multimodal method in sloshing analysis analytical mechanics concept
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Multimodal Method in Sloshing Analysis:Analytical Mechanics Concept
byAlexander Timokha
CeSOS/AMOS, NTNU, Trondheim,
NORWAY
&
Institute of Mathematics,
National Academy of Sciences of Ukraine,UKRAINE
Trondheim, 28. May, 2013
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Multimodal Method in Sloshing Analysis:Analytical Mechanics Concept
Etymology comes from
which is the most cited paper on sloshing of the last two decades
CONCE
PT
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Multimodal Method in Sloshing Analysis:
Analytical Mechanics Concept
The concept originally appeared in XIX century but generalized in 2000-2013 by
the author together withProf. Odd M. Faltinsen
Liquid Sloshing Dynamicsistreated as aconservative mechanical system with infinite degrees of freedomso that
the Lagrange formalism can adopt thegeneralized coordinates andvelocities
responsible for the global liquid modes instead of working with typically-
accepted hydrodynamic characteristics (velocity field, pressure, etc.)
CONCE
PT
Historical aspects & Ideas: Linear & Nonlinear
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IDEAS come from aircraft and spacecraft applications:
when the task consists of describing
the coupled dynamics of
a body with cavities filled by a liquidN.E. Joukowski (1885) paper
`On the motion of a rigid body with cavities
filled by a homogeneous fluid
Joukowski theorem for a completely filled tank:
`The rigid body-ideal irrotational incompressible fluid mechanical
system can modelled as a rigid body with aspecifically-modifiedinertia
tensor
Considering the fluid as frozen is a wrong way (boiled and fresheggs!!!).
The velocity field is described by the so-called Stokes-Joukowski
potentials.
H
ISTORY&
IDEAS
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by the liquid:
ideal, incompressible,
irrotational flow
Translatory velocity:
Instant angular velocity:1 2 2( ) ( ( ), ( ), ( ))Ov t t t t
4 5 6( ) ( ( ), ( ), ( ))t t t t
The liquid velocity potential
0( ) ( )( , , , ) ( , , ), ( , , )Ox y v t tz t x z r x y z y r
the Stokes-Joukowski potential
FULLY FILLED TANK
H
ISTORY&
IDEAS
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Having known the time-independentStokes-
Joukowski potentials makes it possible to find the
fluid flow for any time instant so that
the velocity field the pressure,
the resulting hydrodynamic force and moments,etc.
are explicit functions of the six input generalized
coordinates
What about liquid sloshing (free surface)?
H
ISTORY&
IDEAS
LINEAR SLOSHING
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LINEAR SLOSHING,
50-60s of XX century
Liquid:
ideal
incompressible irrotational flow
1 2 2( ) ( ( ), ( ), ( ))Ov t t t t
4 5 6( ) ( ( ), ( ), ( ))t t t t
0( , , , ) ( )
( , , )
( , , ) ( , , )
( , ,( )) 0
( ) ( ) NO
N
N
NN
N
r x y z x x y z t R t
x y
yv t z
x
t
z t t y
sloshing modesinterpreted as generalized coordinates
(infinite set!!!)
Free surface
generalized velocities
H
ISTORY&
IDEAS
Joukowski solution
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As long as we know
the time-independent Stokes-Joukowski potentials(Neumann problem in the unperturbed liquid domain)
the natural sloshing modes (the spectral boundary problemin the unperturbed liquid domain)then
the free-surface elevations
the hydrodynamic forces and moments (provided by the
corresponding Lukovsky formulas)
are functions of the input and the generalized coordinates
where the latters are the solution of the linear oscillator
problem:
and the hydrodynamic coefficients are integrals over
and
0( , , )x y z
( , , )N
x y z
60( 3)2 1 2
41 5 2 4
( ) ( ) , 1,2,...k mm m
m mkm m
k
m
mg g m
( , , )N x y z
0( , , )x y z
H
ISTORY&
IDEAS
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( )
free surface ( ): , , , 0
subject to volume conservation
(
0
)N
Q t
t Z x y z
dQ
t
NONLINEAR MULTIMODAL METHOD:new life in 00s
Liquid:
ideal
incompressible irrotational flow
1 2 2( ) ( ( ), ( ), ( ))Ov t t t t
4 5 6( ) ( ( ), ( ), ( ))t t t t
0( , , , ) ( , , ,{ ( )} ( )) ( , , )N NON
N t R tx y z t r x y z x y zv
The free surface elevations, the hydrodynamic forces and moments
also remain functions of the six inputandinfinite setof the free-
surfacegeneralized coordinates
Generalized velocities
Generalized coordinatesH
ISTORY&
IDEAS
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THERE ARE
APPLIEDMATHEMATICAL &
PHYSICAL
PROBLEMSTO BE SOLVED
TO IMPLEMENT
THE NONLINEAR
MULTIMODALMETHOD
H
ISTORY&
IDEAS
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The nonlinear multimodal method
