1

Multimodal Method in Sloshing Analysis:

Analytical Mechanics Concept

by Alexander 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

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NC

EPT

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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 with Prof. Odd M. Faltinsen

“Liquid Sloshing Dynamics” is treated as a conservative mechanical system with infinite degrees of freedom

so that the Lagrange formalism can adopt the generalized coordinates and velocities responsible for the global liquid modes instead of working with typically- accepted hydrodynamic characteristics (velocity field, pressure, etc.)

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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 liquid N.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 a specifically-modified inertia tensor’ • Considering the fluid as “frozen” is a wrong way (boiled and fresh

eggs!!!). • The velocity field is described by the so-called Stokes-Joukowski’

potentials.

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& ID

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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 zr x y zy r

the Stokes-Joukowski potential

FULLY FILLED TANK

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& ID

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Having known the time-independent Stokes-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)?

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& ID

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LINEAR SLOSHING, 50-60’s 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

N

N

N

r x y z xx y z t R t

x y

yv t z

x

t

z t t y

sloshing modes

interpreted as generalized coordinates (infinite set!!!)

Free surface generalized velocities

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RY

& ID

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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 problem

in 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 mm m

km m

k

mm

g g m

( , , )N x y z

0( , , )x y z

HISTO

RY

& ID

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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 00’s

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 input and infinite set of the free-surface generalized coordinates

Generalized velocities

Generalized coordinates HISTO

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& ID

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THERE ARE • APPLIED

MATHEMATICAL & • PHYSICAL

PROBLEMS TO BE SOLVED TO IMPLEMENT THE NONLINEAR MULTIMODAL METHOD

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The nonlinear multimodal method 1. The method is of analytical nature.

What are analytical limitations? 2. The Euler-Lagrange equation for

liquid sloshing dynamics 3. Physical and mathematical arguments

for choosing the generalized coordinates

4. How does it work?

The m

ultimodal m

ethod

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A. For any instant must be defined in the time-varied liquid domain and satisfy the tank surface condition

B. Derivation of the Euler-Lagrange equations with respect to the generalized coordinates and velocities

( , , )N x y z

( )Q thand-made product for each tank shape

e.g., the Bateman-Luke variational principle

MAT

HE

MAT

ICA

L LIM

ITATIO

NS

Bateman-Luke variational principle derives the Euler-Largange equation

The following problems must be solved analytically:

Assuming A (we know analytical modes):

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(Kinematic Eq.): , changingd

;d

N NK NK K

K KK

A AA F

tN

(Dynamic Eq.): 1 2 3

1 2 changing1

... 0 2

K KLK K L

K KLN N N N N

A A l l lF NF F

where

* *( ) ( )

1 * 2 * 3 *( ) ( ) ( )

d ; d ,

d ; d ; d

N N N NK N N KQ t Q t

N N NQ t Q t Q t

A Q A Q

l x Q l y Q l z Q

EU

LE

R-L

AG

RA

NG

E E

QU

ATIO

N

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 solved analytically: A. For any instant must be defined

in the time-varied liquid domain and satisfy the tank surface condition

B. Derivation of the Euler-Lagrange equations with respect to the generalized coordinates and velocities

C. Reduction of the infinite-dimensional Euler-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 thand-made product for each tank shape

e.g., the Bateman-Luke variational principle

Normally, asymptotic approaches

results of the last decade

PHY

SICA

L AG

RU

ME

NT

S

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HOW does it work? Let’s consider devils examples for:

1. For 2D flows 2. For 3D rectangular tanks with upright

walls 3. For complex tank shapes H

ow does it w

ork?

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 – Moiseev’s asymptotics • Increasing forcing magnitude, or the critical depth

ratio (h/l=0.3368…) – secondary resonances and adaptive asymptotics

Intermediate and small depths (liquid depth/tank length < 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 modes can be resonantly excited due to the so-called secondary resonances.

What does it mean in terms of leading generalized coordinates?

2D in rectangular tank

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When forcing the lowest natural frequency: Finite liquid depth (liquid depth/tank length>0.2) • A small forcing magnitude – Moiseev’s asymptotics • Increasing forcing magnitude, or the critical depth ratio

(h/l=0.3368…) – secondary resonances and adaptive asymptotics

Intermediate and small depths (liquid depth/tank length < 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 modes can be resonantly excited due to the so-called secondary resonances.

