Natural sloshing frequencies in rigid truncated conical tanks

Natural sloshing frequencies in rigid truncated conical tanks

I. Gavrilyuk

Staatliche Studienakademie Thu¨ ringen- Berufsakademie Eisenach, University of Cooperative Education, Eisenach, Germany

M. Hermann

Friedrich-Schiller-Universita¨ t Jena, Jena, Germany

I. Lukovsky and O. Solodun

Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine, and

A. Timokha

Centre 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 numerical analytical methods for engineering computation of natural sloshing frequencies and modes i the case of truncated circular conical tanks.

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

Findings– Comparative numerical analysis detects the limit of applicability of variational methods in terms of the semi-apex angle and the ratio between radii of the mean free surface and the circular bottom. 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 the lower natural frequencies for relevant tanks. For V-shaped tanks, dependencies of the lowest natural frequency versus the semi-apex angle and the liquid depth are described.

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

Originality/value– Although variational methods have been widely used for computing the natural sloshing 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.

KeywordsNumerical analysis, Frequencies, Liquid flow containers Paper typeResearch paper

1. Introduction

The number of constructions carrying a large liquid mass is enormous. Coupling the structure 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 DFG 436UKR 113/33/00). The last author is grateful for the sponsorship of the Alexander von Humboldt Foundation.

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

Engineering Computations:

International Journal for Computer-Aided Engineering and Software

Vol. 25 No. 6, 2008 pp. 518-540

qEmerald Group Publishing Limited 0264-4401

DOI 10.1108/02644400810891535

(eigen-) frequencies and modes. Being subject of spacecraft applications, the natural sloshing frequencies and modes for conical tanks were studied in 1950-1960’s years of the past century. Special attention was paid to an estimate of the resulting hydrodynamic 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 conicalbottom shapes, typical examples are demonstrated by Damatty and Sweedan (2006). Owing to seismic events, liquid motions in a water tank on the supporting tower cause severe hydrodynamical loads. In the modelling of these loads, equivalent mechanical systems (Damatty and Sweedan, 2006; Duttaet al., 2004; Shrimali and Jangid, 2003) can be used.

These systems relate liquid dynamics to oscillations of a pendulum or a spring-mass system. The eigenfrequencies of the equivalent systems should coincide with the lower natural sloshing frequencies and, therefore, an accurate prediction of the sloshing frequencies and modes is needed (Damattyet al., 2000; Dutta and Laha, 2000; Tang, 1999). This can be done by various Computational Fluid Dynamics (CFD) methods or by using semi-empirical approximate formulae (Damattyet al., 2000; Gavrilyuket al., 2005). Although the CFD-methods demand lots of CPU’s power, they are primary employed in engineering practise to guarantee a substantial precision. When concentrating on linear and nonlinear sloshing in V-shaped pure conical tanks, Gavrilyuket al. (2005) showed that an alternative might consist in a semianalytical solution method. The method keeps the accuracy of the CFD-methods, but remains CPU-efficient and simple in use. The constructed numerical-analytical solutions may facilitate development of nonlinear sloshing theories (Lukovsky and Bilyk, 1985;

Lukovsky, 1990, 2004; Bauer and Eidel, 1988; Faltinsenet al., 2000, 2003; Gavrilyuk et al., 2005, 2006). These theories are of importance for studying a resonant coupled vibration of a tower and the contained liquid.

