Adaptive multimodal approach to nonlinear sloshing in a rectangular tank

DoiftnVeritot TecI*roIogj Ft1

Ship KyGrOmchanics Laboratory

Ubrary

Mekelweg 2-2628 CD Deift The Netherlands

Phofle: 31 15786873- Fax: 31 15 781836

Adaptive Muitirnodal Approach. to Ndnlinear S1oshing in a

Rectangular Tank

Q.M. Faltinsen', A.N. Timokha2

'Department of Marine Hydrodynamics, Faculty. of Marine Techp.o1pgy NTNTJ, Trondheim,N-7491, Norway

2lnstitute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska, 3 str., Kiev, 252601, Ukraine

Abstract

Two'-dimensiona1 nonlinear sloshing of an incompressible fluid with irrotational flow in a rectangular tank is analyzed by a modal theory. Infinite tank roof height ad no over-turnmg waves are assumed The moda,l theory is based on an mflnitedimensional system of nonlinear ordinary differential equations coupling generalized coordinates of free surface and

fluid motiOn associated with amplitude response of natural modes. This modal system is asymptotically reduced to an infinitedimensional system of ordinary dierentiä1 equations with fifth order polynomial nonlinearity by assuming sufficiently small. fluid motion relative to fluid depth and tank. breadth. When introducing inter-modal ordering, the System can be detuned and truncated to describe resonant sloshing in different domains of the excitation period. Resonant sloshing due to surge and pitch sinusoidal excitation of primary mode is considered. By assuming that each mode has only one main harmonic an adaptive pro-cedure, is proposed to describe direct and secondary resonant responses when Moiseevlike relations do not agree with experiments, i.e. when excitation amplilude is not very small, the fluid, depth is close to critical depth and for small depth. Adaptive procedures have been estáblishêd for all excitation periods as long as the mean fluid depth h is larger than 0.24 times the tank breadth 1. Steadystate results for wave elevation, horizontal force and pitch moment are experimentally validated except when heavy roof impact occurs The analysis of small depth requires that many modes have primary order and that each mode may have more than one main harmonic. This is illustrated by an example for h/I. = 0.173, where the previous model by [Faltinsen et al (2000)] failed New model agrees well with experiments

- _{fl'_}

r11

--1

Iiitrodution

A partially filled hip táiik can eperience violent fluid cargo loads (see, [ISSC Report (1997)]). The model of "frozen" fluid and linear theory of slosh.ing are not applicable in thiscase Very

long time simulations are needed to obtain statistical estimates of the fluid cargo response.

Com-plex transients should be accounted for. Examples of direct numerical simulations of the fluid

motions in a tank have been reported by [ISSC Report (1997), EUROSLOSH Report (1995)]. These methods utilize various FD, FE and BE approaches. A difficulty is to perform long time simulations. One reason is problems in satisfying volume (energy) conservation. The long time

simulations give then non-realistic flows. There is also the additional problem of describing

accurately fluid impact inside the tank

Several analytical methods based on potential theory have been developed to study nonl.inear sloshing. MOst of them examine fluid response due to direct or parametric excitation and utilize

asymptoti. and modal technique. Modal representation means that Fourier series with time

dependent coefficients are used to describe the free surface. The natural modes for linear sloshilig are implemented as basis functions. The free surface shape is mathematically expressed as

z

f(x,y,t) =

82(t)(stirfacernode)(x,y). (1)

Here Oxyz coordinate system is fixed with the tank and t is time.

The original free bound.ry problem can be reduced to a finite sequence of asymptotic

approximations of flj by postulating asymptotic relations between 6. Such detuning proce-dure for direct resonant sloshing in a 2D rectangular tank was proposed by [Moiseev (1958)]

and developed by [Faltinsen (1974)] for nonlinear response of lowest primary mode. Moiseev's

asymptotic relations lead to a third order theory in amplitude of primary mode. This theo-ry is not uniformly

'alid for ciltical depth h/i = 0.3374.. (1 is the tank bteadth). The fifth

order fluid response at critical depth was then reported by [Waterhouse (1994)].. If depth is small, the asymptotic relations should be detuned to cover shallow water asymptotics (see,

[Ockendon, Ockendon & Johnson (.1986)]).

The asymptotic relations ifltroduced by Mdiseev can be used for modulated waves in

rectä.-gular and vertical cylindrical tanks. The problem in rectanrectä.-gular tnk is then reduced to single

Duffing-like equations for slowly varying primary mode amplitude (f3) (see, [Shemer (1990)] and [Tsai et at (1990)]). Similar approaches have also been reported for vertical cylindrical tanks of.

'All the frequencies below mean circular frequencies with dimension [rad/sJ

3

circular and rectargular crosssections by [Miles (1984a), Miles (1984b), Henderson & Miles (1991), Miles (1994)] for both directly and parametrically excited surface waves.

The above mentioned asymptotic approaches cannot describe complex transients due to inter-modal interactions. Further the exitation amplitude has to be very small. This limits the

theories to realistically describe sloshing in a ship tank.

[Faltinsen et al (2000)] reported a general modal system of nonlinear _{ordinary differential} equations coupling 8. It was reduced to a threeclimensional system of nonlinear differential

equations to describe resonant sloshing in a rectangular tank of finite fluid depth. The reduction is based on the asymptotic relations of Faltinsen-Moiseev's theory (see, [Faltinsen (1974)]) with

flu 113,132 213,

fl

. Here

is a small parameter characterising the ratjo between

the excitation amplitude and the tank breadth. The main results by [Faltinsen (1974)] were

rederived. The method can describe transients. Simple formulas for hydrodynamic force and moment on the tank were presented and expressed by /32. The procedure ws experimentally

validated for different finite fluid depths, excitatiOn periods and small excitation amplitudes.

Further investigations by this asymptotic model have shown its limitations to simulate steady-state motions and transients when maximum free su1ace elCvation is the order of the tank breadth or fluid depth. This happens if either is not very small, the depth h is close to the critical value h/I = 0.3374... or in shallow water. The numerical, simulations give

a

non-realistic response of higher (driven) modes., The assumption about ulngle dominant mode is then questionable.

A way to solve this problem for surge and pitch excited sloshing in a rectangular tank is proposed in this paper. The failure of the asymptotic approximations is explained as nonlinear fluid interactions causing energy with frequency content at higher natural frequencies. Higher modes can give sufficient effect on the stability of wave motions and thereby determine the behavior of main response and cause a "switch" between asymptotic theories. This point of view allowed to describe in details the mechanism of secondary resonance and explain some

previously found disagreements betWeen theory and experiments.

An example ofsuch mechanisms is as follows. Nonlinearities cause: oscillations with frequency

2o-, where o is the excitation frequency1 of the rigid body motion. If the second natural frequen-cy o-2 of the fluid is clQse to 2o-, secondary resonance will, occur. The generalized coordinate /32

will be amplified and can be of the same order as th. Nonlinear interactions can also cause res-onant oscillations at other natural frequencies. Since the difference between natural frequencies decreases with decreasing fluid depth, this. is more likely to occur at smaller fluid depths. This is a feasôn why th&flnite fluid depth theories by [Faltinsen (1974)] and [Faltinsen et al (2000)] are invalid for small fluid depth. The effective domain of secondary resonance coincides then with primary resonance. If the excitation amplitude is increased, the fluid response becomes large in

an increased frequency domain around the first natural frequency. This increases the possibility

that large nonlinearly excited resonance oscillations at a higher natural frequency can occur. Both the second and third order mode can then be of the same order as the first mode. That

is why the theory by [Faltinsen et al (2000)] is only valid for very small excitation amplitudes. Since the amplification of the fluid motion is relatively larger in the vinicity of the critical depth

then at other fluid depths, the upper bound of tank excitation amplitude where the theory by

[Faltinsen et al (2000)] is applicable for critical depth is relatively small.

The limitation of the asymptotic. theory [Faltinsen et al (2000)] implies that one must de-rive other as ptotic modal equations without a fixed dominant mode. We will still assume e to le small and that the response is asymptotically larger thefl the excitation. The method uses the general modal system presented by [Faltinsen et al (2000)] and Taylor expansion of

volume-varying integrals depending on fit, The derived approximate modal system couples

and translatory and angular body motions. The coefficients of this system depend

unique-ly on the mean depth. This is its main advantage relative to the general modal system by

[Faltinsen et al (2000)]. These coefficients can be computed before time simulation. In addi-tion, we expand the formulas for hydrodynamic forces and moments correctly to 0(e) The

infinite-dimensional system is not convenient for simulation. It should be detuned (truncated)

to obtain a finite-dimensional structhre. The Moiseev asymptotics by [Faltinseri ët al (2000)] is

a particular case.. The secondary resonance needs to consider two or more modes to have the

same order. The derived procedure adapts modal system to simulate sloshing occurring when a "switch" between asymptotic relations occurs.

