Multimodal Method in Sloshing Analysis Analytical Mechanics Concept

8/22/2019 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)


Increasing forcing magnitude, or the critical depth ratio

(h/l=0.3368) secondary resonances and adaptiveasymptotics



Boussinesq ordering




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.



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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Increasing forcing magnitude, or the critical depth

ratio (h/l=0.3368) secondary resonances andadaptive asymptotics



Boussinesq ordering







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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Boussinesq ordering







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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Boussinesq orderingInternal structures (e.g., bulkheads)






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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planar, diagonal (squares)-type, swirling and chaos

excitation amplitude = 0.0078L

3Dslosh

ing



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3Dslosh

ing






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3Dslosh

ing






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3Dslosh

ing






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3Dslosh

ing






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3Dslosh

ing



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



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secondary resonance is far from the r e0.6 angh




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11secondary resonance by (22) a1 t /0 1 33. .0h


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



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111.0 secondary resonance by (22) at / 1.033h



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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


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
http://www.imath.kiev.ua/~tim/PAPERS/JoEM2007.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/POF2011-.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JSV2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JSV2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/ZAMM2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/ZAMM2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/ZAMM2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2010.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2010.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/M3AS2008.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/M3AS2008.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/EC2008.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/EC2008.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2007.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2007.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2007_SM.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JoEM2007.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JoEM2007.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JoEM2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JoEM2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/PoF2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/PoF2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/FDR2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/FDR2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/M3AS2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/M3AS2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFS2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFS2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFS2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFS2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/M3AS2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/M3AS2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/FDR2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/FDR2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/PoF2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/PoF2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JoEM2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JoEM2006.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JoEM2007.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JoEM2007.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2007_SM.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2007.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2007.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/EC2008.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/EC2008.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/M3AS2008.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/M3AS2008.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2010.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2010.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/ZAMM2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/ZAMM2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JSV2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JSV2011.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/POF2011-.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2005.pdf


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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
http://www.imath.kiev.ua/~tim/PAPERS/JFM2005.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2003.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2003.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2002.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2002.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2002.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2001.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2001.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2001.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/PMM2001.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/PMM2001.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/PMM2001.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2000.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2000.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2000.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2000.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/PMM2001.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/PMM2001.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2001.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2001.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2002.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2002.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2003.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2003.pdfhttp://www.imath.kiev.ua/~tim/PAPERS/JFM2005.pdf


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Multimodal Method in Sloshing Analysis Analytical Mechanics Concept

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