in the co-called single-dominant modal system describing the resonant liquid sloshing in a rectangular tank performing harmonic longitudinal motions. Its implementation is demonstrated for the steady-state (periodic) resonance waves when the modal system admits an analytical asymptotic solution depending on the introduced damping rate functions. Recent experiments by Bäuerlein & Avila (2021) are employed to show that the viscous damping of the higher natural sloshing modes matters, and the damping functions depend on the wave amplitude. Запропоновано техніку машинного навчання для отримання функції демпфування у так званій однодомінантній модальній системі, яка описує резонансне хлюпання рідини в прямокутному резервуарі, що перебуває у стані гармонічного поздовжнього збурення. Її реалізація продемонстрована для усталених (періодичних) хвиль, коли модальна система допускає аналітичний асимптотичний розв’язок, який залежить від введених функцій демпфування. Нещодавні експерименти Bäuerlein & Avila (2021) використовуються для того, аби показати, що демпфування вищих власних форм коливання рідини має значення і в’язке демпфування є функцією амплітуди хвилі. 2022 Article Learning the singledominant modal system on resonant sloshing in a rectangular tank / A.O. Milyaev, A.N. Timokha // Доповіді Національної академії наук України. — 2022. — № 6. — С. 46-53. — Бібліогр.: 8 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2022.06.046 http://dspace.nbuv.gov.ua/handle/123456789/187900 532.595 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Механіка Механіка |
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Механіка Механіка Milyaev, A.O. Timokha, A.N. Learning the singledominant modal system on resonant sloshing in a rectangular tank Доповіді НАН України |
| description |
A machine learning technique is proposed to derive the damping rate functions in the co-called single-dominant
modal system describing the resonant liquid sloshing in a rectangular tank performing harmonic longitudinal motions.
Its implementation is demonstrated for the steady-state (periodic) resonance waves when the modal system admits an
analytical asymptotic solution depending on the introduced damping rate functions. Recent experiments by Bäuerlein
& Avila (2021) are employed to show that the viscous damping of the higher natural sloshing modes matters, and the
damping functions depend on the wave amplitude. |
| format |
Article |
| author |
Milyaev, A.O. Timokha, A.N. |
| author_facet |
Milyaev, A.O. Timokha, A.N. |
| author_sort |
Milyaev, A.O. |
| title |
Learning the singledominant modal system on resonant sloshing in a rectangular tank |
| title_short |
Learning the singledominant modal system on resonant sloshing in a rectangular tank |
| title_full |
Learning the singledominant modal system on resonant sloshing in a rectangular tank |
| title_fullStr |
Learning the singledominant modal system on resonant sloshing in a rectangular tank |
| title_full_unstemmed |
Learning the singledominant modal system on resonant sloshing in a rectangular tank |
| title_sort |
learning the singledominant modal system on resonant sloshing in a rectangular tank |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2022 |
| topic_facet |
Механіка |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/187900 |
| citation_txt |
Learning the singledominant modal system on resonant sloshing in a rectangular tank / A.O. Milyaev, A.N. Timokha // Доповіді Національної академії наук України. — 2022. — № 6. — С. 46-53. — Бібліогр.: 8 назв. — англ. |
| series |
Доповіді НАН України |
| work_keys_str_mv |
AT milyaevao learningthesingledominantmodalsystemonresonantsloshinginarectangulartank AT timokhaan learningthesingledominantmodalsystemonresonantsloshinginarectangulartank |
| first_indexed |
2025-07-16T09:39:17Z |
| last_indexed |
2025-07-16T09:39:17Z |
| _version_ |
1837795910470336512 |
| fulltext |
46 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 6: 46—53
Ц и т у в а н н я: Milyaev A.O., Timokha A.N. Learning the single-dominant modal system on resonant sloshing in
a rectangular tank. Допов. Нац. акад. наук Укр. 2022. № 6. С. 46—53.
https://doi.org/10.15407/dopovidi2022.06.046
https://doi.org/10.15407/dopovidi2022.06.046
UDC 532.595
A.O. Milyaev1
A.N. Timokha1,2, 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-mail: amilyaev@gmail.com, tim@imath.kiev.ua, atimokha@gmail.com
Learning the single-dominant modal system
on resonant sloshing in a rectangular tank
Presented by Academician of the NAS of Ukraine A.N. Timokha
A machine learning technique is proposed to derive the damping rate functions in the co-called single-dominant
modal system describing the resonant liquid sloshing in a rectangular tank performing harmonic longitudinal motions.
