Natural sloshing frequenciesin rigid truncated conical tanks

I. GavrilyukStaatliche Studienakademie Thüringen- Berufsakademie Eisenach,

University of Cooperative Education, Eisenach, Germany

M. HermannFriedrich-Schiller-Universität Jena, Jena, Germany

I. Lukovsky and O. SolodunInstitute of Mathematics, National Academy of Sciences of Ukraine, Kiev,

Ukraine, and

A. TimokhaCentre of Excellence “Centre for Ship and Ocean Structures”,

Norwegian University of Science and Technology, Trondheim, Norway

Abstract

Purpose – The main purpose of this paper is to develop two efficient and accurate numericalanalytical methods for engineering computation of natural sloshing frequencies and modes i the caseof truncated circular conical tanks.

Design/methodology/approach – The numerical-analytical methods are based on a Ritz Treftzvariational scheme with two distinct analytical harmonic functional bases.

Findings – Comparative numerical analysis detects the limit of applicability of variational methodsin terms of the semi-apex angle and the ratio between radii of the mean free surface and the circularbottom. The limits are caused by different analytical properties of the employed functional bases.However, parallel use of two or more bases makes it possible to give an accurate approximation of thelower natural frequencies for relevant tanks. For V-shaped tanks, dependencies of the lowest naturalfrequency versus the semi-apex angle and the liquid depth are described.

Practical implications – The methods provide the natural sloshing frequencies for V-shaped tanksthat are valuable for designing elevated containers in seismic areas. Approximate natural modes canbe used in derivations of nonlinear modal systems, which describe a resonant coupling with structuralvibrations.

Originality/value – Although variational methods have been widely used for computing the naturalsloshing frequencies, this paper presents their application for truncated conical tanks for the first time.An original point is the use of two distinct functional bases.

Keywords Numerical analysis, Frequencies, Liquid flow containers

Paper type Research paper

1. IntroductionThe number of constructions carrying a large liquid mass is enormous. Coupling thestructure and liquid requires a precise and efficient computation of the natural sloshing

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/0264-4401.htm

The authors thank the German Research Society (DFG) for financial support (project DFG436UKR 113/33/00). The last author is grateful for the sponsorship of the Alexandervon Humboldt Foundation.

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Received 6 February 2007Revised 12 February 2008Accepted 14 February 2008

Engineering Computations:International Journal forComputer-Aided Engineering andSoftwareVol. 25 No. 6, 2008pp. 518-540q Emerald Group Publishing Limited0264-4401DOI 10.1108/02644400810891535

(eigen-) frequencies and modes. Being subject of spacecraft applications, the naturalsloshing frequencies and modes for conical tanks were studied in 1950-1960’s yearsof the past century. Special attention was paid to an estimate of the resultinghydrodynamic force and moment.

The international standards concerning the megaliter elevated tanks (Eurocode 8,1998) have stated a typical design for concrete tanks of conical and conicalbottomshapes, typical examples are demonstrated by Damatty and Sweedan (2006). Owing toseismic events, liquid motions in a water tank on the supporting tower cause severehydrodynamical loads. In the modelling of these loads, equivalent mechanical systems(Damatty and Sweedan, 2006; Dutta et al., 2004; Shrimali and Jangid, 2003) can be used.These systems relate liquid dynamics to oscillations of a pendulum or a spring-masssystem. The eigenfrequencies of the equivalent systems should coincide with the lowernatural sloshing frequencies and, therefore, an accurate prediction of the sloshingfrequencies and modes is needed (Damatty et al., 2000; Dutta and Laha, 2000; Tang,1999). This can be done by various Computational Fluid Dynamics (CFD) methods orby using semi-empirical approximate formulae (Damatty et al., 2000; Gavrilyuk et al.,2005). Although the CFD-methods demand lots of CPU’s power, they are primaryemployed in engineering practise to guarantee a substantial precision. Whenconcentrating on linear and nonlinear sloshing in V-shaped pure conical tanks,Gavrilyuk et al. (2005) showed that an alternative might consist in a semianalyticalsolution method. The method keeps the accuracy of the CFD-methods, but remainsCPU-efficient and simple in use. The constructed numerical-analytical solutions mayfacilitate development of nonlinear sloshing theories (Lukovsky and Bilyk, 1985;Lukovsky, 1990, 2004; Bauer and Eidel, 1988; Faltinsen et al., 2000, 2003; Gavrilyuket al., 2005, 2006). These theories are of importance for studying a resonant coupledvibration of a tower and the contained liquid.

In order to find the natural sloshing frequencies and modes, a spectral boundaryvalue problem (Lukovsky et al., 1984; Ibrahim, 2005) has to be solved. When the tank isV- and L- shaped, the boundary value problem has no analytical solutions. Isolatedanalytical solutions exist only for pure (non-truncated) conical V-tanks. Such anexample has been specified by Levin (1963) for the two lowest natural modes and thesemiapex angle u0 ¼ 458. These modes are characterised by the wave number m ¼ 1 inangular direction. Dokuchaev and Lukovsky (1968) generalised this result. Theyshowed that analogous analytical solutions exist as u0 ¼ arctan ð

ffiffiffiffim

p Þ. Mikishev andRabinovich (1968) and Feschenko et al. (1969) used these solutions for evaluation oftheir numerical algorithms. Furthermore, simple approximate analytical solutions forpure conical V-tanks can be obtained by replacing the planar waterplane by a sphericsegment. In that case, the spectral boundary value problem admits separation ofvariables in the spherical coordinate system. This has been realised by Dokuchaev(1964) and Bauer (1982). A satisfactory agreement with experimental data by Mikishevand Dorozhkin (1961) and Bauer (1982) was reported as u0 ¼ 158.

The spectral boundary value problem for the linear sloshing modes admits avariational formulation (Feschenko et al., 1969). This variational formulation facilitatesthe Ritz-Treftz numerical scheme, whose practical realisation requires special sets ofanalytical harmonic functional bases. Two of these harmonic bases are used in thepresent paper for engineering computations of the natural frequencies and modes ofliquid sloshing in truncated conical tanks. The first basis is of polynomial type

Natural sloshingfrequencies

519

(harmonic polynomial solutions, HPS). The second one employs the Legendre functionsof first kind; it may be adopted by the mentioned nonlinear multimodal sloshingtheories. Extensive numerical experiments have been done to identify all thegeometrical parameters (semi-apex angle, position of secant plane and liquid depth), forwhich the proposed functional bases guarantee a sufficient number of significantfigures for the lower natural frequencies.

In Section 3, the main focus is on the HPS. For the V-shaped tanks, we show that11-17 HPS provide 4-6 significant digits of the lower natural frequencies for thesemi-apex angles, which are smaller than 758 and larger than 108. For the V- andL-shaped tanks which are characterised by semi-apex angles larger than 758, the samenumber of the HPS guarantees 3-4 significant digits. For the L-shaped tanks withsemi-apex angles smaller than 758, convergence to the natural frequencies depends onthe ratio between the radii of the mean liquid plane and the circular bottom. Themethod is not very efficient (only about 2-3 significant digits can be obtained with17-20 basic polynomials) when the semi-apex angle is smaller than 608 and the ratiobetween the specified radii is smaller than 1/2. The slow convergence for the L-shapedtanks can partially be attributed to a singular asymptotic behaviour of the naturalmodes at the contact line formed by the waterplane and conical walls.