One limitation of proposed approximation is the assumed right angle between wave profile and vertical wall at contat point. If real surface shape is different, this causes bad convergence

This happens when sloshing causes run-up at the wall The latter can be important at smaller

The theory assumes infinite tank roof height, but roof impact is very likely during realistic sloshing in ship tanks.

This can be handled for a horizontal tank roof in a similar way as

done by [Faltinsen & Rognebakke (1999)]. They used the theory by ['altinsen et al (2000)] as

an 'iinbient flow. Further studies are needed, for roof impact, in particular, for a chamfered

tank roof (see, [Rognebakke & Faltinsen (2000)]). If roof impact is included, it is believed that

proposed adaptive multimodal approach can be applied to sloshing in a smooth rectangular or prismatic tank of a ship in realistic seaways. But a strategy ha. to be established that accounts

for simultaneous excitation frequencies as well as vertical tank motions. The coupling with the

ship's rigid body motion is also needed. The procedure allows for that. The findings in this report suggest that the fluid depth has to be finite. When the mean fluid depth is less then 0.24 times the tank breadth, it was not possible to find asymptotic relations between ,8 that applied

for all forced excitation periods near primary resonance

2

Modal sloshing theory

2.1

Statement of the problem.

The free boundary problem on sloshing of an incompressible fluid with irrotational flow is

/.=O inQ(t);

!.=vo.v+w.[rxv]

onS(t),

= t0 . ii ±w. [.i x z.'] + on

onE(t), f dQ=const.

(2)

c9t 2 _Q(t)

Here Q(t) 'is the fluid volume, t is time, (x, y, z, t) is the velocity potential, (x, y, z, t) = 0 is the equation of the free surface and ii is outer normal to Q(t). An Oxyz coordinate system fixed

with respect to body is used. vo(t) is the translatory velocity vector of the origin 0, w(t) is the rigid body angular velocity vector, r IS the positiofl vector relative to 0 and U is the gravity potential. Further S(t) is the wetted body surface and E(t) is the free surface.

In order to develop analytical approaches for description of transients in a resonantly excited

rectangular tank a general Semianalytical modal approach to problem (2) was developed by [Faltinsen et al (2000)]. A rigid smooth tpk is assumed with vertical walls near free surface. Overturning waves are not considered. The free surface z

f(x, t) and

in twodimensional

flows are expressed as

f(

t) =132(t)fj(x); (x, z, t) =VOX + VOZ + w(t) i(x, z, t) + Rk (t)cok(x, z), (3)

where the repeated upperlower indexes mean summation and vo

=

{vo,0vo},w

{0,w,0}.

Further {f2(x)} is a set of complete functions on the mean free siirface and

z)} is a

complete set of harmOnic functions satisQying no-slip &nditibis on the wall. 1l(x,.z, t) is Stokes-Zhukovsky potential defined by Neumann boundary value problem

= 0

in Q(t);

Is(t)+E(t) =zzi

xii,

(4)

where v {v, 0, u } is outer normal to Q (t).

The developed variational procedureis based on the works by [Miles (1976)] and [Lukovsky (1976)].

It gives an inflnit&-dimenMonJ system of noplinear ordinary differntiai eqtia.tions coupling $

and R (general modal system)

d

- AflkR'

ü, n

1; (5)

?+PsYR1C+

+w_

()]

+

013 1 20J2'2

+(vQg1+c1vQz)-+(vozg3wvox)w j=O,

1

where gi are the projections of gravity acceleration vector and dot means time derivative.

A, Ak, 1w, twt, 11,13 and J2

are integrals over the timedependent domain Q(t), and,

therefore, is a function of $. We can write (see, [Faltinsen et al (2000)])

çodQ,

_Aflk=Pf

VVkdQ,

Q(t) Q(t)

fdQ, lt

=

f

-dQ,

-

f f22dSfi;

-Q(t) Ot Eo 013i 1

iac

oc

J22=pj

(z--xIdQ=pi

JQ(t) \ Ox Oz j JS(t)+E(t) Ov where p is fluid density and E is the rrian free surface.

The system of nonlinear ordinary equations (5), (6) is a discrete analogy of the original free boundaiy problem (2) in two-dimensional case. It can be considered as governing equations of a conservative, mechanical system with infinite number of degrees of freedom. It has, also, a lot of advantages in deriving asymptotic theories and creating numerica1 schemes.

fQ(t)

Figure 1: Coordinate system.

2.2

Derivation of third (fifth) order mOdal sloshing theory

We consider a fluid volume Q(t) in a rectangular taiij. Here

Q(t) = {l/2 <x <1/2,h <z < f(x,y)}

(1Q)

where 1 is tank breadth and h is meanfluid depth. The mobile coordinate system Oxz is coupled

with mean free surface E0, so that its equation is z = 0. Tie origin 0 is situated in the tank

middle. (See Fig. 1).

We associate furthermore the modal basis {f1(x)} and {(x, z)} with natural modes, i.e.

cosh(ic(z+h))

in

f(x) =cos(c2(x+l/2));

con(x,z) =fn(x) _{h( h)} ' ' (11)

The asymptotic procedure requires Taylor expansions of the itegrals (7), (8) and (9) It is

only possible if the resulting free surface elevation with respect to mean free surface is sufficiently

small relative to tank bleadth, or fluid depth. Contrary to [Fà.ltinsen et al (2000)] we do not associate the procedure with dominating and nondominating modes.

The integrals A, Ak will be expanded up to fifth order in f3. This allows for instance to

irnrestigate fifth order response at critical depth [Waterhouse (1994)]. Other integrals depending

on 2 will be expanded up to third order. This is due to mathematical problems in asymptotic

approidrnations. The, zeroorder approximation of does not allow high derivatives calculations.

Expansion of A. The integrals A should be expanded in power series by f (free surface

elevation) to calculate it as function, of_fi correctly to fifth order. Taylor series of (7) gives

A rI/2 1 p1/2 3m ₁ ,.i/2 2

=-.=

I çcdQ±

_I

çofdx+ _I

f2dx+

_I f3dx+ p JQ0 Jl/2 2 J-1/2 ôz z=O _{6 J-1/2 t9z} z=O 1 [1/2 4i-I/2 8z3 7 1 1/234 -f4dx± 120 f_j/2 t9Z4 z=Of5dx + o(f5). (12) z Undisturbed water plane

t

/

/ z=-h

x=W2

/

By setting (3) into (12) we get correctly to fifth order

A=

{

±

± CnnAkfih/33/3k±

+

_cflflALäks/

kfipq

} wiere 7rn

E= 0.5ic tarih(,h).= --tanh(tch);

_{Ck =rn'ik}

nk-12

and integer tensors A re given by (70) and (71).

Equations (5) and (6) require and to be known. The explicit expressions for these derivatives are

A 1

{ & +

+ cflnAkjuii$k± cflflEflAk/3i/,1d/3P+

} (15)

ôfiL = p

{o,. +EAj3

+ CflflAM/3u/33+

nnnAkj3u,33/3k±

±6cflflAfl.kP13/3i9/3 }.

(16)

Expansion of Ak. The tensor products AflkRk in (5) and, 4ARThRk in (6) should be cal-culated up to fifth. order in, /3. Since R = O(fiTh), the expansion Of Ank has to be cOrrect to

foirth order. Taylor series gives

=

I

n VçpkdQ ± f112 k)

fdx+

f2

P Qo -1/2

Z0

2 -1/2

1"

Vk)

f3dx + 1 1

a3(v

V)

6 J-L/2 z=O _{24 J_112}

az

It follows that

fd

z=o 2dx± =0

±(f4);

(17)

+ r1,/3 +

+

+

njjpqfii,$i,aj3a),

(18)

where comma separates symmetric sets of indexes. The tensors H are

4CA)

+ 2EEkA,),

= 2Cnk(En +

Ek)A2) + 2(CflflEk.+ CkkEn)A,

[cc ± Ckk + EkE)A

+ (EflEk(Cfl ± Ckk) + 4Ci)Ajpq

kflp ±

(3)

'n/c,ijpq

[cflkEfl(cflfl + 3Ckk) + Ekkk + 3Cnn))A)jpq+

+(EkCnn(Cnn + 3Ckk) + EnCkk( Ckk + 3Cnn))Ajjpq] (19)

where tensors A are defined by (71) and (72)

The partial derivatives of Ak by fi, have the following form

84k

_{[rig, +}

211nk,i,J3 +

3r4,J3u/3i

+

(20)

Asymptotic solution of (5).

Furthermore (5) is considered as a linear system of algebraic

equatiOn with. respect to functions Rk,. i.e.

ARk

(21)

Here Rc can be expressed corret1y to fifth order in flu as

Rc 9k

₊

+

93P + %i p,q1/3i/3Pfi + ₍₂₂₎

Tensors V are found by subStituting (22) into (21) and gathering similar terms in /3. The

introduced tensors V have no symmetry between the index i and other indexes (i corresponds to the summation by 3i other indexes by /3k). The structure of (21) combined with (18) does not guarantee symmetry between indexes jpqr. That is why the commas between i, j, p, q and r are introduced.