In order to find the natural sloshing frequencies and modes, a spectral boundary value problem (Lukovskyet al., 1984; Ibrahim, 2005) has to be solved. When the tank is V- andL- shaped, the boundary value problem has no analytical solutions. Isolated analytical solutions exist only for pure (non-truncated) conical V-tanks. Such an example has been specified by Levin (1963) for the two lowest natural modes and the semiapex angleu0¼458. These modes are characterised by the wave numberm¼1 in angular direction. Dokuchaev and Lukovsky (1968) generalised this result. They showed that analogous analytical solutions exist asu₀¼arctanðpffiffiffiffim

Þ. Mikishev and Rabinovich (1968) and Feschenkoet al. (1969) used these solutions for evaluation of their numerical algorithms. Furthermore, simple approximate analytical solutions for pure conical V-tanks can be obtained by replacing the planar waterplane by a spheric segment. In that case, the spectral boundary value problem admits separation of variables in the spherical coordinate system. This has been realised by Dokuchaev (1964) and Bauer (1982). A satisfactory agreement with experimental data by Mikishev and Dorozhkin (1961) and Bauer (1982) was reported asu0¼158.

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

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

In Section 3, the main focus is on the HPS. For the V-shaped tanks, we show that 11-17 HPS provide 4-6 significant digits of the lower natural frequencies for the semi-apex angles, which are smaller than 758 and larger than 108. For the V- and L-shaped tanks which are characterised by semi-apex angles larger than 758, the same number of the HPS guarantees 3-4 significant digits. For the L-shaped tanks with semi-apex angles smaller than 758, convergence to the natural frequencies depends on the ratio between the radii of the mean liquid plane and the circular bottom. The method is not very efficient (only about 2-3 significant digits can be obtained with 17-20 basic polynomials) when the semi-apex angle is smaller than 608and the ratio between the specified radii is smaller than 1/2. The slow convergence for theL-shaped tanks can partially be attributed to a singular asymptotic behaviour of the natural modes at the contact line formed by the waterplane and conical walls.

Lukovsky (1990) has proposed a non-conformal mapping technique to develop the multimodal method for the nonlinear sloshing problem in a non-cylindrical tank.

Lukovsky and Timokha (2002) and Gavrilyuket al.(2005) have realised this technique for a non-truncated V-tank. The natural sloshing modes were then approximated by a special functional basis (SFB). In Section 4, we use the curvilinear coordinate system proposed by Gavrilyuk et al.(2005) and generalise their results on natural sloshing frequencies to the case of truncated V- and L-shaped conical tanks. For typical geometrical configurations of elevated water tanks (a V-shaped cone with the semi-apex angle between 308and 608), the method shows faster convergence behaviour than in the case of Section 3. Six significant digits of the lowest sloshing frequency can be 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 geometrical shape of truncated conical tanks. Bearing in mind that the relevant water tanks are of a V-shape, we present the lowest spectral parameter (with a accuracy of five significant digits) versus the semi-apex angle and the ratio between the radii of the mean water plane and the bottom.

2. Statement of the problem

2.1 Differential and variational formulations

As it is typically assumed in sloshing theory (Ibrahim, 2005), we consider an ideal incompressible liquid with irrotational flow that partly occupies an earth-fixed rigid conical tank with the semi apex angleu0. The mean (hydrostatic) liquid shape coincides with the domainQ0as it is shown in Figure 1. The gravity acceleration is directed downwards along the symmetry axisOx.The wetted conical walls are denoted byS1. The circle S₂ is the tank bottom, S¼S₁<S₂ and P

0 is the non-perturbed (hydrostatic) waterplane. The origin is superposed with artificial apex of the conical surface. Henceforth, the problem is considered in a sizeless statement assuming thatr₀ (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 byr0implieshUh/r0!0 (his the liquid depth) andr₁Ur₁=r₀. The non-dimensional radiusr1and the angleu0

completely determine the geometric proportions ofQ0. The limitr1!1 implies that h!0, ie. the water becomes shallow. At a fixedr1, the scaled liquid depthhtends to zero asu0!p/2.

