A Finite-Dimensional Modal Modelling of Nonlinear Resonant ...

A Finite-Dimensional Modal Modelling of Nonlinear Resonant Sloshing

by

Alexander Timokha&

Martin Hermann(Jena University, Germany)

Overview• Motivation and Basic Free Boundary Problem• Why not CFD methods? Coupling with Tank

Motions• (2D) Nonlinear Waves and Steady-State

Asymptotic Resonant Solutions• (2D) Pseudo-Spectral and Asymptotic Modal

Methods.• (2D) Transient and Steady-State Solutions of the

Modal Systems• (2D-3D) Survey of Existing Modal Systems• Problems and Perspectives

•Motivation and Basic Free Boundary Problem

•Why not CFD methods? Coupling with Tank Motions•(2D) Nonlinear Waves and Steady-State Asymptotic Resonant Solutions•(2D) Pseudo-Spectral and Asymptotic Modal Methods•(2D)Transient and Steady-State Solutions of the Modal System•(3D) Survey of Existing Modal Systems•Problems and Perspectives

Industry Motivations in Historical OrderIndustry Motivations in Historical Order• Air- and Spacecraft (dynamics and control)• Petroleum Storage Tanks (safety due to

earthquake)• Oil Ship Tanker (dynamics and safety)• (TLD) Tuned Liquid Damper (safety, active

control)• (LNG) Liquefied Natural Gas Carriers (dynamics

and safety)

Gas Transport Containment (aft end)

Fluid Sloshing in a Moving Tank

Experiments are made by Dr.Rognebakke,Centre for Ship and Marine Structure, Trondheim, NTNU, Norway

Free boundary problem

[ ]

[ ]

);(0)()(21

);()(1

);(

.;);(0

02

20

0

)(

tonUrvt

tonf

frrvn

tSonrrvn

constdQtQin

t

tQ

Σ=+⋅+⋅Φ∇−Φ∇+∂Φ∂

Σ∇+

+×⋅+⋅=∂Φ∂

×⋅+⋅=∂Φ∂

==∆Φ ∫

ω

νω

νω

)t,y,x(fz:)t()t(

)t(v)t,z,y,x(U

=Σ=

=general.in FOUND, BE TO :y nk velocitangular ta theis (t)

locity;ry tank ve translato theis (t) potential;gravity theis 0

ψωη

&

&

Why not CFD methods? Coupling with Tank Motions

• Vector-functions vo(t) and ω(t) are coupled by the equations of a whole object, not prescribed, in general

• Hydrodynamic forces and moments F(t) and M(t) are complete functions of f(x,y,t) and Φ(x,y,z,t), their derivatives

• ODE (tank‘s dynamics) + PDE (the fluid sloshing) for coupling (!!!)

Consideration below will for brevity be restricted to

two-dimensional sloshing

• Motivation and Basic Free Boundary Problem• Why not CFD methods? Coupling with Tank



Methods• (2D) Transient and Steady-State Solutions of the

Modal Systems• (3D) Survey of Existing Modal Systems• Problems and Perspectives

TEST PROBLEMResonant (surge for simplicity) Excitations of

the Lowest Natural Frequency

32100 cos 321 ,,i,;;t i ===== ψηητη

frequency. naturalhighest theis 2amplitude; forcing ldimensiona-non is

)( πτ

in non-dimensional form

Transient and steady-state waves

)t,z,x()t,z,x();t,x(f)t,x(f Φ∇=+Φ∇=+ ππ 22 :conditionsy Periodicit

)x(f),x(tf),x(f),x(f 10 00 :conditions Initial =

∂∂

= Useful for coupling

The problem has an Asymptotic Steady-State Solution for Finite Depth

)x(f)tctc(A)x(f)tcc(A)x(ftA)t,x(f 3323

2102

1 3 cos cos2 cos cos :structure l"dimensiona-finite" thehasSolution

++++=

...->...Narimanov (1957)->Moiseyev (1958)->...->Ockendon & Ockendon (1973)-> Faltinsen (1974)->...

modes naturallinear thearecosh

cosh50cos

1

where

322

131

)ih())hz(i()x(f)z,x());.x(i()x(f

),(O);(OA

iii

//

ππφπ

τσσλτ

+=−=

=

−==

Two conclusions: (i) finite number of modes,(ii) Intermodal ordering...