The asymptotic te hnique gives

= Ab - (4EnEaY'll,b,

V = (2Efl)_1CflflAbC - (4EnEaY

'z,b,c,d = cTh7i4nabcd - (4$nEaY1fl,bCd

(2En)1VikH.cd

-

(2En)_iVcH/,d,

Va,b,cdf = 1E_1C2A(4)

_{(4EnEaY'llbcdf}

-

(2En)'V'flci-(2$n)1Vcfl,

(2EY' Va,b,c,dfnk,f (23)

Tensors A are zero when n exceeds the sum of other indexes (for example, ACdf 0, when

m > a+b+c+d+f).

Time derivative of Rc is

= (2Ek) /3k

+ Vi3i3'' + Vp,qfli/3Pfl + Vq,r/33flPfl9/3T)+

9

ip(2) ,' \41J2,kfl(1)

where

=

+ Vj); c.,q = (Vp,q + V;q± T'p,j),

=

(h'p,q,r + Vj,q,r + V,j;r ±

(25)

The asymptotic solution of Stokes-Zhukovsky potential.

The solution of (4) in

time-varying domain Q(t) is required to cá[u1te the integrals 1w, &t and inertia tensor J2. This

problem will be solved asymptotically in accordance with {Narimanov (1957)]. Since the angular velocity is assumed small, he proposed to keep the terms depending on angular velocity in modal equations ordered by w,

w3f3 and w$. It is also preferable to keep inertia term J2, at

least, linear in fl. Since Narimanov assumed a third order theory, these terms agree with general

assumption. Additional terms should be added for fifth order approximation. However this is not possible due to mathematical limitations of asymptotic procedure. This *ill be discussed

below. The following asymptOtics assumes that w =

Asymptotic expansion starts from zero order approximation in mean fluid domain, i e

oci

-in Qo; =-x (z=-h,z=0); z (x=

_9z

Ox

The exact analytical solution of (26) is given by Fourier series

xz - 2af2(x)ij(z);

_{a, = (.)3x2; F(z)}

₍₂₇₎

where

49)

is the primary tensor of X-tensor set (73).

f0 is an infinite differentiable function in domain Qo. Due to behavior of a, for large i the

series (27) allows only first and second derivatives on mean free surface E0. The third derivative can be co idered as a generalized function. Simple analysis shows that seond Order approxima-tion needs the third derivatives of o, while third approximation requires the fothth derivatives. The general theory of boundary problems (see, .[Marti (1986)]) shows that the Neumann problem of third order approximation is incorrect, while the second order approximation problem has a solution in usual sense.

We pose an asymptotic solution of (4) as

=

o +

i +

2;

k(x, z, t) = x(t)f(±)G(z); G(z)

coh(,(z ± h

k = 1,2, (28)

cosh(ich)

±3(V ± 2V3 + 34jp,qI3P/3 + 4,qrI3P/9h/3D),

(24)

1

where x

(t), k

1, 2 are linear and quadratic in (t) respectively and index i implies sum

mation from 1 to infinity.

The expression (28) satisfies Laplace equation and a11 the boundary conditions except Neu-mann boundary- condition on free surface; This condition in linear approximation is

02Q0

1+-

f.

c9Z2 z=O ôx z=O

By setting (28) into (29) we get

This means

=

o(2fik(t)fiP(t)fi(x)G(z)

where (0) 2

-4Eq (29)

l,i)_1 (icA2

2kA) +

(icpA,

- icA))

(32) 11 where (1)

-01 =

Oi3k(t)ffih(x)GIL(Z), ,c,a2T2(,cjA

-T = tanh(,ci).

(30) 2E,

12 satisfies the following bouxiday condition on E0

= -

z=0 2

+

(a2c0

z=O

- 1)

oc1 1. z=O (31)

az z=0

ôz

ff +

-_ =ofx

We can then write

= co +

(oi3i(t)

± O2 $uc(t)I37(t)) f(x)G'(z)

(33)

All the own summation by i, k and p in (30) and (33) are finite due to the mentioned properties of the tensors A.

The terms depending on l

and 1wt When sustituting (9) into (6), we find formally out

that the cubic approximation of 1 is needed to get the terms with wBi,83 in square brackets of (6). The following consequence of integral formulas shows that (33) contains all the required terms. We can write

ai ai d

I

/p

fQfQfQ5fifQ

[L1/2 a2c 1

fdx + f

_{dQ f amatdQ] ±}

r a2c d c ac 1-1/2

8L=1

+

JQ aatdQ -

j

dQ =

.L1/2 1/2 1/2 1

± w L112 -fdx- L1,2

1ftdx, (34)

where we used that o does not depend on I3, and the following differential relations

oc a2c a2c 82c

=

813,hUt =

The last three integrals of (34) can be expanded in power series. This give

p

[()

±

+ ,8cfiL2)

) ± w(jL

+

(35) where

-4x°)l(,c,)3T,, L',j =.O

-

(36)

=

+ EmOkA

-

(37) /-)(1) L

-

2,(2)

n(2)

'-'i,u' k,p,p -

-

±Em(0)kA

-

OAj).

(38)

Tensors X are defined by (74). The sums by m in (36)-(38) are finite due to reasons mentioned above.

Inertia.

Inertia tensor J (scalar J2 for 2D flows) should be expanded by /3 up to quadratic

terms, i.e.

j2 = p1

(j0

+

/31CJ

₊

(39)

Direct asymptotic expansion gives

j(9)

=

(h2

-

0.2512)

-

X°))2[h,cj

_{- 4T],}

₍₄₀₎

=

('ck)_2X)

(41)

i= 1

(-2x(0)[ix

_-

-

OEi1ir_2X

-

₍₄₂₎

where Y-tensors are calculated by (75). The summation symbols are introduced to underthie

that the suTms by i are infinite.

The inertia term in (6) can be expressed as

Modal equations.

Now we can substitute (16), (20)., (), (5), (35), (39) and (43) into (6). We get correctly to the fifth order in /3 and third order for terms containing angular position /

and angular velocity

i that

fla(5 _{+ d/3b + d18b,3c + d,c,d/3bI3c/34}

_{dC,d,f3d/31) ±}

+ t,j3c + t,c,df3cfid ±ta,b,c,d,f/3 13 13f)

₊

+P(iuo + wvo - gsin&) ±

Qfi(i'o -

WvO - g(1 - cosi,b)) -

w2Q1(J') +

±

+

13P4)

+ wQM(/3L

+

PL4)

= 0, 1, (44)

where

Egg, /2 P2j_i

I)2'

P2i =0,

=

2E,, (45)

( is the natural frequency of /2th mode),

Further

d

=

2E(A

+ Vu),

(46)

d'

=

2E,((2Eay' CaaA2bL + EnA

TIj +

(47)

=

2E,h( CaaAd, +

+ EnAV,c + z,b,c,d)'

(48)

,bAd,f= 2E, (

GaE;'A1 .±

cflflEflAfV

+

+EflA'jI',d + "a,b,c,d,f)'

(49)

t

=

2E(V, + (8EaEb)_h11,h),

(50)

=

2E(2Vff + VEA + (

a$bY'fljc.+ (2Ea)1Vllj.),

(51)

ta,,d = 2E(3d + 2CEA +

+

+llE"'

+ (2Eb)'vdrI

±

(52)

ta,l,c,d,f

2Ep(4V,Cdf + 3

dEnA

+

±

+ )1ab,,zcdf (EaEb)1 + HE; V7 + fl,E;'

+

V' Vb+

+fl,p (2Ea)

'

₊

(53)

where tensor summations in (46) (53) by ii and k aie finite.

2.3

Hydrodynamic force and moment on the tank

The formulas below use cooidinates of the fluid mass center in a tatik with infinite tank roof

height, i.e.

XC = + (_1)z±1), _zc

+

(54)

Hydiodynamic. force.

Our force calculations are based on fOrrthila derived by [Lukovsky (1990)]

F mg - m[i0 + w x v0. + w x (w X rC) ±

_{x rc +2w x j + C]}

(55)

(rn is the fluid mass, rc is the radiusvector of mass center in mobile coordinate system). Here mg is the fluid weight. The tefms in square brackets mean: io is the acceleration of the origin 0., w x rc is the tangential acceleration, w x (w x rc) is the centripetal acceleration,

2w x tC is COriolis acceleration, C is the relative acceleration.

The force (Fr, 0, F) in two-dimensional flow can then be titten as

F=rngi-

VOz + WVØ 0)2_{XC ±C)zc ± 2C ± XC),} (56)

= mg3 - m(i'o wvo - W2ZC

_{wxc - 2wc. + ic),}

(57)

where te coordinates ±c, ZC are defined by (54) and the projectionsgi, g of the gravity vector on the axes of Oxz are

gi =.gsin&; 3

gcos&.