Lukovsky (1990) has proposed a non-conformal mapping technique to develop themultimodal method for the nonlinear sloshing problem in a non-cylindrical tank.Lukovsky and Timokha (2002) and Gavrilyuk et al. (2005) have realised this techniquefor a non-truncated V-tank. The natural sloshing modes were then approximated by aspecial functional basis (SFB). In Section 4, we use the curvilinear coordinate systemproposed by Gavrilyuk et al. (2005) and generalise their results on natural sloshingfrequencies to the case of truncated V- and L-shaped conical tanks. For typicalgeometrical configurations of elevated water tanks (a V-shaped cone with thesemi-apex angle between 308 and 608), the method shows faster convergence behaviourthan in the case of Section 3. Six significant digits of the lowest sloshing frequency canbe computed by using only 6-10 basis functions.

A comparative analysis of the two methods is presented in Section 5. In Section 6,we discuss the dependence of the lowest natural sloshing frequency on the geometricalshape of truncated conical tanks. Bearing in mind that the relevant water tanks are of aV-shape, we present the lowest spectral parameter (with a accuracy of five significantdigits) versus the semi-apex angle and the ratio between the radii of the mean waterplane and the bottom.

2. Statement of the problem2.1 Differential and variational formulationsAs it is typically assumed in sloshing theory (Ibrahim, 2005), we consider an idealincompressible liquid with irrotational flow that partly occupies an earth-fixed rigidconical tank with the semi apex angle u0. The mean (hydrostatic) liquid shape coincideswith the domain Q0 as it is shown in Figure 1. The gravity acceleration is directeddownwards along the symmetry axis Ox. The wetted conical walls are denoted by S1.The circle S2 is the tank bottom, S ¼ S1 < S2 and

P0 is the non-perturbed

(hydrostatic) waterplane. The origin is superposed with artificial apex of the conicalsurface. Henceforth, the problem is considered in a sizeless statement assuming that r0(the bottom radius for L-shaped and the water plane radius for V-shaped tanks)

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is chosen as a characteristic geometrical dimension. Scaling by r0 implies h U h/r0 ! 0(h is the liquid depth) and r1 U r1=r0. The non-dimensional radius r1 and the angle u0completely determine the geometric proportions of Q0. The limit r1 ! 1 implies thath ! 0, ie. the water becomes shallow. At a fixed r1, the scaled liquid depth h tends tozero as u0 ! p/2.

Linear sloshing in a motionless tank is governed by the following boundary valueproblem (Lukovsky et al., 1984; Ibrahim, 2005):

Df ¼ 0 inQ0; ›f›x

¼ ›f›t

;›f

›tþ gf ¼ 0 on

X0;

›f

›n¼ 0 on S;

ZP

0

›f

›xdS ¼ 0;

ð1Þwhere f(x,y,z,t) is the velocity potential, x ¼ f( y,z,t) describes the free surface, n is theouter normal to S and g is the gravity acceleration scaled by r0ðg U g=r0Þ. The initialconditions:

f ð y; z; t0Þ ¼ F0ð y; zÞ; ›f›t

ð y; z; t0Þ ¼ F1ð y; zÞ ð2Þ

at an instant time t ¼ t0, determine a unique solution of equation (1). The functions F0and F1 define initial displacements of the free surface and its velocity, respectively.

2.2 Natural sloshing modesThe solution of equation (1) is associated with the free-standing waves:

fðx; y; z; tÞ ¼ cðx; y; zÞexpðistÞ; i 2 ¼ 21 ð3Þwhere a is the natural sloshing frequency and c(x,y,z) is the so-called natural mode.Inserting equation (3) into equation (1) leads to the spectral problem:

Dc ¼ 0 inQ0; ›c›x

¼ kc onX

0;

›c

›n¼ 0 on S;

ZP

0

›c

›xdS ¼ 0 ð4Þ

Figure 1.Hydrostatic liquiddomains in L- and

V-haped tanks

xx

zO

y

S0

S0

r1

r1

r0

r0

S1

Q0

h

S2

h

g

y

Oz

q0

q0

Natural sloshingfrequencies

521

where the eigenvalue k is defined by:

k ¼ s2

g: ð5Þ

The spectral problem (4) has a real positive point wise spectrum (Morand and Ohayon,1995):

0 , k1 # k2 # . . . # kn # . . .

with a unique limit point at infinity, i.e. kn !1; n!1. Together with the constantfunction, projections of the eigenfunctions, f nð y; zÞ ¼ cnj

P0 constitute an orthogonal

basis in the mean-squares metrics. Because the velocity potential f satisfies thevolume conservation condition (see, the last integral equality in equation (1)), gettingknown {kn} and {cn}, one can represent the solution of equation (1) and equation (2) asa Fourier series by fn ¼ cnðx; y; zÞexpðisntÞ, namely, as a superposition of thefree-standing waves.

2.3 On the Ritz-Treftz schem for the spectral problem (4)Problem (4) admits a minimax variational formulation (Feschenko et al., 1969; Morandand Ohayon, 1995), which is based on the positive functional:

KðcÞ ¼RQ0ð7cÞ2dQRP

0

c2dSð6Þ

under the supplementary conditionRP

0

c dS ¼ 0. In that case, the absolute minimumof the functional equation (6) coincides with the lowest eigenvalue of the spectralproblem (4). Furthermore, the necessary condition for an extrema of equation (6) leadsto the variational equation:Z

Q0

ð7c;7hÞdQ2 kZP

0

chdS ¼ 0 ð7Þ

with respect to a non-constant function c, where h is a smooth test-function.Variational problem (7) may be solved by the Ritz-Treftz variational scheme.

Approximate solutions are then posed as the following linear combination of smoothharmonic functions:

cðx; y; zÞ ¼Xqk¼1

akBkðx; y; zÞ: ð8Þ

Substituting equation (8) into equation (7) and using Bi(i ¼ 1, . . . ,q) as test-functions,one obtains the spectral matrix problem:

Xqk¼1

ð{aik} 2 k{bik}Þak ¼ 0; i ¼ 1; . . . ; q ð9Þ

Here, the elements of the non-negative matrices A ¼ {aik}; B ¼ {bik} are computedby the formulae:

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522

aik ¼ZQ0

ð7Bi;7BkÞdQ ¼ZP

0þS

›Bi›n

Bj dS; bik ¼ZP

0

BiBkdS; ð10Þ

and the approximate eigenvalues are roots of the equation:

detðA2 kBÞ ¼ 0; ð11Þ

which appears as the necessary solvability condition of system (9). By increasing thedimension q the non-zero roots of equation (11) converge (from above) to the lowereigenvalues of equation (4). Approximate eigenfunctions (8) are formed by theeigenvectors of equation equation (9), {ak,k ¼ 1, . . . ,q}.

A key difficulty of the Ritz-Treftz scheme consists of establishing a suitableanalytical functional basis {Bk}. The completeness of functional sets significantlydepends on the actual shape of Q0. To the author’s knowledge, there are two types ofanalytical functional sets, which may be adapted to the studied case. The first set isproposed in the book by Lukovsky et al. (1984). It follows from a separation ofvariables in the Laplace equation done in the spherical coordinate system. Theseharmonic solutions are of polynomial structure (harmonic polynomial solutions, HPS)in the Cartesian coordinate system. Their completeness is proved for all star-shapeddomains with respect to the origin. Being rewritten in a cylindrical coordinate system,the HPS admit the separation of the angular coordinate and keep polynomial structurewith regard to the remaining coordinates, i.e. in projections on a meridionalcross-section. A specific functional basis (SFB) is presented by Gavrilyuk et al. (2005).It is derived in the cylindrical coordinate system combined with a non-conformalmapping of the meridional cross-section. The functional set is harmonic and satisfiesthe zero-Neumann condition on the conical walls. Furthermore, we utilise these twofunctional sets in the Ritz-Treftz scheme to solve the spectral problem (4). Thetree-dimensional problem and its variational formulation (7) are thereby reduced to twodimensions by separating the angular-type variable.