Its implementation is demonstrated for the steady-state (periodic) resonance waves when the modal system admits an
analytical asymptotic solution depending on the introduced damping rate functions. Recent experiments by Bäuerlein
& Avila (2021) are employed to show that the viscous damping of the higher natural sloshing modes matters, and the
damping functions depend on the wave amplitude.
Keywords: sloshing, machine learning, damping.
МЕХАНІКА
MECHANICS
Two-dimensional nonlinear resonant sloshing in a rectangular tank is normally adopted as a
benchmark problem for complex experimental-and-theoretical studies and developing & testing
diverse Computational Fluid Dynamics (CFD) methods. The analytical studies assume an in-
viscid incompressible liquid with irrotational flows so that the viscous damping is neglected. A
consequence of this simplification is that the phase lag between the steady-state (periodic) wave
response and harmonic horizontal tank forcing becomes a piecewise function [1, chapter 8] pos-
sessing the values 0 and ±π . Moreover, neglecting the viscous damping theoretically yields the
infinite nonlinear resonant wave amplitude that has no a physical meaning.
Damping of the linear liquid sloshing is naturally associated with the laminar viscous bound-
ary layer on the wetted tank surface. The related constant damping rates are theoretically & ex-
perimentally estimated & measured by Keulegan [2]. Chapter 6 in [1] discusses extra linear &
nonlinear damping mechanisms incl. wave breaking and free-surface contamination. Extensive
experimental studies are needed to judge on when and what from these mechanisms play a non-
negligible role in theoretical analysis. Recent paper [3] represents an example of these experi-
mental studies. Its authors measure the phase lag between the harmonic horizontal forcing and
47ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 6
Learning the single-dominant modal system on resonant sloshing in a rectangular tank
steady-state (periodic) motions of the liquid mass centre. A reason for focusing on the phase lag
is that the viscous damping causes a crucial effect on the phase lag converting it, as mentioned
above, from piecewise to smooth function. The measurements used to estimate a suitable con-
stant damping rate in the damped Duffing equation, which was speculatively adopted by the au-
thors as a phenomenological model of the liquid mass centre motions. Such a constant damping
rate was also employed in the lowest-order equation of the so-called single-dominant nonlinear
modal system [4], which, as it was demonstrated in numerous publications (see, a review in [1]),
satisfactory predicts the steady-state resonant sloshing when the forcing amplitude is relatively
small, the forcing frequency is close to the lowest natural sloshing frequency, the liquid depth is
not small, and the secondary resonance phenomenon [5] does not matter. The measurements in
[3] show that the matched damping ratio constξ = in the Duffing mathematical model should be
too far ( –315 10ξ = ⋅ ) from predictions by Keulegan ( –35,7 10ξ = ⋅ ) to fit the measured phase lag
even for the lowest excitation amplitude when, no doubt, the single-dominant model system is
applicable and the damping should mainly be associated with the viscous boundary layer effect [1,
5]. Adopting this larger damping rate in the lowest-order modal equation (damping of the higher
natural sloshing modes is totally neglected by the authors) shows an insufficient consistency with
the measured data which may wrongly conclude that either the single-dominant modal system is
not applicable (no doubt, it is applicable) or the total energy dissipation is much larger than its
estimate by Keulegan [2].
The single-dominant modal system [2, 4, 5] belongs to the co-called Reduced Order Models
(ROMs) resulted from applying, simultaneously, projective, and asymptotic methods to the origi-
nal free-surface “sloshing” problem. The present paper implements a machine learning algorithm
[6] to restore the damping terms/functions in the modal system from a series of experimental data
on the phase lag. Going this way, one is assumed that (i) damping of the higher natural sloshing
modes cannot be neglected, (ii) the damping terms/functions depend on the wave amplitude [6],
and (iii) the single-dominant modal system becomes inapplicable with increasing the forcing am-
plitude and, surely, it fails when experimental observations [3] report fragmentations of the free
surfaces (e. g., due to the wave breaking). The factors (i-iii) are fully ignored in the theoretical
manipulations [3] with the single-dominant modal system.