(58)

Hydrodyna.mic moment. The hyd±odynamic moment relative to axis Qy can also be

calcu-lated by formula derived by {Lulcovsky (1990)]

Mo=mr.Cx(gwxvovQ)J'.w_J.w_wx(J'.w)-1+lt_wx(1_1t). (59)

For calculation of the hydrodynamic moment relative to other axis through point P we can use the formula

MprpoxF+Mo,

(60)

where hydrodynamic force is given by (55).

The formula (59) does not have a simple structure die to the term l

- i

By using the derivations above we find

where L2 are calculated by (36)-(37). Then pitch moment about 0 in two-dimensional

flows

M = m(xc(g3

kfiPJ(2))

pl(/J'+

+2J13/3)w - pt (mO) +

IPLiT, + . + _{± 2L}

_{rnImI8P), (62)}

where j(°), j(1), j(2) _{are given by (40)-(42) and}

=

3

Surge

and

pitch excited resonant sloshing

3.1

Modal system for resonant sloshing

FOr resonant sloshing with small forced excitation amplitude the rn3gaitlides Of w, , are

associated"with the small parameter . Terms o(c) will be omitted. Modal system (44) takes

then the following form

d /3af3b

dfiafibfic

d' dfiIfi

_:!:

_d,ffififlhfi

ab=1 a,b,c=1 a,b,c,d=1 a,b,c,d,f= 1

+

+

_{ta,b,C,d/3/9/313 +}

i:

t

Cd/fl 3 /3 /3 +

ab=1 a,b,c=1 a,b,c,d=1 _a,b,c,d,f=1

±of3 +P)0 - gI') +

0, u (63)

The terms associated with inertia tensor J2 have higher older in this apprOxiniation. How-ever, inertia terms are present in form'tiias for lateral force and pitch moment. These formulas are correctly to 0(c)

F

mg'çb - m(i'o +çiiz°

+

_ic);

F = -

(ü°Z +

_ic),

(64)

M =

rn(xcg

z('o

- g';)) -

p1thJ°-_p!(mLO

+ mfiPL(l)

k

+ 4)/3m$P + 2Lj,mfi8&),

(65)

where z = h/2, is th vertical coordinate of center of gravity of "frozen" fluid mass.

Modal system (63) implies all iodes to have the same order. Many coefficients d and t of this system are zero. However, the direct ppllcability Of this uniformly structured system

equations. The fifth-dimensional system has for instance more than 1000 non-zero coefficients. Another difficulty is due to high harmonics in primary modes caused by response of high modes. For instance a term proportional to _{will appear iii the first equation of (63). th}

ad

_{/3 have} respective1r _{añd 5cr ä.s nihin harmonics. This gives an oscifiatory term with frequency 21cr in th} and, leads to a very stiff noplinear system of ordinary differential equations. The numerical time integration is then inefficient and easily unstable. Only a few seconds of real timewas possible

to calcnlate with our numerical integrator. It is believed that surface tension and damping _can

be important for higher modes. Since energy dissipation is not present in our theoretical model, additional. coefficients associated with damping of different modes have to be incorporated. The effect of damping due to viscosity and surface tension should be further investigated; The studies by [Ketilegan (1959)], [Miles (1990)], [Cocciaro et al (1991)]) and [Miles & Henderson] are useful in this context.

3.2

tetiming procedure by [Faltirisen et al (2000)]

(63) can be used in a detnning procedure by introducing appropriate asymptotic relations be-tween modal functions. One way to tune the system is to add to all the summations of (63) an analytiëal condition cQupling the indexes of summations. If the sum of indexes does not exceed 3 (order of the theory), we arrive at the modal System by [Faltinsen et al (20O0)] describing the response of the lowest primary mode (or transients)

/3(I + Di'(i, 2)fi2 + D2'(i,l,

_')8?)+ D1'(2, 1)fi2fl1+

+T0'(1, 1,

+T0' (2, 1)/12

+

+ Pi(iox - gçL') + = 0,

D12(1, 1)ë1fi1 + /32 +T02(1, 1)fiifii ±

o$2 = 0.

(66)

The coefficients of (66) (see, Appendi II) coincide with the ones derived by [Faltinsen et al (2000)].

This is an explicit verification of these theories. The third equation of this system contains fi linearly and nonlinear terms in /3i and 82. It means, that thuid mode is considered as

driv-n by primary adriv-nd secodriv-nd.ry modal fudriv-nctiodriv-ns. The modal system of fifth order theory by

3.3

Detuning procedure for secondary resonance

We consider the case when the harmonic mo- is still dominating in each mth thode, but

asymptotic relations between modes cn be changed with little changes of excitation parameters. This happens in beating waves when frequency responsesof modal functions are time dependent. The fluctuation of dominating response /3i leads immediately to changes of the response in higher

modes. These. responses depend on initial conditions. If they are small, it does not make the

dominating motion unstable. However, the higher modes can have large resonant response that makes the various asymptotic and averaging models iirnpplicable. The large. response in higher modes is always caused by their resonant excitation due to interaction with other modes.

Our system has general multidirtiensional structure which can be adapted to multimodal resonant excitation. It requires preliminary analysis to start the simulation. The analysis is connected with an 'a priori' estimate of possible inter-modal resonances due to transients. In simplified form it can be treated as the following adaptive procedure. We consider a series of

natural frequencies o, o-2, 0.3,... and a set of possible frequencies o, 2o-, 3cm,... caused by main excitation frequency a.

In a rectangular tank excited by surge and pitch we expect direct

resonance for odd modes. Secondary resonance can occur for all modes. [Faltinsen et al (2000)] described the secondary resonance of the second mode due to direct resonance of lowest mode.. This happens as

T o I2tanh(irh/l)

TI

=

_V tanh(2irh/1)

= (2, h).

Here T is the excitation period and Ti is the highest natural period for the fluid motions.

Similarly, for secondary resonance of mth mode

T

=

o-1 -

-z(m,h) =

If main frequency a is close to one of the natural frequencies of odd modes and away from other modes, this mode is primary excited. In order tO add the most dangerous secondary

resonance we should find the mode (even or odd) of which the hatural frequency is close to 2a.

Let us assume as an example that these two modes correspond to and /3g. The two first

nonlinear equations of general system give a kernel Qf this interaction. Any other mode /3m can be considered as having lower order or driven (they are linear in /3m and nonlinear in dominating modes).

The secondary resonance causes these modes to have similar order, i.e. /32 th. This means,

17

rn tanh(irh/1)

tanh(mirh/l)

(67)

that additional nonlinear terms in and /32 should be included. Two of the equations (66) take then the following asymptotic form

th(1 + D1'(l, 2)132 + D2'(l, 1, 1)13? + D2'(l, 2, 2)/3) +i92(D11(2,1),8 + D21(2,

2, 1)i38)±

+TO'(1,1,1)131131$1 ±T1'(2,2,i)132132131 ±/31132(TO'(2,1) +T1'(2,1,2)/32)+

±(Tfli + Pi()ô - g1')

+bQ14°

=0,

1(D12(1, 1)/3 + D22(1, 2,1)131/92) + /32(1 + D2(2, 1,1)13? + D22(2., 2, )/3)+

+/11è1(TO2(1, 1) + T12(1, 1,2)) +T12(2, 2,2)132/92/92 + 12(2, 1,i)/fi + 0. (69)

This system is of third order in /3i and /32. It contains all the necessary terms of

Faltinsen-Moiseev theory and a theory considering 131

= O(N).

In additiOn, the third order

terms similar to

fi,

/3/3 are i cluded to describe a LsWitth between these asymptotics duEing transientswhen fl - /32 O(h/3) in framework of

a third order theory. The responses of

third and fourth modes are not included in the presented equ3tions. They can be considered

as dtiveii and follow from (63) when no corresponding high modes secOndary resonance occurs.

Four euations for and /6 are nonlinear in /3

and /32 and linear in /3,/3,/3 and

/36

respectively.

When. the three modes131 13 and /3 have the same order, the corresponding nonlinear system

of differential equation can be derived in a similar way. It couples three modal functions up to

terms of third order. A similar procedure can ä.lso be thade for primary excited. nonlowest

natural mode.

3.4

Adaptive detuning procedure

Modal functions in general case can be associated with an order _fi Q(P1K), where

K is

the order of the theory and p <K. The analytical conditions in summations should couple pj

instead of indexes i. In order to keep only terms of 0(E) in (63) we use the condition

p ., K.