Linear sloshing in a motionless tank is governed by the following boundary value problem (Lukovskyet al., 1984; Ibrahim, 2005):

Df¼0 inQ₀; ›f

›x ¼›f

›t; ›f

›t þgf ¼0 onX

0; ›f

›n¼0 onS;

Z P

0

›f

›xdS¼0;

ð1Þ wheref(x,y,z,t) is the velocity potential,x¼f(y,z,t) describes the free surface,nis the outer normal toSand g is the gravity acceleration scaled byr0ðgUg=r0Þ. The initial conditions:

fðy;z;t0Þ ¼F0ðy;zÞ; ›f

›tðy;z;t0Þ ¼F1ðy;zÞ ð2Þ at an instant timet¼t0, determine a unique solution of equation (1). The functionsF0

andF1define initial displacements of the free surface and its velocity, respectively.

2.2 Natural sloshing modes

The solution of equation (1) is associated with the free-standing waves:

fðx;y;z;tÞ ¼cðx;y;zÞexpðistÞ; i²¼21 ð3Þ wherea is the natural sloshing frequency andc(x,y,z) is the so-called natural mode.

Inserting equation (3) into equation (1) leads to the spectral problem:

Dc¼0 inQ₀; ›c

›x¼kconX

0; ›c

›n¼0 onS;

Z P

0

›c

›xdS¼0 ð4Þ

Figure 1.

Hydrostatic liquid domains inL- and V-haped tanks

x x

O z y

S₀

S₀

r₁

r₁ r₀

r₀

S₁ Q₀

h

S₂

h g

y z O

q₀

q₀

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521

where the eigenvaluekis defined by:

k¼s²

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 constant function, projections of the eigenfunctions,fnðy;zÞ ¼cnjP

0constitute an orthogonal basis in the mean-squares metrics. Because the velocity potential f satisfies the volume conservation condition (see, the last integral equality in equation (1)), getting known {kn} and {cn}, one can represent the solution of equation (1) and equation (2) as a Fourier series by fn¼cnðx;y;zÞexpðisntÞ, namely, as a superposition of the free-standing waves.

2.3 On the Ritz-Treftz schem for the spectral problem (4)

Problem (4) admits a minimax variational formulation (Feschenkoet al., 1969; Morand and Ohayon, 1995), which is based on the positive functional:

KðcÞ ¼ R

Q0ð7cÞ²dQ RP

0

c²dS ð6Þ

under the supplementary conditionRP

0

cdS¼0. In that case, the absolute minimum of the functional equation (6) coincides with the lowest eigenvalue of the spectral problem (4). Furthermore, the necessary condition for an extrema of equation (6) leads to the variational equation:

Z

Q0

ð7c;7hÞdQ2k Z

P

0

chdS¼0 ð7Þ

with respect to a non-constant functionc,wherehis 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 smooth harmonic functions:

cðx;y;zÞ ¼X^q

k¼1

a_kB_kðx;y;zÞ: ð8Þ

Substituting equation (8) into equation (7) and usingB_i(i¼1,. . .,q) as test-functions, one obtains the spectral matrix problem:

X^q

k¼1

ð{aik}2k{bik}Þa_k¼0; i¼1;. . .;q ð9Þ

Here, the elements of the non-negative matricesA¼{aik}; B¼{bik} are computed by the formulae:

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aik¼ Z

Q0

ð7B_i;7B_kÞdQ¼ Z

P

0þS

›B_i

›n B_jdS; bik¼ Z

P

0

B_iB_kdS; ð10Þ

and the approximate eigenvalues are roots of the equation:

detðA2kBÞ ¼0; ð11Þ

which appears as the necessary solvability condition of system (9). By increasing the dimensionq the non-zero roots of equation (11) converge (from above) to the lower eigenvalues of equation (4). Approximate eigenfunctions (8) are formed by the eigenvectors of equation equation (9), {a_k,k¼1,. . .,q}.