Asymptotic Resonant Solutions• (2D) Pseudo-Spectral (i) and Asymptotic Modal

Methods (ii)• (2D) Transient and Steady-State Solutions of the


Two conclusions appear as Postulations ofthe Finite-Dimensional Modal Modelling

∑

∑∞

=

∞

=

+=Ω++=Φ

=

121

1

1

:case)(2Dtion representaFourier

iiinOO

iii

).z,x()t(R,...)n),t(,z,x()t(z)t(vx)t(v)t,z,x(

);x(f)t()t,x(f

ϕβω

β

Pseudospectral methods– NAIVE TRUNCATIONAsymptotic methods – ORDERING and

as a consequence,-- truncation

Pseudospectral methods:ORIGINATED by Perko (1965)

,...,i,Jl)vgv(l)vgv(

ldtdllRRAAR

,...;,n,ARAdtd

.alet

iiOO

iO

i

t

i

t

ikn

k,n i

nk

k i

nn

knkkn

21021

21

21

Faltinsen by formin system sLukovsky'

12223

1331

0311

222

==∂∂

−∂∂

−−+∂∂

+−+

+

∂∂

−∂∂

+∂∂

+∂∂

+∂∂

==

∑∑

∑

βω

βω

βω

βω

βω

βω

ββωωω

&&

&&

itinkn J,l,l,l,A,A βωω of functions alranscedentexplicit t are12222

Asymptotic multimodal systems (2D case)can be derived from pseudospectral ones

• Physically defined ordering between modal functions βi(t) relative to the forcing τ and neglecting o(τ) leads to finite-dimensional structures. Examples (2D case):

• Finite depth – Narimanov-Moiseyev ordering: β1=O(τ1/3), β2=Ο(τ2/3),β3=Ο(τ)

• Secondary resonance ordering with decreasing depth and increasing excitation ε implies: βi(t)=Ο(τ1/3), i=1,N; βi(t)=O(τ2/3), i=N+1,2N; βi(t)=O(τ), i=2N+1,3N

• Boussinesq ordering for intermediate and shallow depth• Existing steady-state asymptotic solutions are derivable

from the modal systems instead of the original free boundary problem(!!!)

COUPLING: Hydrodynamic Forces

and Moments

(!!) Hydrodynamic forces and moments are explicit function of the generalized coordinates -- efficient coupling

Questions:• If an asymptotic modal system uses asymptotic

predictions, can it be used for transient waves and coupling? Since there are no exact solutions -- validation by experiments...

• What about periodic (steady-state) solutions of these models? Based on direct calculations...

• Example of this talk – the modal system based on the Narimanov-Moiseyev ordering: β1=O(τ1/3); β2=Ο(τ2/3); β3=Ο(τ) (!)




Methods: Advantages and Disadvantages• (2D) Transient and Steady-State Solutions of the


Differential equations based on the Moiseyev-Narimanov asymptotics

Faltinsen, Rognebakke, Lukovsky & Timokha (JFM2000)

Transient response, validation by experiments (very popular example from JFM2000)

Asymptotic and Numerical Analysis of the Asymptotic Modal System

forcingarbitrary for ity applicabilstudy to1),( −∞∈λ

Periodical solutions as the perturbed bifurcation problemT(β,λ,τ)=0

Nonlinear free-standing waves – the local solutions of the unperturbed problem

(unperturbed bifurcations τ=0)

Local nonlinear resonant periodical solutions Perturbed bifurcations τ>0

Non-local perturbed bifurcations for h=0.5, τ=0.0001 in small vicinity of λ=0.

Secondary bifurcation indicates ||β1||∼||β2||




Methods: Advantages and Disadvantages• (2D) Transient and Steady-State Solutions of the

Modal Systems• (2D-3D) Survey of Existing Modal Systems• Problems and Perspectives

Importance of filling level

• Finite water depth sloshing (filling height/tank length>0.24)– Resembles standing wave

• Shallow water sloshing (filling height/tank length<0.1)– Hydraulic jump / bore– Thin vertical jet – run-up

• Intermediate depth

Modal system accounting for secondary resonance (Faltinsen & Timokha (2001))

Comparison with experiments and CFD calculations (Smoothed Particles, Flow3D fails)

for large-amplitude forcing; fluid depth/tank length=0.35

Shalow sloshing ordering: comparison with experiments (Faltinsen & Timokha (2002))

Free surface elevation for square based tank withfinite depth.

Modal system by Faltinsen-Timokha (2003)

• ζ=∑∑βik(t) cos(πi(x+0.5L))cos(πk(y+0.5L)

• β10=O(τ1/3) β01=O(τ1/3)

• β20=O(τ2/3) β11=O(τ2/3) β02=O(τ2/3)

• β30=O(τ) β21=O(τ) β12=O(τ) β03=O(τ)

Wave amplitudes A as a function of excitation frequency σ

Perodical solutions (classified) in a square basetank for longitudinal excitation. Effect of fluid depth

Periodical solutions (classified) in a square basetank for longitudinal excitation. Effect of fluid depth

Problems and perspectives• Modal systems exist for different types of the

tanks. However…the problems consist of:• Rigorous bifurcation analysis of periodic

solutions for existing multidimensional systems and related numerical (path-following) schemes.

• Conclusions on applicability, the limits...• Cauchy problem for modal system (transient

waves) – numerical schemes for modal systems of large dimension including stiffness.

• Modal systems for tanks of complex geometry.

http://www.humboldthttp://www.humboldt--foundation.defoundation.de

The speaker’s attendanceat this conference was sponsored

by theAlexander von Humboldt Foundation

The speaker’s attendanceat this conference was sponsored

by theAlexander von Humboldt Foundation

Thank you for your attention

A Finite-Dimensional Modal Modelling of Nonlinear Resonant ...

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