3. Ritz-Treftz method based on the HPSWe use the cylindrical coordinate system (X,j,h) linked with the original Cartesiancoordinates by:

x ¼ X þ X0; y ¼ j cosh; z ¼ j sinh: ð12Þ

Here, the lag X0 along the vertical axis is introduced to superpose the origin of thecylindrical coordinate system with the waterplane. The solution of equation (4) isrepresented in the following form:

cðX ; j;hÞ ¼ wmðX ; jÞsinmh

cosmh

!; m ¼ 0; 1; 2. . . ð13Þ

This makes it possible to separate the angular coordinate h and to reduce thethree-dimensional boundary value problem (4) to the following m-parametric family(m is a non-negative integer) of two-dimensional spectral problems:

Natural sloshingfrequencies

523

›

›Xj›wm›X

� �þ ›

›jj›wm›j

� �2

m 2

jwm ¼ 0 inG; ›wm

›X¼ kmwm on L0;

›wm›n

¼ 0 on L; jwmðX ; 0Þj , 1;ZL0

j›w0›X

dj ¼ 0:ð14Þ

Problem (14) is defined in a meridional plane of Q0and L ¼ L1 þ L2 (Figure 2). Thismeans that the eigenvalues of the original three-dimensional problem constitute atwo-parametric set k ¼ kmiðm ¼ 0; 1; . . .; i ¼ 1; 2; . . .Þ, where i $ 1 enumerates theeigenvalues of equation (14) in ascending order. The corresponding eigenfunctions ofequation (4) take the form equation (13) with wm ¼ wmiðX ; jÞ.

According to Lukovsky et al. (1984), the HPS admit the following form in themeridional plane (after separation of the h-coordinate):

wðmÞk ðX ; jÞ ¼2ðk2mÞ!ðkþmÞ! R

kP ðmÞkX

R

� �; k $ m; R ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX 2 þ j 2

p; ð15Þ

Where P ðmÞk are Legendre’s functions of first kind. The functions {wðmÞk }are the solutions

of the first equation of (14). It can be shown that wðmÞk has indeed a polynomial structurein terms of X and j. The first functions of the set equation (15) take the form:

wð0Þ0 ¼ 1; wð0Þ1 ¼ X ; wð0Þ2 ¼ X 2 2j 2

2; ::: ðm ¼ 0Þ;

wð1Þ1 ¼ j; wð1Þ2 ¼ Xj; wð1Þ3 ¼ X 2j2j 3

4; ::: ðm ¼ 1Þ;

wð2Þ2 ¼ j 2; wð2Þ3 ¼ Xj 2; wð2Þ4 ¼ X 2; j 2 2j 4

6; ::: ðm ¼ 2Þ;

wð3Þ3 ¼ j 3; wð3Þ4 ¼ Xj 3; wð3Þ5 ¼ X 2; j 3 2j 5

8; ::: ðm ¼ 3Þ:

The computation of {wðmÞk } can be realised by the following recurrence relations:

Figure 2.Meridional planes ofL- and V-shaped tanks

X

G

Gh

h

O

O1

1

XL0

L0r1

q0

q0

r1

L1

L1

L2

L2

x

x

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524

›wðmÞk›X

¼ ðk2mÞwðmÞk21; j›wðmÞk›j

¼ kwðmÞk 2 ðk2mÞXwðmÞk21;

ðk2mþ 1ÞwðmÞkþ1 þ ð2kþ 1ÞXwðmÞk 2 ðk2mÞðX 2 þ j 2ÞwðmÞk21;

ðk2mþ 1Þjwðmþ1Þk ¼ 2ðmþ 1Þ ðX 2 þ j 2ÞwðmÞk21 2 XwðmÞk� �

:

By separating the h-coordinate in the variational formulation (7) and in therepresentation equation (8), we arrive at the following m-parametric families (m isnon-negative integer) of approximate solutions:

wmðX ; jÞ ¼Xqk¼1

aðmÞk wðmÞkþm21ðX ; jÞ; ð16Þ

and spectral matrix problems:

Xqk¼1

aðmÞik� �

2 km bðmÞik

� �� �ak ¼ 0 ði ¼ 1; . . . ; qÞ;

det aðmÞik� �

2 km bðmÞik

� �� � ¼ 0;ð17Þ

following from (9). The elements aðmÞik� �

and bðmÞik� �

are computed for the L-cones bythe formulae:

aðmÞij ¼Z r1

0

j›wðmÞiþm21

›XwðmÞjþm21

!X¼0

djþZ 02h

j›wðmÞiþm21

›jwðmÞjþm21

!j¼tanu0X2r1

dX

2tanu0

Z 02h

j›wðmÞiþm21

›XwðmÞjþm21

!j¼tanu0X2r1

dX2

Z 10

j›wðmÞiþm21

›XwðmÞjþm21

!X¼2h

dj;

bðmÞij ¼Z r1

0

jwðmÞiþm21wðmÞjþm21

� X¼0

dj;

and for the V cones by the formulae:

aðmÞij ¼Z 1

0

j›wðmÞiþm21

›XwðmÞjþm21

!X¼0

djþZ 02h

j›wðmÞiþm21

›jwðmÞjþm21

!j¼tanu0Xþ1

dX

2tanu0

Z 02h

j›wðmÞiþm21

›XwðmÞjþm21

!j¼tanu0Xþ1

dX2

Z r10

j›wðmÞiþm21

›XwðmÞjþm21

!X¼2h

dj;

bðmÞij ¼Z 1

0

jwðmÞiþm21wðmÞjþm21

� X¼0

dj:

Natural sloshingfrequencies

525

For each fixed m, the second equation of (17) has q positive roots (n ¼ 1,2, . . . ,q). Themultiplicity should be accounted for. Since the Ritz-Treftz method implies theminimisation of a functional, the approximate values kmn converge from above. Thismakes it possible to check the convergence by the number of significant digits, whichdo not change as increases. The method gives the best approximation for the lowesteigenvalue km1.

Convergence. Our numerical experiments were primary dedicated to eigenvalueskm1, m ¼ 0,1,2,3. These eigenvalues are responsible for the lowest natural modes,which give a decisive contribution to hydrodynamic loads (Ibrahim, 2005; Gavrilyuket al., 2005). In the case of V-tanks, the method shows a fast convergence to km1 andprovides satisfactory accuracy for km2 and km3, too. It shows a slower convergencefor the L-tanks. Furthermore, the convergence depends not only on the tank type(V or L-shaped), but also on the semi-apex angle u0and the dimensionless parameter0 , r1 , 1. The results in Table I (A) exhibit a typical convergence behaviour for theV-tanks with 108 # u0 # 758 and 0.2 # r1 # 0.9. The table shows stabilisation of 5-6significant digits as q $ 14. The best accuracy is established for the lowest spectralparameter k11. The accuracy grows with q for m – 0. However, computations of thevalue k01, which is responsible for the axial-symmetric natural mode, may becomeunstable for q . 17. This explains why we do not present numerical results on thiseigenvalue for q ¼ 20 While m – 1 and 0.2 # r1 # 0.9, increasing u0 . 758 leads to aslower convergence. In this case, q ¼ 17,. . . ,20 guarantees only 3-4 significant digits(necessary engineering accuracy). The same number of basic functions keeps thisnumber of significant digits for the tanks with r1 , 0.2 and 108 # u0 # 758. When r1tends to zero (truncated tank is close to a non-truncated one), the approximations km1were validated by numerical results reported by Gavrilyuk et al. (2005) as well as byexperimental data given in Bauer (1982).