Let the horizontal tank dimension L be the characteristic size and 12T = π σ be the charac-
teristic time ( 1σ is the lowest natural sloshing frequency). Then the “damped” non-dimensional
single-dominant nonlinear asymptotic modal system [1, 4] for the horizontally-forced rectangular
tank takes the following form
1 1 1 1, , 3 1 1 1 2 1 2
2 2 2
2 1 1 1 1 3 2 1 1 2
( ) (2 , |
c
)
(( o )) s ,
m m m
a
d
d d P t
= …+ β + Ξ β β β + β + β β +
+ β + β β
β
+ β = θβ β η σ −
β
σ
2 2
2 2 2 2 2 1, , 3 2 4 1 1 5 12 , | 0,( )m m m d d= …+ σ β + σ Ξ β β β + β + ββ =β
(1)
2 2
3 3 3 3 3 1, , 3 3 1 1 2 2 1 1 3 2 1
2 2
4 1 1 5 1 2 3 2
2 , |
cos( ),
( )m m m
a
q q q
q q P t
= …+ σ β + σ Ξ β β β + β + β + β +
+ β β + β β = η σ σ − θ
β β β β
48 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 6
A.O. Milyaev, A.N. Timokha
where the three hydrodynamic generalized coordinates ( )i tβ , i = 1, 2, 3, come from the functional
(Fourier) representation of the free surface
1
1
1
( , ) ( )cos ;
2i
z y t t i y
∞
=
⎛ ⎞⎛ ⎞= ζ = β π +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
1/3
1 ( );Oβ = 2/3
2 ( );Oβ = 3 ( );Oβ = ( ),n oβ = 4; 1,n (2)
1σ are the natural sloshing frequencies, 1/σ= σ σ ( is the forcing frequency),
= π π −−2 / ( )tanh( ) 1 )1(( )i
iP i i h , 2 tanh( ) / tanh( )m m m h hσ = π π , the hydrodynamic coeffi-
cients nd , nq depend on the non-dimensional liquid depth h , 2aη is the non-dimensional forcing
amplitude, and is the phase lag in the harmonic forcing. The computed values of the hydrody-
namic coefficients are tabled in [1]. A novelty with respect to the original inviscid analysis in [1]
consists of the generally non-constant damping terms 1, , 32 , |( )i i m m m i= …σ Ξ β β β incorporated into
the modal system, where iΞ are a priori unknown damping rate function.
The steady-state resonant sloshing is associated with periodic solutions of (1). Following the
Moiseev’ asymptotic scheme which is described in [1, Chapter 8] derives the asymptotic periodic
solution
3
1 1 2cos( ) cos(3 ) sin(3 ) ( ),( ) ( )t a t a k t k t oβ = σ + σ + σ +
2
2 0 1 2cos(2 ) sin(2 ) ( ),( ) ( )t a l l t l t oβ = + σ + σ + (3)
3 1 2
3
1 2 3 3
[ cos sin ]
cos sin cos3 sin 3 ( ),
( )
[ ]
t n t n t
a N t N t N t N t o
β = σ + σ +
+ σ + σ + σ + σ +
where 1 2 1aP= η , 1/3( )a O= is the dominant wave amplitude, and
4 5
0 2
2
,
2
d d
l
−=
σ
2
4 5 2
1 2 2 2 2
2 2 2
( )( 4)
,
)
( )
2(( 4) 16 ( )
d d
l a
a
+ σ −=
σ − + σ Ξ
4 5 2 2
2 2 2 2 2
2 2 2
2( )
;
(( 4) 16 )
( )
( )
( )
d d a
l a
a
+ σ Ξ=
σ − + σ Ξ
2 2
2 1 1 3 1 1 1 1 2