A key difficulty of the Ritz-Treftz scheme consists of establishing a suitable analytical functional basis {Bk}. The completeness of functional sets significantly depends on the actual shape ofQ0. To the author’s knowledge, there are two types of analytical functional sets, which may be adapted to the studied case. The first set is proposed in the book by Lukovsky et al. (1984). It follows from a separation of variables in the Laplace equation done in the spherical coordinate system. These harmonic solutions are of polynomial structure (harmonic polynomial solutions, HPS) in the Cartesian coordinate system. Their completeness is proved for all star-shaped domains with respect to the origin. Being rewritten in a cylindrical coordinate system, the HPS admit the separation of the angular coordinate and keep polynomial structure with regard to the remaining coordinates, i.e. in projections on a meridional cross-section. A specific functional basis (SFB) is presented by Gavrilyuket al.(2005).

It is derived in the cylindrical coordinate system combined with a non-conformal mapping of the meridional cross-section. The functional set is harmonic and satisfies the zero-Neumann condition on the conical walls. Furthermore, we utilise these two functional sets in the Ritz-Treftz scheme to solve the spectral problem (4). The tree-dimensional problem and its variational formulation (7) are thereby reduced to two dimensions by separating the angular-type variable.

3. Ritz-Treftz method based on the HPS

We use the cylindrical coordinate system (X,j,h) linked with the original Cartesian coordinates by:

x¼XþX₀; y¼jcosh; z¼jsinh: ð12Þ Here, the lagX0 along the vertical axis is introduced to superpose the origin of the cylindrical coordinate system with the waterplane. The solution of equation (4) is represented in the following form:

cðX;j;hÞ ¼wmðX;jÞ

sinmh cosmh

!

; m¼0;1;2. . . ð13Þ

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

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›

›X j›wm

›X

þ ›

›j j›wm

›j

2m²

j wm¼0 inG; ›wm

›X ¼kmwmon L₀;

›wm

›n ¼0 onL; jwmðX;0Þj,1;

Z

L0

j›w0

›X dj¼0:

ð14Þ

Problem (14) is defined in a meridional plane ofQ0andL¼L1þL2(Figure 2). This means that the eigenvalues of the original three-dimensional problem constitute a two-parametric set k¼k_miðm¼0;1;. . .;i¼1;2;. . .Þ, where i$1 enumerates the eigenvalues of equation (14) in ascending order. The corresponding eigenfunctions of equation (4) take the form equation (13) withwm¼w_miðX;jÞ.

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

w^ðmÞ_k ðX;jÞ ¼2ðk2mÞ!

ðkþmÞ! R^kP^ðmÞ_k X

R ;k$m; R¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X²þj²

p ; ð15Þ

WhereP^ðmÞ_k are Legendre’s functions of first kind. The functions {w^ðmÞ_k }are the solutions of the first equation of (14). It can be shown thatw^ðmÞ_k has indeed a polynomial structure in terms ofXandj. The first functions of the set equation (15) take the form:

w^ð0Þ₀ ¼1; w^ð0Þ₁ ¼X; w^ð0Þ₂ ¼X²2j²

2 ; ::: ðm¼0Þ;

w^ð1Þ₁ ¼j; w^ð1Þ₂ ¼Xj; w^ð1Þ₃ ¼X²j2j³

4 ; ::: ðm¼1Þ;

w^ð2Þ₂ ¼j²; w^ð2Þ₃ ¼Xj²; w^ð2Þ₄ ¼X²;j²2j⁴

6 ; ::: ðm¼2Þ;

w^ð3Þ₃ ¼j³; w^ð3Þ₄ ¼Xj³; w^ð3Þ₅ ¼X²;j³2j⁵

8 ; ::: ðm¼3Þ:

The computation of {w^ðmÞ_k } can be realised by the following recurrence relations:

Figure 2.