In Table I (B), typical convergence behaviour for the L-shaped tanks with108 # u0 # 758 is presented. A comparative analysis of parts (A) and (B) illustratesthat the method is in the latter case less efficient. In particular, computations of k01 arenot so precise. Moreover, 18-20 basic functions lead to 4-5 significant digits of km1 onlyfor r1 $ 0.4. This is not the case for lower values of r1. Furthermore, whenr1 # 0.2,q ¼ 17, . . . ,20 can only guarantee 2-3 significant digits for k11. The slowerconvergence for the L-shaped tanks can in part be clarified by the occurrence ofsingular first derivatives of the eigenfunctions cm at an inner vertex between L0and L1(see the mathematical results by Lukovsky et al., 1984). The HPS are smooth in the(j, X)-plane, and, therefore, do not capture this singular behaviour. The singularitydisappears when the corner angle is less than 908. This occurs only for the L-shapedtanks. The V-shaped tanks are characterised by a similar singularity at the vertexformed by L1and L2. However, because the natural modes (eigenfunctions cm) should“decay” exponentially downward, the method may be sensitive with respect to thatsingularity only for shallow water. Our numerical experiments confirm this factasr1 , 0.1.

3.1 Ritz-Treftz method based on the SFBThe nonlinear resonant sloshing is effectively studied by multimodal methods. As it isshown by Lukovsky (1975), Lukovsky and Timokha (2002), these methods requireanalytical expressions for the natural modes, which:

EC25,6

526

AB

qr 1

¼0.

2r 1

¼0.

4r 1

¼0.

6r 1

¼0.

8r 1

¼0.

9Q

r 1¼

0.2

r 1¼

0.4

r 1¼

0.6

r 1¼

0.8

r 1¼

0.9

k01 2

5.03

2415

4.57

7833

4.15

2352

23.

8055

012.

9613

998

70.0

6337

15.1

4094

6.89

9986

4.56

0991

12.

6834

115

3.39

4779

3.39

2286

3.38

4865

3.14

1171

2.20

0342

1050

.467

6211

.124

496.

6722

664.

5511

502.

6831

178

3.38

5603

3.38

5592

3.38

1845

3.13

8861

2.19

7438

1239

.061

1110

.185

636.

6667

044.

5508

772.

6829

3211

3.38

5600

3.38

5590

3.38

1827

3.13

8718

2.19

7238

1432

.075

7210

.017

456.

6656

784.

5505

902.

6827

9614

3.38

5600

3.38

5590

3.38

1822

3.13

8665

2.19

7162

1627

.649

0410

.016

126.

6650

584.

5504

222.

6827

5117

3.38

5600

3.38

5590

3.38

1819

3.13

8644

2.19

7138

1825

.152

9910

.014

266.

6648

864.

5503

492.

6827

30k

11

21.

3436

311.

3357

231.

2999

771.

0072

610.

6073

408

16.5

0485

5.69

4833

3.51

6782

1.66

1815

0.72

6555

51.

3044

131.

3017

931.

2541

570.

9344

120.

5425

0310

13.9

4781

5.63

3730

3.51

5861

1.66

1684

0.72

6507

81.

3043

781.

3016

941.

2540

540.

9338

850.

5422

8412

13.4

8767

5.63

1904

3.51

5547

1.66

1608

0.72

6489

111.

3043

771.

3016

921.

2539

820.

9338

350.

5422

5714

12.2

4724

5.63

0560

3.51

5365

1.66

1559

0.72

6484

141.

3043

771.

3016

871.

2539

720.

9338

230.

5422

5116

11.6

6350

5.62

9898

3.51

5269

1.66

1551

0.72

6482

171.

3043

771.

3016

861.

2539

690.

9338

190.

5422

4918

11.4

5072

5.62

9777

3.51

5237

1.66

1537

0.72

6480

201.

3043

771.

3016

861.

2539

670.

9338

170.

5422

4820

11.3

3183

5.62

9888

3.51

5214

1.66

1532

0.72

6478

k21

22.

4437

392.

4243

902.

8663

32.

1913

711.

5720

488

45.0

2411

9.79

7804

5.95

0925

3.72

4251

1.92

3234

52.

2635

502.

2633

512.

2550

962.

0156

021.

3619

0510

28.6

5250

9.09

6131

5.94

1913

3.72

4049

1.92

3019

82.

2631

512.

2630

872.

2550

042.

0149

231.

3609

2812

25.1

8394

8.97

5383

5.94

1107

3.72

3708

1.92

2950

112.

2631

502.

2630

872.

2549

762.

0148

381.

3608

3714

24.7

7166

8.96

8027

5.94

1002

3.72

3640

1.92

2930

142.

2631

502.

2630

862.

2549

722.

0148

111.

3608

0516

21.5

1273

8.96

6657

5.94

0651

3.72

3591

1.92

2913

172.

2631

502.

2630

862.

2549

692.

0148

011.

3607

9518

19.7

3858

8.96

6531

5.94

0603

3.72

3549

1.92

2899

202.

2631

502.

2630

862.

2549

682.

0147

961.

3607

9020

17.7

6023

8.96

6524

5.94

0589

3.72

3542

1.92

2895

k31

23.

5331

703.

5208

823.

4590

483.

3161

982.

6974

268

139.

8798

15.6

7055

8.14

6078

5.63

4069

3.40

4035

53.

1815

303.

1812

993.

1795

413.

0471

792.

3290

2210

83.8

2254

12.9

5492

8.04

6749

5.63

3996

3.40

3568

83.

1802

513.

1802

493.

1790

803.

0467

422.

3270

4412

64.5

8853

12.2

2964

8.04

4916

5.63

3565

3.40

3460

113.

1802

493.

1802

483.

1790

773.

0466

542.

3268

1214

53.4

1076

12.0

8855

8.04

4459

5.63

3498

3.40

3411

143.

1802

493.

1802

473.

1790

743.

0466

202.

3268

1216

53.1

3873

12.0

7964

8.04

4234

5.63

3377

3.40

3360

173.

1802

493.

1802

473.

1790

733.

0466

062.

3267

9018

38.8

5519

12.0

8400

8.04

4028

5.63

3343

3.40

336

203.

1802

493.

1802

473.

1790

733.

0465

992.

3267

7920

36.4

6749

12.0

8490

8.04

3988

5.63

3321

3.40

3328

Notes:

Col

um

n(A

)is

for

aV

-sh

aped

tan

k,

(B)

corr

esp

ond

sto

aL

-sh

aped

tan

k,u

0¼

308

Table I.Convergence to

km1,m ¼ 0,1,2,3 fordifferent r1 versus the

number of basic functionsq (16)

Natural sloshingfrequencies

527

. are analytically expandable over the waterplane;

. satisfy a zero-Neumann condition at the tank walls and, if the tank isnon-cylindrical; and

. can be transformed to a curvilinear coordinate system (x1,x2,x3), in which the freesurface is governed by the normal form representation x1 ¼ ~fðx2; x3; tÞ.

An example of suitable approximate natural modes for non-truncated V-tanks is givenby Lukovsky (1990) and Gavrilyuk et al. (2005). The present section generalises theseresults.