1 2 2 2 2
1 1 1
1 39 ( 2 )( 9 6 )
2 2 ,)
( 9) 36
( ) ( ) ( )
(
( )
d d d l a l
k a
a
σ − + + σ − − σ Ξ
=
σ − + σ Ξ
2
2 1 1 1 3 2 1 1 1 1
2 2 2 2 2
1 1 1
33 ( 2 )( 9 6 )
2 ;
9
( ) ( ) ( )
( )
(( ) 36 )
d a d d l a l
k a
a
σ Ξ + + σ − + σ Ξ
=
σ − + σ Ξ
2
3 3 3 3
1 2 2 2 2
1 3 3 3
cos ( 1) 2 sin
,
4
( ( ) )
( )
(( ) ( )1 )
P a
n a
P a
θ σ − − σ Ξ θ=
σ − + σ Ξ
49ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 6
Learning the single-dominant modal system on resonant sloshing in a rectangular tank
2
3 3 3 3
2 2 2 2 2
1 3 3 3
2 cos ( 1)sin
;
( ( ) )
( )
(( ))1 4) (
P a
n a
P a
σ Ξ θ + σ − θ=
σ − + σ Ξ
2 2
1 3 3 3 2 1 3 5 2 1 1 0 3
1 2 2 2 2
3 3 3
1 3 1( 1 2 )( 2 ) ( )( 1)
2 4 4 ,
(
( ) ( )
(
(1)
)
4 )
l a l q q q q q q l
N a
a
σ − − σ Ξ + − + − + σ −
=
σ − + σ Ξ
2
2 3 3 3 1 1 3 5 3 3 2 1 1 0
2 2 2 2 2
3 3 3
1 3 1( 1 2 )( 2 ) 2
2 4 4 ,
( ) ( ) ( )
( 1) 4
( )
( )
l a l q q q q q q l
N a
a
σ − + σ Ξ + − + Ξ σ − +
=
σ − + σ Ξ
2 2
1 3 3 3 2 1 3 5 2 4 3
3 2 2 2 2
3 3 3
1 1( 9 6 )( 2 ) ( )( 9)
2 4 ,
6
(
9
) ( )
( )
) 3 ( )(
l a l q q q q q
N a
a
σ − − σ Ξ + + + + σ −
=
σ − + σ Ξ
2
2 3 3 3 1 1 3 5 2 4 3 3
4 2 2 2 2
3 3 3
1( 9 6 )
)
( 2 ) 3( )
2
)
( 9) 3
( )
(
6
( ) (
)
(
l a l q q q q q a
N a
a
σ − + σ Ξ + + + + σ Ξ
=
σ − + σ Ξ
are functions of 1/3( ) 0a O= > and, generally, θ when it comes to 3 ( )tβ . Because the lowest-
order term in (3) is associated with 1 cos( )( )t a tβ = σ , which is purely cosine function, θ is, in
fact, the phase lag between the forcing and the wave response in the lowest-order asymptotic ap-
proximation.
According to the standard Moiseev’s asymptotic procedure [1], the unknown amplitude pa-
rameter a and phase lag θ can be found from the so-called secular system
2 2
1( ( 1)) cos ,ˆm a a+ σ − = θ 2
2 1 ,( )( 2 ) sinm a a a− Ξ = θ (4)
in which
1 1 1 2 2 3 1 1 0 1
1 1
, 2 ,( ( )
2 2
( ))m m a a d d l d l l⎛ ⎞= Ξ Ξ = − − + − +⎜ ⎟⎝ ⎠
2 2 1 2 2 1 3( ( ) ( )) ( ;)
1
, 4
2
m m a a l d d= Ξ Ξ = −
1
1 2 ; .ˆaP −= η σ = σ
Taking the sum of squares in (4) makes it possible to rewrite the secular system to the two equations,
2 2 2 2 2 2 2
1 1 2 2 1 2 1, ( 1) ( ,[( ( ( ) ( )) 2) ( ( ) ( )) )( ) ]ˆa m a a a m a a a aΞ Ξ + − + Ξ Ξ − Ξ =s (5)
and
1 2 1 2 2
2 1 2 1 1 1 2[ ( ) ( ) ( ) .atan2( ( , ) 2 , ( ( , ) ( 1] [ ) ( ) ˆ )])m a a a a m a a a− −θ = Ξ Ξ − Ξ Ξ Ξ + σ − (6)
Consequently solving (5) and (6) outputs the dominant wave amplitude parameter 0a> and the
phase lag θ as a function of σ̂ . The procedure assumes that , 1, 2,( )m a m = are known functions
of a . This is not true in our case.