For example, E,b,,d=1 should be accompanied by condition. Pa + Pb + Pc + Pd

K. These

conditions allow us to avoid analytical manipulations to derive particular cases of uniformly

valid system detuned for different sets of p2.

b 50

1900

1730

Figure 4:

R.ectangular tank used bi

[Faltinsen et al (2000)]. All numbers in

[mm] FS3 means position of wave probe

19 1380 Cl) 510 595 741

4

_{Comparison between theory and experiments}

Different series of experiments on resonant surge and pitch excited nonlinear sloshing are used for validation. [Olsen & Johnsen (1975)] and [Olsen (1970)] presented results for a tank with rectan-gular crosssectional shape. (See Fig. 2). [Abramson et al] (see Fig. 3) aid [Mikeis et al (1984)]

(see Fig. 5) presented results for prismatic tanks. The experiments by [Faltinsen et al (2000)1

for surge excited rectangular tank (See FIg. 4) will be used to analyze the applicability of modal system to small fluid depth sloshing.

Even if our theory assumes a rectangular tank, it will be appijed to the prismatic tanks. The Figure 5: Prismatic tank used by [Mikelis et al (1984)]. All numbers in

[m-m]. means the position of rotation axis. Figure 2: Rectangular tank used

by [Olsen & Johnsen (1975)] and Figure 3: PriSmatic tank

used by

[Olsen (1970)].

All numbers in [mm]

[kbamson et al] All numbers in [mm].

Figure 6: Dimensions of the lower corner in a prismatic tank.

error in doing so can be assessed by exarpining the eignvalues. A rough estimate of eigenyalues & relative to eigenvahies o of virtual rectangular tank with the same fluid depth and maximum tank breadth as in prismatic tank can be made by using Rayleigh-Kelvin variational formula

&

fE0 çodS

(see, [Feschenko (1969)]). Here o is mean volume of fluid in prismatic taflk and çoj are natural

modes in rectangular tank. It gives

-

_{- 2}

in (sm2 (iin6i/i) + smh2(zinö2)'\ 6152

- taiih(iirh/i)

cosb?(iirh/l)

₎

where 5 and 52 are respectively breadth and height of the corners (see Fig. 6). If this is applied

to the lowest mode of the tank in Fig. 3 when h/i = 0.4, it gives (o -&?)/o? 0.0081.... When

applied to the tank in Fig. 5 with h/i = 0.246, it gives (o -&?)/c 0.03. The error decreases with increasing mode number.

4.1

Surge excited resOnant sloshing

Surge excited resonant sloshing in a rectangular tank with mean fluid depth close to critical value h/I - 0.3374 was studied experimentally by [Olsen Jolunsen (1975)]. Steady-State

re-suits were compared with the third order Moiseev's theory by [Faltinsen (1974)]. This theory predicts infinite response as T

Ti at the critical depth. Here Ti is the highest natural

pe-nod of the fluid motions and T is forced excitation period. If the higher order Moise-fflce theory by [Waterhouse (1994)] is used, the response will be finite. But predicted amplitudes

are much larger and unrealistic relative to the experiments by [Olsen & Johnsen (1975)] for the

fluid depth/tank breadth ratio h/i = 0.35 presented in Fig.. 7.

Modal system (66) gives very large response for the second and third thodes in this case.

0.8 0.7 C 0.6 0.5 C 0.4 0.3 0.2 0.1 0 Third ddrtiby_FáItinen' Calculations' 'Model test' 'Tank,,top' Model I 1:/Ti Model II Môdël ifi

Figire 7: Wave elevation near the wall vs "period-first natural period ratio". Rectangular tank

m Fig 2 with h/I

=

0 35, H/I

=

0 025 H is the surge excitation amplitude

also tested.. This gives also large. respotse in high modes for excitation parameters of the experiments in Fig. 7. Since the response of higher modes becomes so laige, it means that secondary resonance can be expected as T/Tl We therefore decided to use our system

with detuning parameters based on occtrtence of secoiidary resonance.

Our calculations are based on long time series. A very smal linear damping term cajfij is incorporated into each ith mOdal equation, where c was varied from 0.005 to iO'-. The

max-imum value 0.005 is consistent with the damping theory by [Keulegan (1959)] when applied to prirna.ry mode. The time integration started normally from zero i.n,itial conditions.. Maximum damping coefficient was used for first time seriCs. When reaching numerically a periodic solu-tion, the damping coefficient was decreased b a factor of 10. The time integration procedure

was continued with initial conditions obtained from previous: simulation. The procedure, was. continued to reach periodic solution With mInimum dawping coefficient iO. The mean time to reach steadystate solution is up to 20mm of real time and depends on. the excitation ampli-tude and &equency. Adams-Bashforth-MOuiton Predictor-Corrector integrator of varied order (from one to twelve) was used. The simulations Were made by a Pentium-Il 266 computer. The simulation time depends on the chosen approximate modal model and excitation parameters. It varied between 1/3 to .1/100 of real time.

The procedure can be illustrated by using the results in Fig. 7. The first stage of the analysis was to locate four possible resonances for T/Ti. between 0.45 and 1.65. The primary resonances

21

of the first and third mode occur at respectively T/T1 = 1 and T/T1 = 0.55 The secondary resonance of the second mode is predicted at T/T1 = 1.28. The secondary resonance of the third mode is at T/T1 = 1.55.

Three models applicable fof different èriôd domains were used. They are indicated as

Model I, _{II and Ill in Fig. 7. It was controlled that the models overlap with each other in a small}

domain. Model I was used for 0.5 T/T1 0.65. The expected resonances are due to primary

excitation of the third and first mOde. They have the same main frequency response

o. No

secondary resonance is expected. This causes the relations _$s = Q(h/3). _{This means that}

the secondary modes have the main harmonic 2a. Such modes are _{$2 $6} Q(2/3) _Other modes (up to 9th) are considered as driven and having 0(e).

The modal system based on (69) (Model II) was used for 0.6

T/T1

1.25. The modes

$s,

$,

$s,$6 were included as driven. If response is not too large, the modal System (69) gives the same results as shown by third order response by [Faltinsen (1974)] orby (66). The predicted values in Fig. 7 belong to different btaiiches of the steadystate period domain solution. The con-cept of branches of the solutions was for instance extensively discussed by [Faltinsen et al (2000)1.

There exist in their solution an upper and lower branch. The lower branch is divided into a stable and unstable subbranch with a turning point between theth. A jump in the solution will happen at an excitation period corresponding to the turning point. A jump from one branch to another is expected in the neighborhood of T/T1 = 1. If we start the solution from zero initial conditions, the position of the jump period is difficult to predict. The reason is that transients can cause the solution to jump from one branch to another. We therefore selected a different

strategy close to expected jump period. Let us consider first the case when T is lower than the jump period. When the predicted wave elevation was larger than 0.5m, we increased the period

with small steps and used the steadystate solution at previous period as initial conditions. The

fifth order terms in _fi were added for large amplitude response. This Stabilised the calculations.

When starting from zeroinitial conditions in neighborhood of turning point of the lower

branch, i.e. fOr T/T1 > 1.11, it is also easy to obtain wrong result due to large amplitude

transients.. These transients will iii reality cause a series of heavy roof inipacts which damp the

system. However, the presented iodel does not account for this. We used the same stepwise

changing of the period on the lower branch described for the upper branch. But we now started

0.8 0:7 0.8 0.5 0.4 0.3 02 0.1 0 'Thk-.ordec- b Faltinsen -ank_ top

Figure 8: Wave elevation near the wall and dimensionless lateral force 1000F / (pgi2 b) vs

"period-first natural period ratio". Retgiila- tank in Fig. 2 with h/i = 0.35, H/i

0.05. H is the surge excitation amplitude.

0.8 0.7 E 0.6 0.5 0.4 0.3 02 0.1 0.6 0.8 hs, Faflinsen -Thir&.order_theorY_,djctilaIjoh, 'MeLtasr 1 1.2 T!T1 Taiik_tóp 0 1.4 .1.6 0.6 08 1 1.2 in 23 a, C 0 250 200 150 100 50, rd_order_ eo!y_y_Falt!nsen -Calculations 'Model_test X 1.4 1.6 0.6 0.8 1 12 1.4 1.6 0.6 0.8 12 1.4 1.6 TIn T/T1

Figure 9: Wave elevation near the wall and dimensionless lateral force 1000F / (pgl2 b) rs

"period-first natural period ratio". Rectangular tank in Fig.

2 with h/i = 0.35, H/i = 0.1.

H is the surge excitation amplitude.

decreased the excitation period

When T/T1 > 1.28, the third mode response was assumed to have the same order as and /32 (Model Ill). The reason is the iflfluence of the secondary resonance of third mode at T/T1 = 1.55. Model III was used for 1.28 <T/Tl <1.65.