Curvilinear coordinate system. The non-Cartesian parametrisation proposed byGavrilyuk et al (2005) links the x,y,z coordinates with x1,x2,x3 as follows:

x ¼ x1; y ¼ x1x2 cos x3; z ¼ x1x2 sin x3: ð18ÞThus, the variable x3 ¼ h is the polar angle in the Oyz-plane and corresponds to h inequation (12). Figure 3 demonstrates that the hydrostatic liquid domain Q0 takes in the(x1,x2,x3)-system the form of an upright rectangular base cylinder ðx0 # x1 # x10; 0# x2 # x20; 0 # x3 # 2pÞ. The domain G* represents a rectangle with thesides h ¼ x10 2 x0and x20 ¼ tan u0 in theOx2x1-plane. Here, the radius of theundisturbed water plane is rt ¼ 1 for the V-tanks andrt ¼ r1 for the L-tanks. Havingpresented:

wðx1; x2; x3Þ ¼ cmðx1; x2Þsinmx3

cosmx3

!; m ¼ 0; 1; 2; . . . ð19Þ

and following Gavrilyuk et al. (2005), one obtains that the original three-dimensionalproblem (4) admits separation of the spatial variable x3. Furthermore, thetransformation (18) generates the following m-parametric family of spectralproblems with respect to:

p›2cm

›x21þ 2q ›

2cm›x1›x2

þ s ›2cm

›x22þ d ›cm

›x22m 2ccm ¼ 0 in G*; ð20Þ

p›cm›x1

þ q ›cm›x2

¼ kmpcm on L*0 ; ð21Þ

Figure 3.Meridional cross-sectionsof the original andtransformed domains

x x1 x20 r0 L0 x1 L0

x10

x2

x

x20

x2

y

y

L0

q0

q0 L0

L1

L2 L2

L2

L2

L1

L1L1

r1 r1

G G∗

gG∗

r0

x0x0

x1000

O O

∗

∗

∗

∗

∗

∗

EC25,6

528

s›cm›x2

þ q ›cm›x1

¼ 0 on L*1 ; ð22Þ

p›cm›x1

þ q ›cm›x2

¼ 0 on L*2 ; ð23Þ

jcmðx1; 0Þj , 1; m ¼ 0; 1; 2; . . . ; ð24ÞZ x20

0

c0x2dx2 ¼ 0; ð25Þ

where G* ¼ {ðx1; x2Þ : x0 # x1 # x10; 0 # x2 # x20}, �Wð0Þk , d ¼ 1 þ 2x22; c ¼ 1=x2 andthe boundary of G* consists of the portions L*0 ;L

*1 and L

*2 .

Particular solutions of equation (20) and equation(22). Gavrilyuk et al. (2005) studieda spectral problem, which is similar to equations (20)-(25). Following results byEisenhart, they proved that equation (20) and equation (22) allow together for theseparation of the spatial variables x1 and x2. For our problem, this separation leads tothe following particular solutions:

xn1TðmÞn ðx2Þ and

TðmÞn ðx2Þx1þn1

; n $ 0: ð26Þ

In order to determine T ðmÞn , we have to consider the following homogeneous boundaryvalue problem, which depends on the real parameter n :

x22ð1 þ x22ÞT00ðmÞn þ x2ð1 þ 2x22 2 2nx22ÞT

00ðmÞn þ ½nðn2 1Þx22 2m 2�T ðmÞn ¼ 0; ð27Þ

T00ðmÞn ðx20Þ ¼ n

x20

1 þ x220TðmÞn ðx20Þ; jT ðmÞn ð0Þj , 1: ð28Þ

It can be shown that the problem (27) and (28) has only nontrivial solutions for acountable set of values n ¼ nmn . 0ðm ¼ 0; 1; . . .; n ¼ 1; 2; . . .Þ:

The second class of functions, TðmÞn , appears only in the case of x0 – 0, i.e. when the

conical tank is truncated. Computation of TðmÞn leads to the following n-parametric

problem:

x22ð1 þ x22ÞT00ðmÞ þ x2ð1 þ 4x22 2 2nx22ÞT 0n þ ½ðnþ 1Þðnþ 2Þx22 2m 2�T

ðmÞ ¼ 0; ð29Þ

T 0ðmÞðx20Þ þ ðnþ 1Þ x201 þ x220

TðmÞðx20Þ ¼ 0: ð30Þ

Obviously, nontrivial solutions of equation (29) and equation (30) exist only for acountable set of nonnegative values n.

Let us now show that the solution of equations (27) and (29) can be expressed interms of the spheroidal harmonics and the set {nmn} is the same for the problems (27)and (28) and the problems (29) and (30). For this purpose, we change the variables inequations (27) and (29) by m ¼ ð1 þ x22Þ2ð1=2Þ and substitute yðmÞ ¼ mnTðmÞ and

Natural sloshingfrequencies

529

yðmÞ ¼ m212nTðmÞ into equation (27) and equation (29), respectively. This reduces thetwo equations to the same well-known differential equation:

ð1 2 m 2Þy00ðmÞ2 2my0ðmÞ þ nðnþ 1Þ2 m2

1 2 m 2

�yðmÞ ¼ 0;

whose solutions coincide with the Legendre function of first kind, i.e. yðmÞ ¼ P ðmÞn ðmÞ.Furthermore, treating the boundary conditions (28) and (30) in the same way and

using the substitution m ¼ cos u, the following common equation is obtained:›P ðmÞn ðcos uÞ

›u

����u¼u0

¼ 0: ð31Þ

This equation can be considered as a transcendental equation for the computation ofthe values {nmn}- Appendix (Figure A1) presents the first 12 values of {nmn}(m ¼ 0,1,2,3) versus u0.

In conclusion, with the technique described above, we get the following nontrivialparticular solutions

TðmÞnmk ðx2Þ ¼ ð1 þ x22Þnmk

2 P ðmÞnmk1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ x22q

0B@

1CA; ð32Þ

TðmÞnmk

ðx2Þ ¼ ð1 þ x22Þ212nmk

2 P ðmÞnmk1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ x22q

0B@

1CA: ð33Þ

Particular solutions as a functional basis. Let the particular solutions (26), (32) and (33)be rewritten in the form:

W ðmÞk ðx1; x2Þ ¼ N ðmÞk xnmk1 T ðmÞnmkðx2Þ; �WðmÞk ðx1; x2Þ ¼ �N

ðmÞk x

212nmk1 T

ðmÞnmk

ðx2Þ: ð34ÞHere, N ðmÞk and �N

ðmÞk are multipliers which are chosen to satisfy the following condition:

1 ¼ kW ðmÞk jj2L*2þL*

0

¼ k �WðmÞk jj2L*2þL*

0

¼Z x20

0

x2½ðW ðmÞk jx1¼x10 Þ2 þ ðW ðmÞk jx1¼x0 Þ2�dx2

¼Z x20

0

x2½ð �WðmÞk jx1 ¼ x10Þ2 þ ð �WðmÞk jx1¼x0 Þ2�dx2:

ð35Þ

equation (35) says that W ðmÞk and �WðmÞk . have the unit norm (in the mean-squares

metrics) on the boundary L*2 þL*0 , where equations (21) and (23) should beapproximately satisfied. Explicit formulae for these normalising multipliers havethe form:

EC25,6

530

N ðmÞk ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2nmk10 þ x2nmk0q 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR x20

0 ð1 þ x22Þnmk P ðmÞnmk� 2

dx2

r ;

�NðmÞk ¼

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2222nmk10 þ x2222nmk0

q 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR x200 ð1 þ x22Þ212nmk P ðmÞnmk

� 2dx2

r :

The case m ¼ 0 requires, in addition, the volume conservation condition (25).This implies a re-definition of the functions W ð0Þk and �W