1. The method is ofanalytical nature.
What are analytical limitations?2. The Euler-Lagrange equation for
liquid sloshing dynamics
3. Physical and mathematical argumentsfor choosing the generalized
coordinates
4. How does it work?
Themultimodalmethod
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A. For any instant must be defined
in the time-varied liquid domain
and satisfy the tank surface conditionB. Derivation of the Euler-Lagrange
equations with respect to the
generalized coordinates and velocities
( , , )N x y z
( )Q t
hand-made
product for each
tank shape
e.g., the
Bateman-Luke
variational
principle
MATH
EMATICALL
IMITATIONS
Bateman-Luke
variational principle
derives the Euler-
Largange equation
The following problems must be solvedanalytically:
Assuming A (we know analytical modes):
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(Kinematic Eq.): , changingd
;d
N N K NK K K KK
A AA F
tN
(Dynamic Eq.):
1 2 31 2 changing1 ... 02K KLK K L
K KLN N N N N
A A l l l F NF F
where
* *( ) ( )
1 * 2 * 3 *( ) ( ) ( )
d ; d ,
d ; d ; d
N N N NK N N K Q t Q t
N N NQ t Q t Q t
A Q A Q
l x Q l y Q l z Q
EULE
R-LAGRANG
EEQUATION
Application needs a finite-dimensional form in application,
so how to use the Euler-Lagrange equation, e.g., for
Steady-state and transient response? Wave elevations? Forces and moments? Coupling? Realistic clean tanks? Dissipation? Internal structures effect?
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Problems to be solvedanalytically:
A. For any instant must be defined
in the time-varied liquid domain
and satisfy the tank surface conditionB. Derivation of the Euler-Lagrange
equations with respect to the
generalized coordinates and velocities
C. Reduction of the infinite-dimensionalEuler-Lagrange equations to a finite-
dimensional form
D. Accounting for specific phenomena
neglected by the physical model
(damping, inner structures, wall/roof
impact, perforated bulkhead, and so on)
( , , )N x y z
( )Q t
hand-made
product for each
tank shape
e.g., the
Bateman-Luke
variational
principle
Normally,
asymptotic
approaches
results of the last
decade
PH
YSICALAGR
UMENTS
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HOW does it work?
Lets considerdevils examples for:
1. For 2D flows2. For 3D rectangular tanks with upright
walls
3. For complex tank shapesHowdoesit
work?
What should be solved?
Choosing the leading and negligible
generalized coordinates Modifying the modal equation due to
damping, specific phenomena and internal
structures
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When forcing the lowest natural sloshing frequency:Finite liquid depth (liquid depth/tank length>0.2)
A small forcing magnitude Moiseevs asymptotics
Increasing forcing magnitude, or the critical depth
ratio (h/l=0.3368) secondary resonances andadaptive asymptotics
Intermediate and small depths (liquid depth/tanklength < 0.2)
Multiple secondary resonance
Boussinesq ordering
Internal structures (e.g., bulkheads)
Small forcing amplitude quasilinear theory
Increasing forcing amplitude secondary resonance
Nonlinearity implies an energy transfer from the lowest,primary excited, to a set of higher modes. The lattermodes can be resonantly excited due to the so-calledsecondary resonances.