50 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 6
A.O. Milyaev, A.N. Timokha
In the forthcoming analysis, we suppose that there exist experimental measurements of the
phase lag versus the forcing frequency, i. e.
2
(( ) , ) ,
ai i ησ θ
2
1, ,
a
i N η= … for the fixed non-dimen-
sional liquid depth h and several forcing amplitudes 2aη . A machine learning technique can then
be utilised to get an approximation of ( )m aΞ using these measurements. Because 1/3( )a O= is
the small parameter, an adequate approximation of 1 ( )aΞ and 2 ( )aΞ takes the polynomial form
=Ξ = ∑1 { 0};{ }( ) N m
m m ma x x a , −
=Ξ = ∑ 1
2 { 0}( });{ N m
m m ma x y a , (7)
where we accounted for that 1/3 1/3
2 1 2 1( ) ( )O Oβ =β ⇒Ξ = Ξ .
Substituting (7) into (5) and extracting the forcing frequency gives
2
2 2 2 2
2 1 2 12
ˆ ,; { }, { }; 1 ; { } ( ; { } 2 ;{( } )( ) ( ) ) ( )m m a m m ma x y m a y a m a y a a x
a
σ η = − ± − − Ξ
which has a physical meaning when both the right-hand side and expression under the square
root are non-negative. The existence conditions yield the interval min 2 , { }, { }( )a m ma x y aη
max 2 , { }, { )}( a m ma x yη depending on { }, { }m mx y and 2aη . Changing a in the interval outputs
2
2 ); { },ˆ };( {m m aa x yσ η as a function of the lowest-order amplitude parameter for the fixed values
of { }, { }m mx y and 2aη . Inserting 2
2 ); { },ˆ };( {m m aa x yσ η into (6) derives
2 2
2 2 1 1( ; { }, { }; ) atan2[ ; { } 2 ; {( ) ( ) ({ } , ; })m m a m m ma x y m a y a a x m a y aθ η = − Ξ +
2
2( ; { }, { }; 1)]ˆ ( )m m aa x y+ σ η − (8)
as a function of a on the interval min 2 max 2, { }, { , { }, {( }) ( )}a m m a m ma x y a a x yη η for the fixed
{ }, { }m mx y and 2aη .
Using the exact solution (8) and the machine learning makes it possible to compute
{ }, { }m mx y . For this purpose, we introduce the distance function 2( , ; { }, { })a m mD i x yη between
the theoretical phase-lag curve by (8) with the fixed { }, { }m mx y ; 2aη , and a fixed experimental
point
2
(( ) , )
ai i ησ θ :
2
min 2 max 2
2 2
2 2}( ), { , { } , { }, { )}(
, ; { }, { }( )} min ( ( ; { , { }; ) ( ) )[ a
a m m a m m
a m m m m a ia x y a a x y
D i x y a x y η
η η
+η = σ η − σ
2 2
2( ( ; { }, { }; ) ) ]a
m m a ia x y η+ θ η − θ .
Utilising the distance function determines the loss function
2
2
2
1
({ }, { }) ( , ; { }, { })
a
a
N
m m a m m
i
C x y D i x y
η
η =
= η∑ ∑ , (9)
which characterises the summarised distance between the measured values ησ θ
2
,(( ) , )
ai i
2
1 ,,
a
i N η= … and the theoretical curve by (8). Minimisation of the loss function (9) can be done
by using the gradient descent.
Example. The experimental data on the liquid mass centre (amplitude and phase lag) were
reported in [3] for the non-dimensional depth 0.8h = , which implies [1] 1 3.142,d = 2 2.533,d =
51ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 6
Learning the single-dominant modal system on resonant sloshing in a rectangular tank
3 –0.021,d = 4 –0.042,d = 5 –3.225.d = The measurements were done for the forcing non-dimen-
sional amplitudes equal to 2 0.0009, 0.0017, 0.0032aη = , and 0.0064 . The liquid mass centre
non-dimensional motions in the horizontal direction are theoretically described by the formula
[1, Eq. (8.73)]
1 32
2 1( ) ( )
9
( ) ( ) ( )Cy t t t o
h
= − β + β + =
π
3 3
1 1 2 22
2 1 1([ ( )]cos [ ]sin
9 9
a a N n t a N n t
h
=− + + σ + + σ +
π
3
1 3 2 4
1 1+ cos3 ( )sin3 ) ( ).