ð0Þk by: W

ð0Þk U W

ð0Þk 2 c

ð0Þk ,

�Wð0Þk U

�Wð0Þk 2 �c

ð0Þk ;where:

cð0Þk ¼2

x220

Z x200

x2Wð0Þk ðx10; x2Þdx2; �cð0Þk ¼

2

x220

Z x200

x2 �Wð0Þk ðx10; x2Þdx2:

Variational method. In accordance with the Ritz-Treftz scheme, we representapproximate solutions of equations (20)-(25) in the form:

cmðx1; x2Þ ¼Xq1k¼1

aðmÞk WðmÞk þ

Xq2l¼1

�a ðmÞl �WðmÞl : ð36Þ

By separating the x3 coordinate in variational formulation (7) (after substitution ofequation (19)), representation (36) leads to the m-parametric family of spectralproblems:

XQk¼1

aðmÞik� �

2 km bðmÞik

� �� �ak ¼ 0 ði ¼ 1; . . . ;QÞ;

det ~aðmÞijn o

2 km ~bðmÞik

n o� ¼ 0:

ð37Þ

The spectral problem (37) has Q ¼ q1 þ q2 eigenvalues. Because the representation(36) contains two types of functions, namely W ðmÞk and, �W

ðmÞl there exist four

sub-matrices of ~aðmÞijn o

and ~bðmÞij

n osuch that:

~aðmÞij ¼aðmÞij1 a

ðmÞij2

aðmÞij3 aðmÞij4

0@

1A; ~bðmÞij ¼

bðmÞij1 bðmÞij2

bðmÞij3 bðmÞij4

0@

1A:

The elements aðmÞijsn o

and bðmÞijsn o

, s ¼ 1, . . . ,4, are computed by the formulae:

aðmÞij1 ¼Z x20

0

x21x2›W ðmÞi›x1

2 x1x22

›W ðmÞi›x2

!x1¼ht

W ðmÞj dx2

2

Z x200

x21x2›W ðmÞi›x1

2 x1x22

›W ðmÞi›x2

!x1¼hb

W ðmÞj dx2;

Natural sloshingfrequencies

531

aðmÞij2 ¼Z x20

0

x21x2›W ðmÞi›x1

2 x1x22

›W ðmÞi›x2

!x1¼ht

�WðmÞj dx2

2

Z x200

x21x2›W ðmÞi›x1

2 x1x22

›W ðmÞi›x2

!x1¼hb

�WðmÞj dx2;

aðmÞij3 ¼Z x20

0

x21x2› �W

ðmÞi

›x12 x1x

22

› �WðmÞi

›x2

!x1¼ht

W ðmÞj dx2

2

Z x200

x21x2› �W

ðmÞi

›x12 x1x

22

› �WðmÞi

›x2

!x1¼hb

W ðmÞj dx2;

aðmÞij4 ¼Z x20

0

x21x2› �W

ðmÞi

›x12 x1x

22

› �WðmÞi

›x2

!x1¼ht

�WðmÞj dx2

2

Z x200

x21x2› �W

ðmÞi

›x12 x1x

22

› �WðmÞi

›x2

!x1¼hb

�WðmÞj dx2;

bðmÞij1 ¼ h2tZ x20

0

x2 WðmÞi W

ðmÞj

� x1¼ht

dx2; bðmÞij2 ¼ h2t

Z x200

x2 WðmÞi

�WðmÞj

� x1¼ht

dx2;

bðmÞij3 ¼ h2tZ x20

0

x2 �WðmÞi W

ðmÞj

� x1¼ht

dx2; bðmÞij4 ¼ h2t

Z x200

x2 �WðmÞi

�WðmÞj

� x1¼ht

dx2:

In the case ofL- and V-tanks, we have ht ¼ r1= tan u0, hb ¼ 1= tan u0 and ht ¼ 1= tan u0,hb ¼ r1= tan u0, respectively.

Convergence. Column A in Table II shows a typical convergence behaviour in thecase of V-tanks with 108 # u0 # 758 and 0.2 # r1 # 0.9. When 0.2 # r1 # 0.55, themethod generates 4-5 significant digits of km1 for q ¼ q1 ¼ q2 ¼ 7, . . . ,10 (14, . . . ,20basic functions). This is consistent with the convergence results in Section 3. However,the SFB keeps also a fast convergence to km1 for r1 , 0.2. This includes the case of k01,which has not been satisfactory handled by the HPS. Moreover, when 0 , r1 # 0.4,the number of significant digits is larger (for the same number of basic functions) for158 # u0 , 308, but it is only marginally smaller for 158 # u0. Gavrilyuk et al. (2005)related such a slower convergence of an analogous method for smaller semi-apexangles to the asymptotic behaviour of the exact solution along the vertical axis. Theirconclusion is that the eigenfunctions cm should exponentially decay downward Oxfor a circular cylindrical tank, to which the conical domain tends as u0 decreases.However, W ðmÞk and �W

ðmÞk do not capture this decaying. Furthermore, decreasing

the dimensionless liquid depth h(r1 ! 1 or u0 ! 908) may cause a lower accuracy(3-4 significant digits for 18-24 basic functions).

EC25,6

532

A BQ r1 ¼ 0.2 r1 ¼ 0.4 r1 ¼ 0.6 r1 ¼ 0.8 r1 ¼ 0.9 r1 ¼ 0.2 r1 ¼ 0.4 r1 ¼ 0.6 r1 ¼ 0.8 r1 ¼ 0.9

2 3.385676 3.385666 3.382064 3.148405 2.224870 20.82985 10.41489 6.934470 4.781070 2.8688943 3.385606 3.385596 3.381920 3.143923 2.211744 20.29240 10.14616 6.754682 4.629653 2.7489544 3.385601 3.385591 3.381878 3.141937 2.206179 20.15184 10.07588 6.707583 4.588863 2.7159115 3.385600 3.385590 3.381859 3.140888 2.203283 20.09711 10.04852 6.689248 4.572788 2.7025146 3.385600 3.385590 3.381847 3.140267 2.201583 20.07070 10.03531 6.680396 4.264916 2.6958177 3.385600 3.385590 3.381840 3.139870 2.200501 20.05610 10.02801 6.675501 4.560515 2.6920068 3.385600 3.385590 3.381835 3.139601 2.199769 20.04724 10.02358 6.672529 4.557819 2.6896379 3.385600 3.385590 3.381832 3.139410 2.199252 20.04147 10.02070 6.670596 4.556053 2.688066

10 3.385600 3.385590 3.381829 3.139270 2.198872 20.03753 10.01872 6.669272 4.554837 2.68697311 3.385600 3.385590 3.381827 3.139164 2.198585 20.03471 10.01732 6.668327 4.553964 2.68618112 3.385600 3.385590 3.381826 3.139082 2.198364 20.03263 10.01628 6.667629 4.553317 2.685590

k012 1.304378 1.301707 1.254338 0.935957 0.544861 11.33533 5.647088 3.528677 1.672867 0.7324103 1.304377 1.301695 1.254148 0.934864 0.543483 11.31577 5.637221 3.521265 1.666870 0.7292464 1.304377 1.301691 1.254073 0.934437 0.542976 11.30879 5.633703 3.518590 1.664611 0.7280935 1.304377 1.301689 1.254036 0.934226 0.542729 11.30558 5.632078 3.517343 1.663527 0.7275366 1.304377 1.301688 1.254016 0.934106 0.542589 11.30384 5.631202 3.516665 1.662926 0.7272247 1.304377 1.301687 1.254003 0.934032 0.542502 11.30281 5.630678 3.516258 1.662559 0.7270328 1.304377 1.301687 1.253994 0.933983 0.542445 11.30214 5.630340 3.515995 1.662319 0.7269069 1.304377 1.301687 1.253988 0.933949 0.542405 11.30168 5.630110 3.51515 1.662153 0.726818