What does it mean in terms of leading
generalized coordinates?
2Dinrectangulartank
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When forcing the lowest natural frequency:Finite liquid depth (liquid depth/tank length>0.2)
A small forcing magnitude Moiseevs asymptotics
Increasing forcing magnitude, or the critical depth ratio
(h/l=0.3368) secondary resonances and adaptiveasymptotics
Intermediate and small depths (liquid depth/tanklength < 0.2)
Multiple secondary resonance
Boussinesq ordering
Internal structures (e.g., bulkheads)
Small forcing amplitude quasilinear theory
Increasing forcing amplitude secondary resonance
Nonlinearity implies an energy transfer from the lowest,primary excited, to a set of higher modes. The latter modescan be resonantly excited due to the so-called secondaryresonances.
What does it mean in terms of leading
generalized coordinates?
2Dinrectangulartank
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Example 1: The Moiseev-type multimodal theory
The modal solution
Moiseev proved for the steady-state (periodic) sloshing due toexcitation of the lowest frequency when there are no secondary
resonances
01
1
12
( , , ) ( , ) ( , ),
( , ) ( ) ( , 0),
cosh ( )cos ( )
cosh( )
O n nn
n nn
n
y z t v r y z R y z
z y t t y
n z hn y
nh
1/3 2/3
1 1
4
3
2
2 2
3
3
( ), ( ),
( ); ( ), 4
/ / ( ); ( ) 1,
n n
R O R O
R O R O n
l l O O
Example1:
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2
1 1 2 3
22
2 2
1 1 1 2 1 2 1 1 1 1 2 1
22 2 1 1 1
2 2
3 3 1 2 1 1 2 1 1 1 1
4 5
2
3 1 2 3 4 2 35
1( ) ( ) ( ) ,
( ) 0,
(
)
.
(
) ( )
d d d
d d
q
K t
q q tq Kq
Modal equations
2 , 4,.( ..,)iii i
iK Nt
24 4
( )( ) ( ) ( )
i ii
gP S
l l
tK t t t
Forcing term:
Example1:
Coefficients are analytically found as functions ofthe liquid depth to the tank breadth ratio
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Tran
sients:
Experiments
Theorywithdifferent
initialscenarios
Steady-state waves with the horizontal/angular harmonicforcing,
the dominant amplitude parameter:
(frequency , nondimensional amplitude ),
/ 0.3368...h l / 0.3368...h l
Example1:
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When forcing the lowest natural frequency:Finite liquid depth (liquid depth/tank length>0.2)
A small forcing magnitude Moiseevs asymptotics
Increasing forcing magnitude, or the critical depth
ratio (h/l=0.3368) secondary resonances andadaptive asymptotics
Intermediate and small depths (liquid depth/tanklength < 0.2)
Multiple secondary resonance
Boussinesq ordering
Internal structures (e.g., bulkheads)
Small forcing amplitude quasilinear theory
Increasing forcing amplitude secondary resonance
Nonlinearity implies an energy transfer from the lowest,primary excited, to a set of higher modes. The latter modescan be resonantly excited due to the so-called secondaryresonances.
What does it mean in terms of leading
generalized coordinates?
Example2:
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Example2:
Exam
ple2:
Adaptivemodalsystemsfor
criticaldepthand
increasing
forcin
gamplitude
Nonlinear multiple frequency effects excite higher natural frequencies
Increased importance with decreasing depth and increasing forcingamplitude
Reason for decreasing depth importance is that when the
liquid depth goes to zero
Implies that more then one mode is dominant
nn
Diff d i f h li d di d
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Different ordering of the generalized coordinates due to
secondary resonances with a finite liquid depth
Example2:
Rectangular tank, 1x1m, nearly critical
depth, h/l = 0.35
Subharmonicregimes are
predicted
I i th f i lit d d ti f f i t h/l 0 4
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Increasing the forcing amplitude and accounting for roof impact, h/l=0.4
Example2:
0.01
0.1
impact neglected impact accounted forFlow 3D
CFD: Symbols and represent numerical results by the viscous
CFD-code FLOW-3D obtained for fresh water with different internal
parameters of the code: for alpha=0.5, epsdj=0.01 and foralpha=1.0, epsdj=0.01.