The calculations accounting for secondary resonances are in good agreement with experi-ments. A jump at. T/T1 1.3 is predicted i additiofl to the jump period T/T1 = 1.11. Since the excitation amplitude is small in the case Of Fig. 7, the jump in the solutions at secondary

resonance is not large. Figs. 8 and 9 present comparisons between experiments and calculations

0.7 0.6 E 0.5 C 0.4 0 0.3 a, 0.2 0.1 0

Model II Model ill

100 60 40 E 80 20 0 06 'Calculations' 'ModoLtestwater)'. X 'Modek_tast_(reginol_oil)' lIE 08 x 0.8 1 1.2 1.4 16 Till

Figure 10: Wave elevation near the wall and dimensionless lateral force l000F/(pgi2b) vs "period-first natural period ratio". Prismatic tank in Fig.

3 with h/i = 0.4, H/i = 0.01.

H is the surge excitation amplitude.

corner impact roof impact

b

Calcujaoons'

'Model_test_(watery X

lodel test (reOinoI oil)' IC

corner impact roof impact

1.4 1.6

Figure 11: Wave elevation near the wall and dimensionless lateral force 1000F / (pgl2b) vs "period-first natural period ratio". Prismatic tank in Fig.

3 with h/i = 0.4, H/i = 0.1. H

is the surge excitation amplitude. -'ta1culations'

-'Tank_top'

...

'Upper_come? f ...

Model II Model Ill

1.6 180 1.4 160 Upper_corner' ; -140 1.2 C 120 a, C 0.8 a, 0.6 V 100 80 60 a, 0.4 ₄₀ 0.2 20 0 0 08 12 1.4 1.6 06 06 Till 08 06 1.4 1.6

Both wave elevation at the tank wall and lateral fluid force are exaniied. The symbol b used in expression for dimensionless force means the tank length. The simulation strategy in using Models I, II and III did not change.

The calculated responses for small excitation amplitude presented in Fig. 7 differ from third

order Moiseevliké theories in period domain 1.06 < T/Ti < 1.11. But the effective domain

of secondary resonance increases with increasing excitation amplitude and cover the range from

0.6 to L28 in Fig. 9. The jump in wave elevation respOnse also increases at T/T1 = 1.3. At

the same time the effective domain of the secondary resonance by the third mode also increases.

We had to switch to Model Ill beginning from T/T1 = 1.3 in the case of Fig. 9.

Since only odd modes contribute to the lateral force, the secondary resonance of the. second mode is generally more important for wave elevation than for lateral force. However, for suffi-ciently large excitation arnplitude the effect of secondary resonance becomes important also for lateral force. The reason is inter-mod.i interaction between nearest od modes /3i and /33 which due to secondary resonance of the third mode interact nonlinearly with each other and have the same order.

Our theory was also compared with experimental results for steadystate sloshing in the prismatic tank shown in Fig. 3. This tank has a chamfered tank roof. Figs. 10 and 11 with wave elevation response contain therefore in addition two horizontal lines illustrating respec-tively the heights above mean free surface of lower point of upper corner and horizOntal tank roof. Fig. 10 presents the calculated a.n4 measured values of lateral force for small excitation amplitude. Experiments for two different fluids are presented. This illustrates that viscosity is unimportant. The period domains for described models are indicated. Calculated wave el-evation and lateral force are shown in parkilel pictures to estirnate ai effect of corner or roof impact on disagreement between theory and experiments. Impact on tank roof can give impor-tant damping of fluid motions (see, [Faltinsen & RognebaLkke (1999)]). The details of the tank

roof matters. The influence of impact on a chamfered tank roof is less severe then impact on a

hotizontal rooL [Rognebakke & Faltinsen (2000)] studied also the case in Fig. 10 by using the model by [Faltinsen et al (2000)] as ambient flow and including roof impact. The response was larger than our predicted values. For instance the maximum predicted dimensionless horizontal force was 120. If tank roof impact is not included in model by [Faltinsen et al (2000)], the

max-imum predicted horizontal force was l4 times larger. Our maxmax-imum predicted value is 90 and

0.8 0.7 0.6 0.5 0.4 0.3 02 0.1 0 _Ji 'CalcuiLdons' MoeLtesr x Tank_top'

(.X.

Model I Model II Model III

Figure 12: Wave eleyation near the wail vs "period-first natural period ratio". Rectangular

tank in Fig. 2 with h/i O35, ''o O.lrad. 'o is pitch excitalion amplitude.

does not account for roof impact. But the roof effect with using new model as ambient flow will be less then the model by [Faltinsen et al (2000)]. The reason is simply that new model predicts

lower wave amplitudes. Fig., 11 illustrates that the disagreement between theot and

experi-ments due to roof impact can increase drastically for large excitation amplitude. The forced surge amplitude is now H = 0.11 while H was 0.011 in. the previous example. The results by

[Faltinsen (1974)j gave then uurea tic predictions. The predicted force amplitudes in resont

conditions could for instance be 13 times larger than experiments. Our predictions are far more reasonable. The experiments as well as our theory predict fluid to hit horizontal tank roof in the

period range 0.85 < T/T1 < 1.25. Disagreement between theory and experiment occurs when

the horizontal fluid motion is in opposite direction of forced tank motion, i.e. for T/T1 < 1.14. Fu±ther studies including tank roof impact are needed to eplain the difference between theory and experiments.

4.2

Pitch excited resonant sloshing

The experiments by [Olsen & Johnsen (1975)] and [Olsen (1970)] were used to validate the.

the-ory with pitch rèson nt excitation of the tank. The experiments presented in Figs. 12, 13 and 14 were made with the. rectangular tank shown in Fig. 2. Fig 12 gives computed and mea-sured maximum wave amplitudes of steady-state sloshing for angular amplitude o 0. lrad.

Since the excitation amplitude is sufficiently small, the wave elevation never hits the roof in

26

1.2 0.9

steady-state motion. However, the calculations show that "beating" waves during transients

have sometimes amplitude up to 1.6 times the breadth. Such transientscan cause heavy roof

impact. The computational strategy did not change. The domains of Models I, II and III are

indicated for all the considered examples.

The theory is in good agreement with experiments in Fig. 12. The secondary resonance of the second mode is also well predicted. A jump between two branches _{occurs theoretically}

at T/T1 = 1.111.

Since experiments thow a value betseen these branches (point J1), it is believed that this is due to improper modeling of damping. This will be mOre evident for a larger excitation amplitude. One could of course also state by looking at this as an isolated case

that the error in predictions at the jump period is small and not larger than at other periods. The minima at T/T1 = 0.74 is not consistent with minima shown in Fig. 7 for the same fluid depth. The reason is the difference in the type of excitation. The modal approach gives

the inhomogeneous periodic terms PmH Cr2 COS at in mth modal equation for surge excitation. Corresponding periodic term is Pm'çbo(zo

-

tanh(!h) ±

.)y2 cos crt for pitch excitation. There exiSt values of: zo and o that can cancel this periodic term in modal equation for the first

mode. Then this mode is not excited and we should consider the sloshing due to inter-modal

interaction with other modes.

Fig. 13 presents calculated and measured values of steady-state wave elevation, lateral

force and pitch moment. This is for h/i = 0.5 which is far away from the critical depth.

Faltinsen (1974)] compared also steady-state wave elevation with experimental recordings fOr

this case and showed good predictions for selected periods. A priori we may have expected the largest response close to critical depth, i.e. in Fig. 12. But theoretical and experimental

amplitude response are larger in Fig. 13 The reason is the increase of angular excitation due to increased height between rotating axis and mean free surface; This leads o larger values of the pitch excitation term Pmu/J0cr2(zo

-

tanh('h) +

) cos at in mth modal equation of (63).

The theory agrees well with experiments. But the calculations do not predict the

experimen-tal response found at the jump period T/T1 = 1.111 (point J1). This is believed to be due to

damping,. The calculations describe the effect of secondary resonance of the second mode on the lateral force. This influence is very small in Fig. 12. When the excitation amplitude, is increased

to 0.2rad, this secondary resonance gives an additional jump at T/T1 = 1.45 (see, Fig. 13). We

note that the experimental values Ji and J2 at the jump period are not well predicted. The

0.8

I

Model H 'Calcutations' 'ModeL test' x .J1 ---f---i .... ___ .... + X - - ---12 1.4 1:6 T/T1 0.055 0.05 0.045 0.04 0.035 0.03 S 0.025 'I, 0.02 E ' 0.015 0.01 0.005 0 Model III 'Cations' 'MoaeL test X

-r-H'

---H'.

0.8 1 1.2 1.4 1.6' T/ll

Figure 13: Wave elevation near 'the wall, dimensionless lateral force l000F/(pgi2b) and

dimen-sioriless pitch moment M/(pgl3b) with respet to rotation axj

"ëtiOdfirst natural period

ratio". Rectangular tank in Fig. 2 with h/i = 0.5, o = 0.lrad. '&o is pitch excitation amplitude. The ft s'urements of lateral' force and pitch moment were made' by [Olsen (1970)].

28 0.8 0.7 0.6 0 Ji 0.4 0.3 02 0.1 x I.. + L 5c 0.8 1 1.2 Tin 200 180 6 120 - 100 .9 E. 40 20 +.