9 9
[( ) ]a k N t k N t o+ σ + + σ + (10)
Let us demonstrate the learning technique by assuming the linear regression for the first damping
rate function, i. e., 1 0 1 0 1; ,( )a x x x x aΞ = + and the constant damping rate for the second natural
sloshing mode (generalised coordinate), 2 0 0( ; )a y yΞ = , where the constant values 0x and 0y
correspond to the damping rates for the first and second natural sloshing modes whose asymp-
totic estimate (from below) was given by Keulegan [2]. Using three measurement series with
2 0.0009, 0.0017,aη = and 0.0032 from [3] computes 0 10.00717, 0.00189x x= = , and 0 0.01y = .
The phase lag (a) and the maximum deviation of the liquid mass centre (b) for the steady-state sloshing in a
rectangular tank with input parameters from the experimental paper [3]. The single-dominant nonlinear modal
system with the damping rate functions (1) is used to derive the steady-state wave solution (3). The measured
phase lag with the lowest forcing amplitudes 2 0.0009, 0.0017,aη = and 0.0032 are used to learn by the
linear regression for 1Ξ and the constant approximation of 2Ξ by the measured phase lag in the panel a. The
experimental data for the largest forcing amplitude 2 0.0064aη = (when the single-dominant modal system is,
generally, inapplicable) are used in a to test the learnt damping rate functions. The panel b also tests the learnt
damping rate functions by comparing the theory and the measured maximum steady-state displacement of the
liquid mass centre in the horizontal direction
52 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 6
A.O. Milyaev, A.N. Timokha
The fourth experimental series with 2 0.0064aη = was characterised by amplification of higher
harmonics and the wave breaking phenomenon on the free surface that indicates inapplicability
of the single-dominant modal system.
The first test for the learnt 0x and 0y consists of comparison with predictions by Keulegan
[3]. According to the Keulegan’s estimate of the laminar viscous boundary layer effect, the viscous
damping rates should be 0 0.0057x = and 0 0.008y = , respectively. Because Keulegan’s formula
gives the estimate from below, this is rather good agreement with the computed values 0 0.00717x =
and 0 0.01y = by the machine learning algorithm. The experimental tank breadth was rather small
(0.05 m) and, therefore, the dynamic contact angle effect on the damping rates cannot be neglected.
Why the dynamic contact angle gives a contribution to the damping rates in narrow tanks was dis-
cussed in [3] and [1, chapter 6]. On the other hand, Bäuerlein & Avila [3] concluded that the damp-
ing rate 0x should be much larger, equal to 0.015 to fit the experimental measurements with the
Duffing mathematical model. Their error is caused by neglecting the second natural sloshing mode
effect and its damping as well as they ignored the dependence on a in 1.
Theoretical and experimental values of the phase lag and the maximum deviations of the
mass centre by (10) are compared in the panels a and b of Figure, respectively. As we mentioned
above, three experimental series in a were used to learn the damping functions but the experi-
mental case with 2 0.0064aη = was adopted for testing the results. For the learning amplitudes
2 0.0009, 0.0017,aη = and 0.0032 in Figure, a, incorporating the computed damping functions
made it possible to perfectly fit the experimental points. Figure, a with 2 0.0064aη = shows that
the theoretical results for the phase lag are not consistent with experiments in the frequency do-
main where the single-dominant modal system (1) is not applicable (in this frequency domain, [3]
reports amplification of higher harmonics and surface wave phenomena alike the wave breaking).
Far from this zone, the damping rate functions are well-predicted and the single-dominant non-
linear modal system is no doubt applicable.
Figure, b tests the learnt damping function by the measured maximum deviations of the liquid
mass centre, which can theoretically be computed by (10). A discrepancy is obviously observed
for the largest forcing amplitude when, as we mentioned above, the single-dominant modal system
is not applicable whilst the case with the lowest forcing amplitude shows an almost perfect agree-
ment with the learnt damping functions. The discrepancy for 2 0.0017aη = and 0.0032 is only
serious in the vicinity of the maximum amplitude response and it looks like in [8] where analogous
discrepancy was explained by amplification of the higher generalised coordinates that requires an
adaptive multimodal modelling.