Experiments: The influence of viscosity: , fresh water; , reginol-
oil; , glycerol-water 63%, , glycerol-water 85%
h f i h l l f
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When forcing the lowest natural frequency:Finite liquid depth (liquid depth/tank length>0.2)
A small forcing magnitude Moiseevs asymptotics
Increasing forcing magnitude, or the critical depth ratio
(h/l=0.3368) secondary resonances and adaptiveasymptotics
Intermediate and small depths (liquid depth/tanklength < 0.2)
Multiple secondary resonance
Boussinesq ordering
Internal structures (e.g., bulkheads)
Small forcing amplitude quasilinear theory
Increasing forcing amplitude secondary resonance
Nonlinearity implies an energy transfer from the lowest,primary excited, to a set of higher modes. The latter modescan be resonantly excited due to the so-called secondaryresonances.
What does it mean in terms of leading
generalized coordinates?
Example3:
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Intermediate and small depths normally cause amplification of higher
modes and a series of local wave breaking & overturning
Example3:
A B i t d i b
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As a consequence, a Boussinesq-type ordering can be
proven being applicable to with1/4( / ) ( ), 1
i iO h l O i R
Transients:
The measured and calculated wave elevations near the wall for
horizontal forcing . The solid and dashed
lines correspond to experiments and the Boussinesq-type
multimodal method, respectively.
/ 0.173, 0.028h l
Example3:
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Example3:
Steady-state due to harmonic forcing = experiments
Chester & Bones
Theory by ChesterMultumodal theory
/ 0.083333h l
0.001254 0.002583
Agreement is almost ideal when no wave breaking occurs and,
therefore, damping does not matter.
Multi-peak response curves anddamping are important whendealing with secondary resonances.
Associated damping becomes important with
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Associated damping becomes important with
Decreasing depth and increasing forcing amplitude
Roof impact
Internal structuresNormally, viscous boundary layer effect is less
important
Damping terms can be incorporated into the modalequations following the strategy in Chapter 6 ofSloshing book.
An open problem is damping due to the local free-surface phenomena:
Overturning and impact on underlying fluid
Breaking waves in the middle of the tank
Examplesvs.damping
Wh f i th l t t l f
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When forcing the lowest natural frequency:Finite liquid depth (liquid depth/tank length>0.2)
A small forcing magnitude Moiseevs asymptotics
Increasing forcing magnitude, or the critical depth ratio
(h/l=0.3368) secondary resonances and adaptiveasymptotics
Intermediate and small depths (liquid depth/tanklength < 0.2)
Multiple secondary resonance
Boussinesq orderingInternal structures (e.g., bulkheads)
Small forcing amplitude quasilinear theory
Increasing forcing amplitude secondary resonance
Nonlinearity implies an energy transfer from the lowest,primary excited, to a set of higher modes. The latter modescan be resonantly excited due to the so-called secondaryresonances.
What does it mean in terms of leading
generalized coordinates?