10 1.304377 1.301686 1.253984 0.933924 0.542376 11.30136 5.629947 3.515687 1.662034 0.72675411 1.304377 1.301686 1.253981 0.933906 0.542354 11.30112 5.629826 3.515592 1.661945 0.72670712 1.304377 1.301686 1.253978 0.933892 0.542337 11.30094 5.629735 3.515520 1.661878 0.726671

k212 2.263162 2.263100 2.255147 2.019323 1.371212 18.03918 9.019177 5.977616 3.761864 1.9527593 2.263151 2.263088 2.255060 2.017249 1.366295 17.98215 8.990657 5.958186 3.742630 1.9381824 2.263150 2.263087 2.255026 2.016335 1.364234 17.96014 8.979646 5.950674 3.734876 1.9322155 2.263150 2.263087 2.255007 2.015850 1.363151 17.94951 8.974330 5.947040 3.731012 1.9291676 2.263150 2.263086 2.254996 2.015563 1.362510 17.94361 8.971384 5.945023 3.728818 1.9274007 2.263150 2.263086 2.254989 2.015378 1.362099 17.94003 8.969588 5.943793 3.727457 1.9262858 2.263150 2.263086 2.254985 2.015252 1.361819 17.93768 8.968417 5.942990 3.726556 1.9255379 2.263150 2.263086 2.254981 2.015163 1.361620 17.93607 8.967612 5.942438 3.725930 1.925011

10 2.263150 2.263086 2.254979 2.015097 1.361473 17.93492 8.967036 5.942042 3.725477 1.92462711 2.263150 2.263086 2.254977 2.015047 1.361362 17.93407 8.966610 5.941750 3.725140 1.92433812 2.263150 2.263086 2.254976 2.015008 1.361276 17.93342 8.966287 5.941527 3.724882 1.924115

k312 3.180280 3.180279 3.179147 3.051144 2.346969 24.36124 12.18061 8.115651 5.700453 3.4738763 3.180251 3.180250 3.179101 3.049247 2.338329 24.25430 12.12714 8.079896 5.667935 3.4415874 3.180249 3.180248 3.179090 3.048337 2.334334 24.21030 12.10514 8.065186 5.654291 3.4273525 3.180249 3.180249 3.179085 3.047828 2.332117 24.18824 12.09411 8.057806 5.647308 3.4198036 3.180249 3.180247 3.179082 3.047514 2.330753 24.17570 12.08787 8.053610 5.643277 3.4153197 3.180249 3.180247 3.179080 3.047306 2.329853 24.16792 12.08395 8.051009 5.640748 3.4124408 3.180249 3.180247 3.179078 3.047161 2.329228 24.16278 12.08138 8.049290 5.639061 3.4104819 3.180249 3.180247 3.179077 3.047056 2.328776 24.15922 12.07960 8.048098 5.637882 3.409090

10 3.180249 3.180247 3.179077 3.046978 2.328438 24.15666 12.07832 8.047239 5.637026 3.40806611 3.180249 3.180247 3.179076 3.046918 2.328179 24.15475 12.07736 8.046600 5.636386 3.40729012 3.180249 3.180247 3.179076 3.046871 2.327976 24.15329 12.07664 8.046112 5.635894 3.406689

Notes: Column (A) is for a V-shaped tank; (B) corresponds to a L-shaped tank, u0 ¼ 308

Table II.Convergence to km1,

m ¼ 0,1,2,3, for differentr1 versus the number of

basic functionsq ¼ q1 ¼ q2 in (36)

Natural sloshingfrequencies

533

Column B in Table II shows convergence for a L-tank. The same r1 and u0 as in thecolumn A are chosen. It can be seen that the numerical results may be less precise thanthose in Section 3. For instance, the same number of basic functions gives only 2-3significant digits when 0.2 # r1 # 0.9. However, in contrast to the HPS, the SFBprovides reliable computations for the case of an axial symmetric mode k01. Inaddition, whereas 0.05 # r1 # 0.4, the lowest eigenvalue k11 is calculated with a betteraccuracy. For the same r1, increasing the semi-apex angle may lead to a slowerconvergence. If q1 ¼ q2 ¼ 12, the number of significant digits also decreases as r1 ! 1.This “shallow water” case is handled with 2-3 significant digits as q1 ¼ q2 ¼ 12, . . . ,14.

The presence of the two types of basic functions in the representation (36) makes itpossible to vary q1 and q2 to obtain a better approximation with the same total numberof basic functions Q ¼ q1 þ q2 Variations of q1 and q2 with a fixed Q $ 16 showed thata better accuracy of km1 can be expected for q2 . q1. In particular, this is true forsmaller liquid depths. For example, when the V-shaped tank is characterised byu0 ¼ 308 and r1 ¼ 0.9, the approximate k11 ¼ 54233738 can be obtained with eitherq1 ¼ q2 ¼ 12(Q ¼ 24) or q1 ¼ 7,q2 ¼ 12(Q ¼ 19).

4. Comparative analysis of the two methodA comparison of numerical experiments performed with the two different functionalbases shows, that the method based on the HPS is more accurate for smaller liquiddepths (0.6 # r1). However, larger liquid depths (r1 # 0.4) are better treated with thesecond method. This can clearly be seen for the L-shaped tanks: the calculations withthe method by the SFB keep robustness as the number of basic functions is increased,while the first method fails for larger dimensions. Generally speaking, the accuracy ofboth methods is similar only for V-tanks with 0:2 # r1 # 0:55; u0 . 10

0.Even though the number of basic functions is small, the two proposed methods give

an accurate approximation of the lowest eigenvalue k11. The lowest eigenvaluedetermines the lowest natural frequency by s11 ¼ ffiffiffiffiffiffiffiffiffigk11p . This frequency is of primaryinterest for modelling tower vibrations with a V-shaped tank. Therefore, we placedspecial emphasis on a comparison of the numerical results obtained by the two methodsfork11. The results are illustrated in Figure 4(a) and (b). Here, domains in the (r1,u0)-planeare identified, for which each of the methods gives the same number of significant digitswith twenty basic functions. One can see that the accuracy of the first method (HPS) maybecome low only for small u0 and r1, e.g. for large liquid depths. In the other cases, themethod guarantees a fast convergence and high accuracy. On the other hand, small u0

Figure 4.The number of significantfigures of k11 obtained forV-tanks with 20 basicfunctions by the methodsbased on the HPS (Case a)and the SF(Case b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0r1

10

7

7

5

5

5 4 3

66

6

7

7

8

8920

30

40

50

60

70

80

q 0 q 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

10

20

30

40

50

60

70

80

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0r1

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

10

20

30

40

50

60

70

80

(b)

EC25,6

534

and r1 are satisfactory handled by the second method (SFB). However, this methodconverges slowly as r1 ! 1 andu0 . 458, e.g. for small liquid depths.

5. The lowest natural sloshing frequencyThe natural sloshing frequencies sm1 are functions of the liquid depth h, the semi-apexangle u0 and the radii r1 and r0. For the V-tanks, an increasing of u0decreases thenon-dimensional eigenvalues km1 (the non-dimensional natural sloshing frequenciess2m1r0=g ¼ km1). For the L-tanks, an increasing of u0increases km1.

Dependence of km1 on the ratio r1/r0 is illustrated in Figure 5(a) and (b). Thesedemonstrate that the non-dimensional frequencies decrease as the ratio r1=r0 ¼ð1 þ tanðu0Þh=r1Þ decreases. In term of the fixed dimensional values of r1 and u0, thismeans that the non-dimensional sloshing frequencies decrease with decreasing theliquid depth h.