80

. .--- * 0 'Calcu1ans' 'ModeL_test' X Tank_top' -: 1.4 1.6

0.03 0.025 0 0.015 0 C 0.01 0.005 0.5 29 1.5 iations' _iesr x x

Figure 15: Dimensionless pitch moment M/(pgl3b) with respect to rotation axis vs dimensionless

forced excitation frequency ol/g, where has dimension [rad/s]. Prismatic tank in Fig. 5 with

h/i = 0.182, ''o = 0.lrad. ''o is pitch excitation amplitude. Models I, II and III were respectively

used in domains 1.75

<a/7

< 2.5, 1.19 < < 1.75 and 0.6

<a./7 <

1.19. 1.2 E 0.8 C 5 0.6 0.4 02

Model II Model Ill

350 300 200 150---100 50 Calculations' ii 'Calculations' ModeItost X Tank_top' 1 _..*' 0.8 1 12 1.4 1.6 08 1 1.2 1.4 1.6 im un

Figure 14: Wave elevation near the wall and dLi.mensionless lateral force lO0OF/(pgi2b) with

respect to rotation axis vs "periodfirst natural period ratio".

Rectangular tanlc with 1 =

1.Om, h/i 0.5, 'o = 0.2rad. 'o is pitch excitation amplitude. The measurements of lateral

force were made by [Olsen (1970)].

reason was previously discussed.

We tried to validate the theory also for a pitch excited prismatic tank. An appropriate

example was found in [Mikelis et al (1984)]. This prismatic tank is presented in Fig. 5. Fig.

15 gives calculated and measured hydrodyamic pitch rnoxents. Model I, II and III were used

in different forced excitation frequency domains as indicated in the figure caption. They are in good agreement with experiments except in the frequncy range where calculated wave elevation

indicates roof impact. The response is presented as function of instead of T/T1. This is

convenient when studying asymptotic moment values for a 0. This value is caused by static moment on inclined tank. Our computations for a 0 give the dimensionless value 0.005

against experimental value 0.00518. (The calculations for small a were done with a model of

seven nonlinearly coupled modes based on (63).) The reason to this small difference is simply our modal presentation of free surface. This means, that static inclined tanks in our approximation

has nonfiat free surface.

4.3

Sloshing in tank

with small fluid depth

We applied adaptive system for different values of fluid depth. Models I, II and III gave well predicted values of wave responses in the period domain of the primary mode for flurid depth

h/i

0.24. When h/i

< 0.24, these models could not cover this period domain completely.

That is why, we do not preseht the comparison with experiments by [Olsen & Johnsen (1975)] for pitch excited resonant sloshing with h/i

=

0.2. Even if this fluid depth is not sufficiently small to use shallow water theory, we speculatively tested different possible shallow water asymptotics including asymptotics given by [Ockendon, Ockendon & Johnson (1986)], i.e. f3 Q(h/2).

Test calculations were made with six modes in (63). In addition, the asymptotics fi

was also tested with six modes. In both cases either calculated responses in period domain near T/T1

=

1 were not consistent with experiments or modal system (66) gave only a few

seconds of real time integration befOre numerical problems occurred due to complex inter-modal

interaction. Te situation, seems to be explained by weak convergence of modal expression of

free surface Smail clange of excitation neds a new ordering between modes and gives differeit

dominating harmonics in each mode. One way to explain this is the change of wave patterns

relative to finite fluid depth case with h/i 0.24. When the fluid depth is small, wave patterns associated with "traveling" -waves and "run-up" phenomena occur. Many modes are required to

0,7 0.6 9.5 0.4 0.3 0.2 0.1 0 -0.1 -'-..---0.2 -0.3 0 t (s)

Figure 16: Measured surface elevation near the wail. Rectangular tank in Fig. 4 with h/i 0.173, H/i = 0.029,T/T.1 = 0.8508. H is th surge excitation amplitude.

approximate such wave profiles. Damping due to run-up may also matter.

An example below illustrates main difficulties with modeling sloshing for small fluid depth.

It is based on resonant sloshing experiments in the rectangular tank hown in Fig. 5. The free surface elevation recording at wave probe FS3 is presented in Fig. 16. The modal system (66)

gives' wave elevation as shown in Fig. 17. The contribution to wave elevation from each mode

is also presented in Fig. 17. Similar resuts were presented by [Faitinseh et al (2000)]. These' results suggest that inter-modal resonances between the primary resonance at T/T1 1, the secondary resonance of the second mode at T/T1 = 1.1446, the direct resonance of the third. mode at T/T1 = 0.56 and the secondary resonance of the third mode at T/T1 = 1.32275 are

important. Since these resonance periods are very close, they should all be accpunted for. Harmonic analysis of the cOntributions gives main harmonic H1 cos cit in the primary mode

and main harmonic H2 cos 2cit in the second mode. The graph 'Mode 3' in 'Fig. 17 shows that

direct and secondary resonance give two main harmonics H3cos cit and H4cos 3cit for the third

mode'. The reason is that T/T1

0.8508 is equally close to 0.56 and L32275. Thus, three modes should be considered having the same order Q(1/3). One of the modes has two main

harmonics. This contradicts our previous assumption 'i'n finite fluid depth cases. This means that

three nonlinear modal equations give three equations with four unknown variables Hi, H2,H3

and H4 in a steadystate analysis. A way to solve this contradiction is to add a mode or4ered

as Q(2/3) that is interacting with dominant modes. This mode will be included nonlinearly

in the equations but its main harmonics will be higher order than the dominant /3i, /32,/93. We

j

'Experfrnent

5 10 15 20 25 30 35 40

07 0.0 05 S 104 03 02 1 0? 0 0i 02 .03 0 5 to '5 30 30 '(a) 30 35 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0.7 0.6 0.5 0.4 0.3 0.2 0 -0.1 -0.2 -0.3 0 ...IIIIIII1! JILl

LLIlIII liii llt!ILL1lIuiIl 11111

,1itIlIIIIII I IIIIIlllIIIIIJhJillOhlIl 11111 A!AUhIIIIIIII I IIIIIIIllItI!&!II!IIEIIIJIIIIIIII [I!ItIII!IIIlI' 40 45 07 0.0 05 0.4 03 02 0t .0? .02 .03 10 15 20 25 30 35 t (s)

r

II IijlJ Iii

1hlIIl!Il1llllI I IFiFiIII llL!fl!llIIIII liii

t!TilIIIII'IlilII IlIllIl IIl I IlIIIIII!p!lIIIIIIIIpIIIIIl

iIIlIJIIIIllIIIIILIl t1IIL'iiIJII'llIiJl

____"

_{S to is 30}

25 00 05 40 45

05)

Figure 17: Calculated, free surface elevaioi near the wail. Contributions from mode 1, 2 and

3. Rectangular tank in Fig. 4 with h/i

0.173, H/i = 0.029,T/T1 0.8508; Uis the surge

excitation amplitude. The modal system by (66) and .[Faitinsen et al (2000)].

Adaptive_model_(no_daniping) I_by_Faltinsen_(2à001 1 I 07 OS 0.0 04 OS 02 0i .0? .02 .03 ml I I 11111 T 1 UIflhIF . L I II,1InrIlI1I[lL iPi1iJIiiI1i

IFIIl ill II II III Ill

I I IllIi III

I

Figure 18: Calculated free surface elevation near the wail. Rectangular tank in Fig. 4 with h/i = 0.173, H/i = 0.029, T/T1 0.8508. H is the surge excitation amplitude. Modal system

includes f3 = Q(e"3),/32 O(h/3),/3 Q(1/3),j3 = Q(eI3). No damping.

0 10 15 20 25 30 35 t (s)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 Adaptivé_modai_systern_with_amping_by_KeuIegan...(1 959) -33

Figure 19: Calculated free surface elevation near the wall. Rectangular tank in Fig. 4 with

h/i

0.173, H/i = 0.029,T/T1 = 08508. H is the Surge excitation amplitude, Modal system

includes

= O('/),8 =

Q(1/3),fi3 =0(E/),f34

= 0(62/3). Damping is based on the theory by [Keulegan (1959)].

used for time simulation an adaptive system with fi ' /33 = O(E1/3),Ø4 Q(2/3).

The calculated wave elevation is shown in Fig. 18. It gives realistic approximation for both minimum and maci.nu wave elevation and "beating" period. However, the higher modes have "high harmonics" noise. This causes numerical problems in vety lông time imuIatiOns.

However, these high modes are believed to be highly damped due to energy dissipation. That is why, additional linear damping terms ao were incorporated in modal system. 'the introduced

damping coefficient c was not high since it was based on the theory by [Keuigan (1959)] for the primary mode. Damping improves numerically the time integration. The calculated wave

elevation is presented in Fig. 19.