Example4:
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Example4:
Examp
le4:Middle
screen
The middle-screen causes: (a) migration of the resonances and super-
multipeak response curves; (b) damping; (c) disappearance of the
primary resonance with increasing the solidity ratios
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Planar waves waves keeps symmetry with respect to the
excitation plane: occur far from the primary resonance. Swirling exactly at the primary resonance Irregular (chaotic) waves. Weak chaos? For the rectangular shape, a diagonal-type (squares-like) waves
waves with an angle to the excitation plane
Three-dimensional: nearly steady-state wave response
3Dslosh
ing
M i t d l t f b t k
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Moiseev-type modal system for square base tank
3Dslosh
ing
2 2 2 1 51 1, 0 1 1 1 2 1 2 2 1 1 1 1 3 2 1 1, 0 1,0 5
1 1
2 26 1 1 1 7 1 8 1 1 9 1 1 10 1 1 11 1 1 1 12 1 1
( ) ( )
( ) 0,
ga a d a a a a d a a a a d a a P S L L
d a b b d c d a b d c b d b a d a b b d b c
2 2 2 2 41 0,1 1 1 1 2 1 2 2 1 1 1 1 3 2 1 0,1 0,1 4
1 1
2 26 1 1 1 7 1 8 1 1 9 1 1 10 1 1 11 1 1 1 12 1 1
( ) ( )
( ) 0,
gb b d b b b b d b b b b d b b P S
L L
d b a a d c d a b d c a d a b d a b a d a c
1,0 1 2,0 2 0,1 1 0,2 2 1,1 1 3,0 3 2,1 21 1,2 12 0,3 3; ; ; ; , ; ; ;a a b b c a c c b
2 2
2 2, 0 2 4 1 1 5 10;a a d a a d a
2 2
2 0,2 2 4 1 1 5 10;b b d b b d b
21 1 1 1 2 1 1 3 1 1 1,1 1
0,c d a b d b a d a b c
2 2 2 1 53 3,0 3 1 1 2 2 1 3 2 1 4 1 1 5 1 2 3,0 3,0 5
1 1
( ) 0,g
a a a q a q a q a a q a a q a a P S
L L
2 2 2
21 2,1 21 1 6 1 7 1 1 1 8 2 9 1 10 2 1 11 1 1 12 1 1 13 1 1 1 14 1 1 15 2 1( ) ( ) 0,c c a q c q a b b q a q a q a b q c a q a b q a b a q a c q a b
2 2 2
12 1,2 12 1 6 1 7 1 1 1 8 2 9 1 10 2 1 11 1 1 12 1 1 13 1 1 1 14 1 1 15 1 2( ) ( ) 0,c c b q c q a b a q b q b q b a q c b q b a q a b b q b c q a b
2 2 2 2 43 0, 3 3 1 1 2 2 1 3 2 1 4 1 1 5 1 2 0, 3 0,3 4
1 1
( ) 0.g
b b b q b q b q b b q b b q b b P S
L L
Modal equations with nine degrees of freedom
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For harmonic forcing, one can find analytically
approximate steady-state solutions and study their stability
(also analytically!!!). This makes it possible to classify the
steady-state regimes and establish the frequency ranges
where the regimes exist and stable.
One can distinguish:3Dslosh
ing
the order (stable steady-state),
the strong chaos (treated as nostable steady-state for leading generalized coordinates), the weak chaos (here, irregular for higher-
order generalized coordinates)
Classification for longitudinal forcing:
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Classification for longitudinal forcing:
planar, diagonal (squares)-type, swirling and chaos
excitation amplitude = 0.0078L
3Dslosh
ing
Classification for longitudinal forcing:
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3Dslosh
ing
Classification for longitudinal forcing:
planar, diagonal (squares)-type, swirling and chaos
excitation amplitude = 0.0078L
Classification for longitudinal forcing:
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3Dslosh
ing
Classification for longitudinal forcing:
planar, diagonal (squares)-type, swirling and chaos
excitation amplitude = 0.0078L
Classification for longitudinal forcing:
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3Dslosh
ing
Classification for longitudinal forcing:
planar, diagonal (squares)-type, swirling and chaos
excitation amplitude = 0.0078L
Classification for longitudinal forcing:
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3Dslosh
ing
Classification for longitudinal forcing:
planar, diagonal (squares)-type, swirling and chaos
excitation amplitude = 0.0078L
Classification for longitudinal forcing:
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3Dslosh
ing
Classification for longitudinal forcing:
planar, diagonal (squares)-type, swirling and chaos
excitation amplitude = 0.0078L (relatively large!!!)
Local phenomena for 3D, but the classification is Ok.
Why? Why is the Moissev-type model still applicable?