Meridional planes of L- and V-shaped tanks

X

G

G h

h O

O 1

1

X L₀

L₀ r₁

q₀

q₀ r₁

L₁

L₁

L₂

L₂

x

x

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›w^ðmÞ_k

›X ¼ ðk2mÞw^ðmÞ_k21; j›w^ðmÞ_k

›j ^¼kw

ðmÞ

k 2ðk2mÞXw^ðmÞ_k21;

ðk2mþ1Þw^ðmÞ_kþ1þ ð2kþ1ÞX w^ðmÞ_k 2ðk2mÞðX²þj²Þw^ðmÞ_k21; ðk2mþ1Þjw^ðmþ1Þ_k ¼2ðmþ1Þ ðX ²þj²Þw^ðmÞ_k212X w^ðmÞ_k

:

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

wmðX;jÞ ¼X^q

k¼1

a^ðmÞ_k w^ðmÞ_kþm21ðX;jÞ; ð16Þ

and spectral matrix problems:

X^q

k¼1

a^ðmÞ_ik

2kmb^ðmÞ_ik

a_k¼0 ði¼1;. . .;qÞ;

deta^ðmÞ_ik 2kmb^ðmÞ_ik ¼0;

ð17Þ

following from (9). The elementsa^ðmÞ_ik andb^ðmÞ_ik are computed for theL-cones by the formulae:

a^ðmÞ_ij ¼ Z r1

0

j›w^ðmÞ_iþm21

›X w^ðmÞ_jþm21

!

X¼0

djþ Z 0

2h

j›w^ðmÞ_iþm21

›j ^w

ðmÞ jþm21

!

j¼tanu₀X2r1

dX

2tanu0

Z 0 2h

j›w^ðmÞ_iþm21

›X w^ðmÞ_jþm21

!

j¼tanu₀X2r1

dX2 Z 1

0

j›w^ðmÞ_iþm21

›X w^ðmÞ_jþm21

!

X¼2h

dj;

b^ðmÞ_ij ¼ Z r1

0

jw^ðmÞ_iþm21w^ðmÞ_jþm21

X¼0dj; and for the V cones by the formulae:

a^ðmÞ_ij ¼ Z 1

0

j›w^ðmÞ_iþm21

›X w^ðmÞ_jþm21

!

X¼0

djþ Z 0

2h

j›w^ðmÞ_iþm21

›j ^w

ðmÞ jþm21

!

j¼tanu0Xþ1

dX

2tanu₀ Z 0

2h

j›w^ðmÞ_iþm21

›X w^ðmÞ_jþm21

!

j¼tanu0Xþ1

dX2 Z r1

0

j›w^ðmÞ_iþm21

›X w^ðmÞ_jþm21

!

X¼2h

dj;

b^ðmÞ_ij ¼ Z 1

0

jw^ðmÞ_iþm21w^ðmÞ_jþm21

X¼0dj:

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For each fixed m, the second equation of (17) hasqpositive roots (n¼1,2,. . .,q). The multiplicity should be accounted for. Since the Ritz-Treftz method implies the minimisation of a functional, the approximate valueskmnconverge from above. This makes it possible to check the convergence by the number of significant digits, which do not change as increases. The method gives the best approximation for the lowest eigenvaluekm1.

Convergence. Our numerical experiments were primary dedicated to eigenvalues km1, 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; Gavrilyuk et al., 2005). In the case of V-tanks, the method shows a fast convergence tokm1and provides satisfactory accuracy forkm2andkm3, too. It shows a slower convergence for the L-tanks. Furthermore, the convergence depends not only on the tank type (V orL-shaped), but also on the semi-apex angleu0and the dimensionless parameter 0 , r1,1. The results in Table I (A) exhibit a typical convergence behaviour for the V-tanks with 108 #u0#758and 0.2#r1#0.9. The table shows stabilisation of 5-6 significant digits as q$14. The best accuracy is established for the lowest spectral parameterk11. The accuracy grows withqform–0. However, computations of the value k01, which is responsible for the axial-symmetric natural mode, may become unstable forq.17. This explains why we do not present numerical results on this eigenvalue forq¼20 Whilem–1 and 0.2#r1#0.9, increasingu0.758leads to a slower convergence. In this case,q¼17,. . .,20 guarantees only 3-4 significant digits (necessary engineering accuracy). The same number of basic functions keeps this number of significant digits for the tanks withr₁,0.2 and 108 #u0#758. Whenr₁ tends to zero (truncated tank is close to a non-truncated one), the approximationskm1

were validated by numerical results reported by Gavrilyuket al.(2005) as well as by experimental data given in Bauer (1982).