A detailed study of the experiments for this fluid depth shows that wave run-up near the

vertical, wall occurs for all excitation periods in the effective domain of primary resonance.. A typical wave profile is presented in Fig.. 20. The ma.ximum length and thickness of the vertical

wall jet depends on excitation period. Many modes are needed to describe this jet. Since the

intersection angle between body si±face and fluid surface is not 900 in this case, it indicates bad

convergence of our model system and that a atematica1 singularity will occur at the body-fluid system contact point. This implies that a matching with a local solution at the contact

Figure

Ri-up phenorenon. Resonant sloshing in the rectangular tank jn Fig. 5. Surge

excitation. Experiments by [Rognebakke (1999)].

5

Conclusions

Nonlinear sloshing in a rectangular tank is analytically studied. Irrotational flow of incompress-ible fluid, infinite tank roof height and no overturning waves are assumed.

The basis of the theory is an infinite-dimensional system of onjinear ordinary differential

equations coupling generalized coordinates f3 of a modal system. This is derived from a

varia-tioal procedure b [Faltinsen et al (2000)]. By assuming the fluid response to be small relative

to fluid depth and tank breadth this modal system is asymptotically reduced to an

infinite-dimensional system of ordinary differentiai equations with polynomial. nonlinearity of fifth order in fij. It contains also third order terms coupling and time-varying functions describing rigid body motions. No ordering between /3 is assumed. The coefficients of this system are unique functions of fluid depth. However, the number of non-zero coefficients increases drastically with increasing dimension.

The surge and pitch excitation of the tank is considered. The tank is forced to oscillate with

period in inicity of the highest natural period of fluid motions. y jntroducing asymptotic

relations between /3 the derived system can be detuned to particular cases of nonlinear sloshing occurring due. to direct and secondary resonances. By secondary resonance is meant that higher

harmonics in the fluid motion cause resonant motion at other natural periods than primary

resonance periods;

The theory based on Moiseev-like inter-modal relations by [Faltinsen et al (2000)1 will be a special case. The latter asymptotic theory is. invalid when, the excitation amplitude is not very

amplitude response in both primary and higher modes. it is demonstrated that inter-modal relations depend on excitation amplitude, period and fluid depth. If each mode has only one

lowest order harmonic, the choice of thse relations are rationally motivated by locating, primary

and secondary resOnance periods. The iethod has been 'validated by corrparing with model tests. Adaptive procedures have been established for all excitation periods as long as the mean

fluid depth. is larger than 0.24 times the. tank 'breadth. Steady-=state results for wave elevation, horizontal force and pitch. moment due .to forced surge ad pitch excitation are. validated except when heavy roof impact occurs. Different stable branches of the analytical solutions re located.

When h/i < 0.24 and depth is not shallow, good agreement with experiments has been

achieved for isolated excitation petiods. An example for h/i 0.173, where the previous model by [Faltinsen et al (2000)] failed, demonstrated this. When the fluid'depth is small, many modes have the same order and each mode may have more than one main harmonic.

Acknowledgments. Authors are grateful to H.Olsen for providing the unpublished mea-sured data on lateral force and pitch moment in Figs. 13 and 14. The work of second author is supported by the Stsrong Point Center on Hydroelasticity at NTNU/SINTEF in Trondhéim,

Norway.

6

Appendix I. introduced tensors

The set of tensors A is given, by

AJ

2

ij"=0

(70)

äj

for other case

(6ij is Kronecker symbol). The following recurrence formulas have been obtained

A2

- A'

A1

±

_{In+kli' nkjp - 'In-kIjp}

+

_In+kIip'

and A(3) nkjpq A(1) 2nkj (2) A(2) A4 A3 A3

'inkjp,q

+

n±'k(jpq' nkjpqz -. rikIjpqi

±.

n+kjpqi'

A'

_{nk,i -}

-

A°

_Inku

-

A°_'In+kli'

A2

_nk,ij

-

A'

_Inklij

-

A'

_In+klii'

A()

_A2

_A4

' A

nk,ijp - InkIiip

In±'klijp' .nk,ijpq - n-k Jijpq in+kiijpq 35

(71)

The sets of X and Y tensors are defined by

7

Appendix. IL Modal system jn symmetric form

The d and t tensors are not symmetric for the complete set of indexes. It is important to note that d are not symmetric in bcdf and t in cdf,since these coefficients are near the products in 3

and their derivatives. In order. to obtain analytical symmetric sructure we rewiite (63) in the

form

N N

Nb

Nbc

L

i8a(öam + i3bDlm(a, b) + /3b/3cD2m (a, b, c) + /9b/3C13dD3m(a,b, c, d)+

a=1 b=1 b=lc=1 b=1c=Id=1

Nb cd

+

_i >i $b$fidD4m(a,b,c,

d, f)fi)+

b=lc=ld=lf=1

N a N

/3I3bTOm(a,b) + _> /3a/3bficT1m(a,b,.c)±

a=lb=1 a=lb=lc=1

NäNc

NaNc d

/3a/3 b/3c/3dT2m (a, b, c, d) + T3m(a, b, c, d, f)/b[3c/3d/3f

a151c1di

a=1b=1c=1d=if=1

+O/3 + Pm(i'ox - g) + thQinL

(76)

where

D1n(a,b)=d,

( 2,rn D2m(a,b,c) = a, 42,m 2,rn L a,b,c

I

ab

TOm(a, b) a,a

t'+t

ab

b=c

bc

3m aa,b,b,b, -b = c d D3m (a, b, c, d) 3,m 3,m 3in

aa,b,b,d + da,b,d,b + ao,d,b,b, b = c, c d

+d

b,c+d:C,b, b

c, c

d

dd

± dd,C

± d:b,d + d'd,b + d$bC ± d'b,

b , c d

x1

_ik

-= (_1)i

x°

:i-M

1,

x° = (-1) +1,

x(2) (1) (1)

i-kO'

ikp - ilkpI± ilk+pI'

(73) (74) (i k)2

(i k)2

i,k (0) i+k

4ik

(0)

ikO'

i,kp

-

_i,IkpI(l) i,k+p (75)

Titm (a.b,c) = )

Itl,Tn

a,a,c

a-

-1 lm Om

I..

tj,;+t

ab

4,m

-'-a,b,b,b,f

+

a,b,b,f,b

+

a,b,f,b,b I a,f,b,b,b'

4,m 4,m 4,m

Ua,c,c,b,c ± Ua,ccc +

+

d4,m 4,m

a,c,c,d,d

+

a,d,d,c,c

+

a,c,d,c,d I a,d,c,d,c±

4,rn

Uo,c,d,d,c +

4,m 4,m 4,m

(ha,b,b,d,f ± aa,b,b,f,d + a,b,d,f,b ± a,b,f,d,b+

d4,m -

-a,d,f,b,b

+

a,f,d,b,b

+

a,f,b,b,d + a,d,b,b,f+

d,f,fbC ±

_f,f,c,b

+

+

4,m 4,m

(habcjf + Ua,cbff + ua,b,f,f,c +

4,rn

a,f,c,f,b

+

'4a,f,b,f,c + a,b,f,c,f

+

-a,c,f,b,f

d4,m 4,m 4,m 4m

a,b,c,d,f a,c,b,d,f ' a,b,c,f,d a,c,b,f,d

d4,m

_+d4'

₊

a,b,d,f,c ' a,c,d,f,b a,b,f,d,c a,c,f,d,b

+

+

_,d,f,b,c

+

,f,d,b,c+

d4,m 4,Tm - 4,m

a,b,d,c,f

+

a,c,d,b,f

+

a,b,f,c,d

+

_ã,c,f,b,d

J

a,d,c,f,b I

a,f,b,d,c ' a,d,b,f,c I

a,f,b,d,c

d4,m d4,m d4,m

-a,d,b,c,f

±

a,f,c,b,d

+

a,d,c,b,f a,f,c,b,d

37

b=c=d=f

b = c = d, d

_f

b

c, c

_{d = f}

b = c, c

d, d = f

b

e, c $ d, d = f

b

c, c

d, d

f

d4,m d4,m

a,b,d,b,f + (ha,b,f,b,d + a,f,b,d,b

+

- a,d,b,f,b b = c, c

d, d

f

D4m(a,b,c,d,f)

4,m l.La,c,c,b,f + d4,m a,b,f,c,c

+

d4,m a,c,b,c,f ' 24;m + aa,c,b,f,c ± (ha,c,f,b,c+

a,f,b,c,c+ a,f,c,c,b I _a,b,c,c,f

4,in _+d4,m

a,c,f,c,b ' a,f,c,b,c a,b,c,f,c b

c, c = d, d

f

#2,m

a,a,c,c

a =

c = d

T2m(a,b,c,d) = a,b,c,c-'--' b,a,c,c a c = d

42,m 42;m

1'a,a,c,d + ba,a,d,c

a =

c d

2,m 2,m