What does it mean in terms of leading generalized coordinates?

2D in rectangular tank

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Example 1: The Moiseev-type multimodal theory The modal solution

Moiseev proved for the steady-state (periodic) sloshing due to excitation 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 2/31 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

Exam

ple 1:

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21 1 2 322

2 21 1 1 2 1 2 1 1 1 1 2 1

22 2 1 1 1

2 23 3 1 2 1 1 2 1 1 1 1

4 523 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: Exam

ple 1:

Coefficients are analytically found as functions of the liquid depth to the tank breadth ratio…

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Tran

sien

ts:

Expe

rimen

ts

Theo

ry w

ith d

iffer

ent

initi

al sc

enar

ios

Steady-state waves with the horizontal/angular harmonic forcing, the dominant amplitude parameter:

(frequency , nondimensional amplitude ),

/ 0.3368...h l / 0.3368...h l

Exam

ple 1:

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When forcing the lowest natural frequency: Finite liquid depth (liquid depth/tank length>0.2) • A small forcing magnitude – Moiseev’s asymptotics • Increasing forcing magnitude, or the critical depth

ratio (h/l=0.3368…) – secondary resonances and adaptive asymptotics

Intermediate and small depths (liquid depth/tank length < 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 modes can be resonantly excited due to the so-called secondary resonances.

What does it mean in terms of leading generalized coordinates?

Exam

ple 2:

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Exam

ple 2:

Exa

mpl

e 2:

A

dapt

ive

mod

al sy

stem

s for

cr

itica

l dep

th a

nd in

crea

sing

fo

rcin

g am

plitu

de

• Nonlinear multiple frequency effects excite higher natural frequencies • Increased importance with decreasing depth and increasing forcing

amplitude • Reason for decreasing depth importance is that when the

liquid depth goes to zero • Implies that more then one mode is dominant

n n

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Different ordering of the generalized coordinates due to secondary resonances with a finite liquid depth

Exam

ple 2:

Rectangular tank, 1x1m, nearly critical depth, h/l = 0.35

Subharmonic regimes are predicted

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Increasing the forcing amplitude and accounting for roof impact, h/l=0.4

Exam

ple 2:

0.01

0.1

impact neglected impact accounted for Flow 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 for “alpha=1.0, epsdj=0.01”. Experiments: The influence of viscosity: ■, fresh water; ○, reginol-oil; □, glycerol-water 63%, ♦, glycerol-water 85%

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When forcing the lowest natural frequency: Finite liquid depth (liquid depth/tank length>0.2) • A small forcing magnitude – Moiseev’s asymptotics • Increasing forcing magnitude, or the critical depth ratio

(h/l=0.3368…) – secondary resonances and adaptive asymptotics

Intermediate and small depths (liquid depth/tank length < 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 modes can be resonantly excited due to the so-called secondary resonances.

What does it mean in terms of leading generalized coordinates?

Exam

ple 3:

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Intermediate and small depths normally cause amplification of higher modes and a series of local wave breaking & overturning

Exam

ple 3:

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As a consequence, a Boussinesq-type ordering can be proven being applicable to with

1/4( / ) ( ), 1i i

O h l O iR

Tran

sien

ts:

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

Exam

ple 3:

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Exam

ple 3:

Steady-state due to harmonic forcing ◊ = experiments Chester & Bones

Theory by Chester Multumodal 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 and damping are important when dealing with secondary resonances.

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Associated damping becomes important with Decreasing depth and increasing forcing amplitude Roof impact Internal structures Normally, viscous boundary layer effect is less

important Damping terms can be incorporated into the modal equations following the strategy in Chapter 6 of «Sloshing» 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

Exam

ples vs. damping

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When forcing the lowest natural frequency: Finite liquid depth (liquid depth/tank length>0.2) • A small forcing magnitude – Moiseev’s asymptotics • Increasing forcing magnitude, or the critical depth ratio

(h/l=0.3368…) – secondary resonances and adaptive asymptotics

Intermediate and small depths (liquid depth/tank length < 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 modes can be resonantly excited due to the so-called secondary resonances.

What does it mean in terms of leading generalized coordinates?