One interesting fact is that k01 < k21 for the studied geometric parameters inthe case of L-tanks. This is the same as for an upright circular cylindrical tank.

Figure 5.Eigenvalues km1 versus r1for L-shaped (Case a) and

V-shaped tanks (Case b)

0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

60

70

80

90

100q0 = π/6

r1

km1

0 0.2 0.4 0.6 0.8 1.00

0.5

1.0

1.5

2.0

2.5

3.0

3.5

θ0 = π/6

r1

κm1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.10

5

10

15

20

25

30

35

40θ0 = π/4

r1

κm1

0 0.2 0.4 0.6 0.8 1.00

0.5

1.0

1.5

2.0

2.5

3.0q0 = π/4

r1

km1

0.1 0.2 0.3 0.4 0.5 0.6 0.7

(a) (b)

0.8 0.9 1.00

5

10

15

20

25

30

35

40q0 = π/3

r1

km1

0 0.2 0.4 0.6 0.8 1.00

0.5

1.0

1.5

2.0

2.5q0 = π/3

r1

km1m=0m=1m=2m=3

m=0m=1m=2m=3

m=0m=1m=2m=3

m=0m=1m=2m=3

m=0m=1m=2m=3

m=0m=1m=2m=3

Natural sloshingfrequencies

535

Further, the natural sloshing frequencies are very close to those for the non-truncatedconical V-shape tank as 0 # r1 , 0.6. Truncation matters for r1 ! 1, i.e. forshallow-water sloshing.

The natural sloshing frequency an s11 of practical importance for the design ofwater towers (Damatty and Sweedan, 2006). Having in mind this fact, we present inTable III the values of k11 versus u0and r1. The corresponding computations have beendone to guarantee up to five significant digits. The dimensional natural sloshingfrequency s11 is computed from k11 by the formula:

s11 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigk11ðu0; r1=r0Þ

r0

s; ð38Þ

where g and r1 are not scaled by r0. The numerical data from Table III can therefore beused in both the structural design and the validation of other numerical methods.

6. Concluding remarksWe have proposed two efficient numerical-analytical methods for the computationof the natural sloshing frequencies and modes in truncated conical tanks. Thesemethods are based on the Ritz-Treftz variational scheme. Extensive numericalexperiments showed that these methods have different domains of applicability interms of the semi-apex angle, the liquid depth and the tank type (V- or L-shaped).

The methods may have slow convergence behaviour and even diverge for someL-shaped tanks. This fact can be explained by the singular behaviour of thenatural modes at the contact line formed by the waterplane and the conical walls.From a mathematical point of view, if the smooth bases would be augmented by aharmonic function, which has the specified singular behaviour, the convergencecan be considerably improved. Lukovsky et al. (1984) gave examples of such anaugmented function for two-dimensional spectral sloshing problems. Dedicatedmathematical studies are needed to specify a suitable singular function for ourcase.

Bauer (1982) and Dokuchaev (1964) compared the numerical data presented inthis paper with simplified analytical approximation, which were obtained for pureconical tanks with small semi-apex angles. Satisfactory agreement was observedonly for a small angle (u0 , 158). This agreement is consistent with theassumptions used by Bauer (1982) who replaced the planar mean liquid surface byspheric segment.

In future work, an emphasis should be placed on the shallow water sloshing. Thiscase requires a dedicated study, which has to be based on a nonlinear dissipativesloshing model. Further, a special analysis is needed for approximate natural modes,which enable handling singular behaviour at contact line formed by the mean freesurface and the rigid walls. Accounting for this singularity should improveconvergence.

Nonlinear phenomena are also of importance for resonant sloshing. Results ofthe present paper may be utilised to improve the multimodal technique (Lukovsky,1990; Faltinsen et al., 2000; Gavrilyuk et al., 2005) and to study the nonlinearsloshing in a truncated conical tank. This will be the main purpose of ourforthcoming studies.

EC25,6

536

r 1\u

0108

158

208

258

308

358

408

458

508

558

608

658

708

0.15

1.67

431.

5862

1.49

501.

4011

1.30

444

1.20

521.

1037

1.00

000.

8943

0.78

680.

6777

0.56

710.

4553

0.10

1.67

431.

5862

1.49

501.

4011

1.30

441.

2052

1.10

371.

0000

0.89

430.

7868

0.67

760.

5670

0.45

510.

151.

6743

1.58

621.

4950

1.40

111.

3044

1.20

521.

1037

1.00

000.

8941

0.78

650.

6772

0.56

650.

4547

0.20

1.67

431.

5862

1.49

501.

4011

1.30

441.

2051

1.10

350.

9996

0.89

350.

7857

0.67

630.

5655

0.45

360.

251.

6743

1.58

621.

4950

1.40

101.

3043

1.20

591.

1030

0.99

860.

8922

0.78

400.

6742

0.56

340.

4517

0.30

1.67

431.

5862

1.49

501.

4010

1.30

411.

2043

1.10

170.

9966

0.88

940.

7806

0.67

060.

5598

0.44

840.

351.

6743

1.58

621.

4950

1.40

081.

3034

1.20

271.

0990

0.99

270.

8845

0.77

500.

6647

0.55

410.

4433

0.40

1.67

431.

5862

1.49

491.

4002

1.30

171.

1994

1.09

380.

9857

0.87

620.

7660

0.65

560.

5456

0.43

590.

451.

6743

1.58

621.

4945

1.39

871.

2980

1.19

301.

0846

0.97

430.

8633

0.75

250.

6425

0.53

350.

4256

0.50

1.67

431.

5860

1.49

351.

3952

1.29

081.

1816

1.06

960.

9566

0.84

420.

7332

0.62

420.

5171

0.41

170.

551.

6743

1.58

561.

4908

1.38

771.

2774

1.16

251.

0461

0.93

040.

8171

0.70

670.

5996

0.49

540.

3936

0.60

1.67

431.

5842

1.48

421.

3730

1.25

401.

1320

1.01

100.

8932

0.78

000.

6715

0.56

760.

4676

0.37

070.

651.

6740

1.57

991.

4697

1.34

561.

2153

1.08

560.

9607

0.84

230.

7309

0.62

610.

5271

0.43

300.

3425

0.70

1.67

271.

5683

1.43

951.

2973

1.15

441.

0181

0.89

150.

7751

0.66

810.

5693

0.47

750.

3910

0.30

860.

751.

6674

1.53

911.

3808

1.21

721.

0632

0.92

420.

8002

0.68

970.

5905

0.50

070.

4183

0.34

170.

2691

0.80

1.64

721.

4709

1.27

411.

0918

0.93

390.

7994

0.68

440.

5850

0.49

780.

4202

0.34

990.

2851

0.22

410.

851.

5781

1.32

511.

0945

0.90

840.

7606

0.64

160.

5438

0.46

140.

3906

0.32

840.

2727

0.22

180.

1741

0.90

1.37

021.

0471

0.81

990.

6599

0.54

230.

4521

0.38

000.

3207

0.27

040.

2267

0.18

790.

1526

0.11

970.

950.

8631

0.59

550.

4461

0.35

110.

2849

0.23

560.

1970

0.16

570.

1394

0.11

670.

0966

0.07

840.

0615

Table III.k11 versus u0and r1 for

V-shaped tanks

Natural sloshingfrequencies

537

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Appendix(The Appendix Figure follows overleaf.)

Natural sloshingfrequencies

539

Corresponding authorA. Timokha can be contacted at: alexander.timokha@ntnu.no

Figure A1.Values of nmn versus u0

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