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The method was developed in
2012, after the book issued
Sphericaltanks
3Dslosh
ing
( and ) radius 0 1205 m ( and ) 0 2615 m ( and ) 0 4065
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11secondary resonance by (01) a0 t /2 0 44. .9h
(and ) radius 0.1205 m ( and) 0.2615 m ( and) 0.4065
Classificationof3Dsloshing
( and ) radius 0 1205 m ( and ) 0 2615 m ( and ) 0 4065
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(and ) radius 0.1205 m ( and) 0.2615 m ( and) 0.4065
secondary resonance is far from the r e0.6 angh
Classificationof3Dsloshing
( and ) radius 0 1205 m ( and ) 0 2615 m ( and ) 0 4065
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(and ) radius 0.1205 m ( and) 0.2615 m ( and) 0.4065
11secondary resonance by (22) a1 t /0 1 33. .0h
Classificationof3Dsloshing
Swirling wave patterns taken for these input parameters from
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g p p p
T.Hysing (1976) Det Norske Veritas, Hvik, Norway
Specifically,
splashing,steep wave patterns,
local breaking.Classificationof3Dsloshing
( and ) radius 0 1205 m ( and ) 0 2615 m ( and ) 0 4065
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(and ) radius 0.1205 m ( and) 0.2615 m ( and) 0.4065
111.0 secondary resonance by (22) at / 1.033h
Classificationof3Dsloshing
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`planar splashing: ``...drops splashed from the tank wall and
showered through the ullage...''but wave patterns remain planar
for 1 splashing (planar & swirling type) is all the frequency rangeh
Classificationof3Dsloshing
Swirling always causes secondary resonances and
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Swirling always causes secondary resonances and
local phenomena when forcing amplitude increases,
but classification remains correct. Again, why?
This is explained by the concept of the weak chaos:Here, a clearly steady-state by a subset of [leading]
generalized coordinates andirregular motions by other
[infinite set] higher-order coordinates caused by higher
resonances.
Weakc
haosinslosh
ingproblem
s
The weak chaos concept for low-dimensional Hamiltonian
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Weakc
haosinslosh
ingproblem
s
p
(conservative) systems (see, e.g. Henning et al. (2013),
Physica D, 253):
ORDERWEAK CHAOS CHAOSLyapunov exponent: 0, but small >0, finite in domains of weak chaos trajectories (by higher-order generalized
coordinates) slowly diffuse into thin chaotic layers and wander through a
complicated network of higher order resonances
For ouralmost-conservative mechanical system with an infinite set of
generalized coordinates (g.c.), this implies:
ORDER WEAK CHAOS CHAOSstable by all g.c. stable by dominant g.c., unstable by all g.c.
but chaos in higher-order g.c.
unless a strong damping occurs
For 3D sloshing, the weak chaos is the reality well modelled by
the multimodal method
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Longitudinal resonant excitation with h/L=0.5 and
for a square-base tank. Higher-order (secondary) resonances.
0.00817 =
Fromt
heordertoweakchaosa
nd,
thereafter,strongchaosforswir
lingwith
increasingtheforcingamplitude
Swirling(weakchaos)
Swirling(weakchaos)
stron
gchaos
stron
gchaos
order
o
rder
order
order
Weakc
haosinslosh
ingproblem
s
Open problems within the framework of the
-
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Open problems within the framework of the
same paradigm, i.e., six degrees of freedomfor
the rigid tank andgeneralized coordinatesfor
the free surface motions:
Openproblems:intensive
1. Different internal structures.
2. Accounting for damping due to wave
breaking, overturning, etc.
3. Complex tank shapes.
4. Order weak chaos chaos.
5. Importance of weak chaos for coupledmotions
CFD are normally unapplicable on the long-time scale!
Input has more than six
-
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p
(infinite) degreesof freedom:
inflow-outflow & sloshing
(damaged ship tank, waveenergy, etc.);
elastic/hyperelastic tank
walls (fish farms,
membrane tanks, etc.);
ship collapse
Openproblems:extensive
REFERENCES
B k
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Books:
1. Faltinsen, O.M., Timokha A.N. (2009): Sloshing. Cambridge University Press. 608pp. (ISBN-13:
9780521881111) Chinese Version of the book issued in 2012: P.R.C.:National Defense Industry Press. 783pp.