Exam

ple 4:

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Exam

ple 4:

Exa

mpl

e 4:

Mid

dle

scre

en

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

3D sloshing

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Moiseev-type modal system for square base tank

3D sloshing

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 22 2,0 2 4 1 1 5 1

0;a a d a a d a

2 22 0,2 2 4 1 1 5 1

0;b b d b b d b

21 1 1 1 2 1 1 3 1 1 1,1 1

ˆ ˆ ˆ 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 SL L

2 2 221 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 2 212 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 SL 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: 3D

sloshing

• the order (stable steady-state), • the strong chaos (treated as no stable steady-

state for leading generalized coordinates), • the weak chaos (here, irregular for higher-

order generalized coordinates)

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Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = 0.0078L

3D sloshing

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3D sloshing

Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = 0.0078L

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3D sloshing

Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = 0.0078L

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3D sloshing

Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = 0.0078L

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3D sloshing

Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = 0.0078L

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3D sloshing

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?

41

The method was developed in 2012, after the book issued

Sphe

rica

l tan

ks

3D sloshing

42

11 secondary resonance by (01) a0 t /2 0 44. .9h

(▲ and ∆) − radius 0.1205 m (♦ and ◊) − 0.2615 m (• and ο) − 0.4065

Classification of 3D

sloshing

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(▲ and ∆) − radius 0.1205 m (♦ and ◊) − 0.2615 m (• and ο) − 0.4065

secondary resonance is far from the r e0.6 angh

Classification of 3D

sloshing

44

(▲ and ∆) − radius 0.1205 m (♦ and ◊) − 0.2615 m (• and ο) − 0.4065

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

Classification of 3D

sloshing

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Swirling wave patterns taken for these input parameters from T.Hysing (1976) Det Norske Veritas, Høvik, Norway

Specifically, splashing, steep wave patterns, local breaking.

Classification of 3D

sloshing

46

(▲ and ∆) − radius 0.1205 m (♦ and ◊) − 0.2615 m (• and ο) − 0.4065

111.0 secondary resonance by (22) at / 1.033h

Classification of 3D

sloshing

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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

Classification of 3D

sloshing

48

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 and irregular motions by other [infinite set] higher-order coordinates caused by higher resonances.

Weak chaos in sloshing problem

s

49

Weak chaos in sloshing problem

s

The weak chaos concept for low-dimensional Hamiltonian (conservative) systems (see, e.g. Henning et al. (2013), Physica D, 253): ORDER WEAK CHAOS CHAOS Lyapunov exponent: <0 >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 our almost-conservative mechanical system with an infinite set of generalized coordinates (g.c.), this implies: ORDER WEAK CHAOS CHAOS stable 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

50 Longitudinal resonant excitation with h/L=0.5 and for a square-base tank. Higher-order (secondary) resonances.

0.00817ε =

From

the

orde

r to

wea

k ch

aos a

nd,

ther

eaft

er, s

tron

g ch

aos f

or sw

irlin

g w

ith

incr

easi

ng th

e fo

rcin

g am

plitu

de

Swirl

ing

(wea

k ch

aos)

Swirl

ing

(wea

k ch

aos)

stro

ng c

haos

stro

ng c

haos

orde

r

orde

r

orde

r

orde

r

Weak chaos in sloshing problem

s

51

Open problems within the framework of the same paradigm, i.e., six degrees of freedom for the rigid tank and generalized coordinates for the free surface motions:

Open problem

s: 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 coupled

motions CFD are normally unapplicable on the long-time scale!

52

Input has more than six (infinite) degrees of freedom: • inflow-outflow & sloshing

(damaged ship tank, wave energy, etc.);

• elastic/hyperelastic tank walls (fish farms, membrane tanks, etc.);

• ship collapse

Open problem

s: extensive

53

REFERENCES 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, 69-82

7. 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

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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 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, 1490–1503

11. Barnyak, M., Gavrilyuk, I., Hermann, M., Timokha, A. (2011): Analytical velocity potentials in cells with a rigid spherical wall. ZAMM, 91, No 1, 38–45

12. Faltinsen, O.M., Timokha, A.N. (2010): A multimodal method for liquid sloshing in a two-dimensional circular 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 resonant three-dimensional sloshing in a square base tank with a finite fluid depth. Physics of Fluids, 18, Art. No. 012103, 1-14

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