Conclusions. An accurate implementation of the single-dominant nonlinear modal system with
the learnt damping terms shows that conclusion in [3] that the actual damping is much larger than
Keulegan’s estimate by the laminar viscous boundary layer effect is generally wrong. To account cor-
rectly for the viscous damping in the modal system, one must introduce the damping functions in the
higher order modal equations and these damping functions should depend on the wave amplitude.
The second author acknowledges the financial support of the National Research Foundation of
Ukraine (Project number 2020.02/0089) and a partial support of Centre of Autonomous Marine
Operations & Systems (AMOS) whose main sponsor is the Norwegian Research Council (Project
number 223254-AMOS).
53ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 6
Learning the single-dominant modal system on resonant sloshing in a rectangular tank
REFERENCES
1. Faltinsen, O. M. & Timokha, A. N. (2009). Sloshing. Cambridge University Press.
2. Keulegan, G. H. (1959). Energy dissipation in standing waves in rectangular basins. J. Fluid Mech., 6, pp. 33-50.
https://doi.org/10.1017/S0022112059000489
3. Bäuerlein, B. & Avila, K. (2021). Phase lag predicts nonlinear response maxima in liquid-sloshing experiments.
J. Fluid Mech., 925, A22, pp. 1-29. https://doi.org/10.1017/jfm.2021.576
4. Hermann, M. & Timokha, A. (2005). Modal modelling of the nonlinear sloshing in a circular tank I: A single-
dominant model. Math. Models Methods Appl. Sci., 15, No 9, pp. 1431-1458.
https://doi.org/10.1142/S0218202505000777
5. Hermann, M. & Timokha, A. (2008). Modal modelling of the nonlinear sloshing in a circular tank II: Second-
ary resonance. Math. Models Methods Appl. Sci., 18, No 11, pp. 1845-1867.
https://doi.org/10.1142/S0218202508003212
6. Ahmed, S. E., Pawar, S., San, O., Rasheed, A., Iliescu, T. & Noack, B. R. (2021). On closures for reduced order
models — A spectrum of first-principle to machine-learned avenues. Phys. Fluids, 33, Art. 091301, 32 p.
https://doi.org/10.1063/5.0061577
7. Raynovskyy, I. A. & Timokha, A. N. (2018). Damped steady-state resonant sloshing in a circular container.
Fluid Dyn. Res., 50, Art. 045502, 20 p. https://doi.org/10.1088/1873-7005/aabe0e
8. Faltinsen, O. M. & Timokha, A. N. (2001). Adaptive multimodal approach to nonlinear sloshing in a rectangular
tank. J. Fluid Mech., 432, pp. 167-200. https://doi.org/10.1017/S0022112000003311
Received 14.06.2022
A.O. Міляєв1
О.М. Тимоха1,2, 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-mail: amilyaev@gmail.com, tim@imath.kiev.ua, atimokha@gmail.com
НАВЧАННЯ МОДАЛЬНОЇ СИСТЕМИ З ОДНІЄЮ ДОМІНАНТНОЮ
УЗАГАЛЬНЕНОЮ ГІДРОДИНАМІЧНОЮ КООРДИНАТОЮ, ЩО ОПИСУЄ
РЕЗОНАНСНІ КОЛИВАННЯ РІДИНИ В ПРЯМОКУТНОМУ РЕЗЕРВУАРІ
Запропоновано техніку машинного навчання для отримання функції демпфування у так званій однодомі-
нантній модальній системі, яка описує резонансне хлюпання рідини в прямокутному резервуарі, що пере-
буває у стані гармонічного поздовжнього збурення. Її реалізація продемонстрована для усталених (періо-
дичних) хвиль, коли модальна система допускає аналітичний асимптотичний розв’язок, який залежить від
введених функцій демпфування. Нещодавні експерименти Bäuerlein & Avila (2021) використовуються для
того, аби показати, що демпфування вищих власних форм коливання рідини має значення і в’язке демпфу-
вання є функцією амплітуди хвилі.
Ключові слова: хлюпання рідини, машинне навчання, демпфування.
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