(ISBN-13: 978-7-118-08608-3)
2. Gavrilyuk, I.P., Lukovsky, I.A., Makarov, V.L., Timokha, A.N. (2006): Evolutional problems of the contained
fluid. Kiev: Publishing House of the Institute of Mathematics of NASU. 233pp. (ISBN 966-02-3949-1)
Selected papers in peer-reviewed journals:
1. Faltinsen, O.M., Timokha, A.N. (2013): Multimodal analysis of weakly nonlinear sloshing in a spherical
tank.Journal of Fluid Mechanics, 719, 129-164
2. Faltinsen, O.M., Timokha, A.N. (2012):Analytically approximate natural sloshing modes for a spherical
tank shape.Journal of Fluid Mechanics, 703, 391-401
3. Faltinsen, O.M., Timokha, A.N. (2012): On sloshing modes in a circular tank.Journal of Fluid Mechanics,
695, 467-477
4. Gavrilyuk, I., Hermann, M., Lukovsky, I., Solodun, O., Timokha, A. (2012): Multimodal method for linear
liquid sloshing in a rigid tapered conical tank.Engineering Computations, 29, No 2, 198-220
5. Lukovsky, I.A., Ovchynnykov, D.V., Timokha, A.N. (2012): Asymptotic nonlinear multimodal method for
liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations.Nonlinear Oscillations, 14,
No 4, 512-525
6. Lukovsky, I.A., Timokha, A.N. (2011): Combining Narimanov--Moiseev' and Lukovsky--Miles' schemes for
nonlinear liquid sloshing.Journal of Numerical and Applied Mathematics, 105, No 2, 69-827. Faltinsen, O.M., Firoozkoohi, R., Timokha, A.N. (2011): Effect of central slotted screen with a high solidity
ratio on the secondary resonance phenomenon for liquid sloshing in a rectangular tank.Physics of Fluids, 23,
Issue 6, Art. No. 062106, 1-13
8. Faltinsen, O.M., Firoozkoohi, R., Timokha, A.N. (2011): Analytical modeling of liquid sloshing in a two-
dimensional rectangular tank with a slat screen.Journal of Engineering Mathematics, 70, 1-2, 93-109
9. Faltinsen, O.M., Firoozkoohi, R., Timokha, A.N. (2011): Steady-state liquid sloshing in a rectangular tank
with a slat-type screen in the middle: Quasilinear modal analysis and experiments. Physics of Fluids, 23, Issue
http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521881111http://www.imath.kiev.ua/~tim/PAPERS/book2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/book2006.pdfhttp://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8834748http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8834748http://www.imath.kiev.ua/~tim/PAPERS/JFM2012b.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2012b.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2012b.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2012.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/EC_2012.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/EC_2012.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/NO_2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/NO_2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JNAM_2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JNAM_2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/POF2011+.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/POF2011+.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JEM_2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JEM_2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JEM_2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JEM_2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/POF2011+.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/POF2011+.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JNAM_2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JNAM_2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/NO_2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/NO_2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/EC_2012.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/EC_2012.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2012.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2012b.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2012b.pdfhttp://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8834748http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8834748http://www.imath.kiev.ua/~tim/PAPERS/book2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/book2006.pdfhttp://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521881111http://www.imath.kiev.ua/~tim/PAPERS/POF2011-.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/POF2011-.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/POF2011-.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/POF2011-.pdf -