In Table I (B), typical convergence behaviour for the L-shaped tanks with 108 #u0#758 is presented. A comparative analysis of parts (A) and (B) illustrates that the method is in the latter case less efficient. In particular, computations ofk01are not so precise. Moreover, 18-20 basic functions lead to 4-5 significant digits ofkm1only for r1$0.4. This is not the case for lower values of r1. Furthermore, when r1#0.2,q¼17,. . .,20 can only guarantee 2-3 significant digits for k11. The slower convergence for the L-shaped tanks can in part be clarified by the occurrence of singular first derivatives of the eigenfunctionscmat an inner vertex betweenL0andL1

(see the mathematical results by Lukovskyet al., 1984). The HPS are smooth in the (j, X)-plane, and, therefore, do not capture this singular behaviour. The singularity disappears when the corner angle is less than 908. This occurs only for theL-shaped tanks. The V-shaped tanks are characterised by a similar singularity at the vertex formed byL1andL2. However, because the natural modes (eigenfunctionscm)should

“decay” exponentially downward, the method may be sensitive with respect to that singularity only for shallow water. Our numerical experiments confirm this fact asr1,0.1.

3.1 Ritz-Treftz method based on the SFB

The nonlinear resonant sloshing is effectively studied by multimodal methods. As it is shown by Lukovsky (1975), Lukovsky and Timokha (2002), these methods require analytical expressions for the natural modes, which:

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AB qr1¼0.2r1¼0.4r1¼0.6r1¼0.8r1¼0.9Qr1¼0.2r1¼0.4r1¼0.6r1¼0.8r1¼0.9 k01 25.0324154.5778334.15235223.8055012.961399870.0633715.140946.8999864.56099112.683411 53.3947793.3922863.3848653.1411712.2003421050.4676211.124496.6722664.5511502.683117 83.3856033.3855923.3818453.1388612.1974381239.0611110.185636.6667044.5508772.682932 113.3856003.3855903.3818273.1387182.1972381432.0757210.017456.6656784.5505902.682796 143.3856003.3855903.3818223.1386652.1971621627.6490410.016126.6650584.5504222.682751 173.3856003.3855903.3818193.1386442.1971381825.1529910.014266.6648864.5503492.682730 k11 21.3436311.3357231.2999771.0072610.607340816.504855.6948333.5167821.6618150.726555 51.3044131.3017931.2541570.9344120.5425031013.947815.6337303.5158611.6616840.726507 81.3043781.3016941.2540540.9338850.5422841213.487675.6319043.5155471.6616080.726489 111.3043771.3016921.2539820.9338350.5422571412.247245.6305603.5153651.6615590.726484 141.3043771.3016871.2539720.9338230.5422511611.663505.6298983.5152691.6615510.726482 171.3043771.3016861.2539690.9338190.5422491811.450725.6297773.5152371.6615370.726480 201.3043771.3016861.2539670.9338170.5422482011.331835.6298883.5152141.6615320.726478 k21 22.4437392.4243902.866332.1913711.572048845.024119.7978045.9509253.7242511.923234 52.2635502.2633512.2550962.0156021.3619051028.652509.0961315.9419133.7240491.923019 82.2631512.2630872.2550042.0149231.3609281225.183948.9753835.9411073.7237081.922950 112.2631502.2630872.2549762.0148381.3608371424.771668.9680275.9410023.7236401.922930 142.2631502.2630862.2549722.0148111.3608051621.512738.9666575.9406513.7235911.922913 172.2631502.2630862.2549692.0148011.3607951819.738588.9665315.9406033.7235491.922899 202.2631502.2630862.2549682.0147961.3607902017.760238.9665245.9405893.7235421.922895 k31 23.5331703.5208823.4590483.3161982.6974268139.879815.670558.1460785.6340693.404035 53.1815303.1812993.1795413.0471792.3290221083.8225412.954928.0467495.6339963.403568 83.1802513.1802493.1790803.0467422.3270441264.5885312.229648.0449165.6335653.403460 113.1802493.1802483.1790773.0466542.3268121453.4107612.088558.0444595.6334983.403411 143.1802493.1802473.1790743.0466202.3268121653.1387312.079648.0442345.6333773.403360 173.1802493.1802473.1790733.0466062.3267901838.8551912.084008.0440285.6333433.40336 203.1802493.1802473.1790733.0465992.3267792036.4674912.084908.0439885.6333213.403328 Notes:Column(A)isforaV-shapedtank,(B)correspondstoaL-shapedtank,u0¼308