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with a slat type screen in the middle: Quasilinear modal analysis and experiments.Physics of Fluids, 23, Issue
4, Art. No. 042101, 1-19
10. Faltinsen, O.M., Timokha, A.N. (2011):Natural sloshing frequencies and modes in a rectangular tank with a
slat-type screen .Journal of Sound and Vibration, 330, 14901503
11. Barnyak, M., Gavrilyuk, I., Hermann, M., Timokha, A. (2011):Analytical velocity potentials in cells with a
rigid spherical wall.ZAMM, 91, No 1, 3845
12. Faltinsen, O.M., Timokha, A.N. (2010): A multimodal method for liquid sloshing in a two-dimensionalcircular tank.Journal of Fluid Mechanics, 665, 457-479
13. Hermann, M., Timokha, A. (2008): Modal modelling of the nonlinear resonant fluid sloshing in a rectangular
tank II: Secondary resonance.Mathematical Models and Methods in Applied Sciences, 18, N 11, 1845-1867
14. Gavrilyuk, I., Hermann, M., Lukovsky, I., Solodun, O., Timokha, A. (2008):Natural sloshing frequencies in
rigid truncated conical tanks.Engineering Computations, 25, Issue 6, 518-540
15. Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2007): Two-dimensional resonant piston-like sloshing in
a moonpool.Journal of Fluid Mechanics, 575, 359-397 [Supplementary material]
16. Gavrilyuk, I., Lukovsky, I., Trotsenko, Yu., Timokha, A. (2007): Sloshing in a vertical circular cylindrical
tank with an annular baffle. Part 2. Nonliear resonant waves.Journal of Engineering Mathematics, 57, 57-78
17. Gavrilyuk, I., Lukovsky, I., Trotsenko, Yu., Timokha, A. (2006): Sloshing in a vertical circular cylindrical
tank with an annular baffle. Part 1. Linear fundamental solutions.Journal of Engineering Mathematics, 54,
71-88
18. Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2006): Resonant three-dimensional nonlinear sloshing in
a square-base basin. Part 3. Base ratio perturbations.Journal of Fluid Mechanics, 551, 93-116
19. Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2006): Transient and steady-state amplitudes of resonantthree-dimensional sloshing in a square base tank with a finite fluid depth.Physics of Fluids, 18, Art. No.
012103, 1-14
20. Gavrilyuk, I.P., Lukovsky, I.A., Timokha, A.N. (2005): Linear and nonlinear sloshing in a circular conical
tank.Fluid Dynamics Research, 37, 399-429
21. Hermann, M., Timokha, A. (2005): Modal modelling of the nonlinear resonant sloshing in a rectangular tank
I: A single-dominant model.Mathematical Models and Methods in Applied Sciences, 15, N 9, 1431-1458
22. Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2005): Classification of three-dimensional nonlinear
sloshing in a square-base tank with finite depth.Journal of Fluids and Structures, 20, Issue 1, 81-103
23. Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2005): Resonant three-dimensional nonlinear sloshing
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in a square base basin. Part 2. Effect of higher modes.Journal of Fluid Mechanics, 523, 199-218
24. Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2003): Resonant three-dimensional nonlinear sloshing
in a square base basin.Journal of Fluid Mechanics, 487, 1-42
25. Faltinsen, O.M., Timokha, A.N. (2002):Asymptotic modal approximation of nonlinear resonant sloshing in a
rectangular tank with small fluid depth.Journal of Fluid Mechanics, 470, 319-357
26. Lukovsky, I.A., Timokha, A.N. (2002): Modal modeling of nonlinear sloshing in tanks with non-verticalwalls. Non-conformal mapping technique.International Journal of Fluid Mechanics Research, 29, Issue 2,
216-242
27. Gavrilyuk, I., Lukovsky, I.A., Timokha, A.N. (2001): Sloshing in a circular conical tank.Hybrid Methods in
Engineering, 3, Issue 4, 322-378
28. Faltinsen, O.M., Timokha, A.N. (2001):Adaptive multimodal approach to nonlinear sloshing in a rectangular
tank.Journal of Fluid Mechanics, 432, 167-200
29. Lukovsky, I.A., Timokha, A.N. (2001):Asymptotic and variational methods in nonlinear problems on
interaction of surface waves with acoustic field.J. Applied Mathematics and Mechanics. 65, Issue 3, 477-485
30. Faltinsen, O.M., Rognebakke, O.F., Lukovsky, I.A., Timokha, A.N. (2000): Multidimensional modal
analysis of nonlinear sloshing in a rectangular tank with finite water depth.Journal of Fluid Mechanics,
407, 201-234
31. Gavrilyuk, I., Lukovsky, I.A., Timokha, A.N. (2000): A multimodal approach to nonlinear sloshing in a
circular cylindrical tank.Hybrid Methods in Engineering, 2, Issue 4, 463-483
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