Table I.

Convergence to km1,m¼0,1,2,3 for differentr1versus the number of basic functions q (16)

Natural sloshing frequencies

527

. are analytically expandable over the waterplane;

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

. can be transformed to a curvilinear coordinate system (x1,x2,x3), in which the free surface is governed by the normal form representationx1¼fðx~ ₂;x3;tÞ.

An example of suitable approximate natural modes for non-truncated V-tanks is given by Lukovsky (1990) and Gavrilyuket al.(2005). The present section generalises these results.

Curvilinear coordinate system. The non-Cartesian parametrisation proposed by Gavrilyuk et al (2005) links the x,y,z coordinates withx1,x2,x3as follows:

x¼x1; y¼x1x2cosx3; z¼x1x2sinx3: ð18Þ Thus, the variablex3¼his the polar angle in theOyz-plane and corresponds tohin equation (12). Figure 3 demonstrates that the hydrostatic liquid domainQ0takes 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 the sides h¼x₁₀2x₀and x₂₀ ¼ tanu0 in theOx₂x₁-plane. Here, the radius of the undisturbed water plane isrt¼1 for the V-tanks andrt¼r1for theL-tanks. Having presented:

wðx₁;x2;x3Þ ¼cmðx₁;x2Þ

sinm x₃ cosm x3

!

; m¼0;1;2;. . . ð19Þ

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

p›²cm

›x²₁ þ2q ›²cm

›x1›x2

þs›²cm

›x²₂ þd›cm

›x2

2m²ccm¼0 in G*; ð20Þ

p›cm

›x₁ þq›cm

›x₂ ¼kmpcm on L*

0; ð21Þ

Figure 3.

Meridional cross-sections of the original and transformed domains

x x₁ x₂₀ r₀ L₀ x₁ L₀

x₁₀

x₂ x

x₂₀ x₂ y

y L₀

q₀

q₀ L

0

L₁

L₂ L₂

L₂

L₂ L₁

L₁ L₁

r₁ r₁

G G∗

g G∗

r₀

x₀

x₀ x₁₀ 0

0

O O

∗

∗

∗

∗

∗

∗

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s›cm

›x2

þq›cm

›x1

¼0 on L*

1; ð22Þ

p›cm

›x1

þq›cm

›x2

¼0 on L*

2; ð23Þ

jcmðx₁;0Þj,1; m¼0;1;2;. . .; ð24Þ

Z x20 0

c0x₂dx₂¼0; ð25Þ

whereG*¼{ðx₁;x₂Þ:x₀#x₁#x₁₀;0#x₂#x₂₀}, W^ð0Þ_k , d¼1þ2x²₂;c¼1=x₂ and the boundary ofG* consists of the portionsL*

0;L*

1 andL*

2.

Particular solutions of equation(20) and equation(22). Gavrilyuket al.(2005) studied a spectral problem, which is similar to equations (20)-(25). Following results by Eisenhart, they proved that equation (20) and equation (22) allow together for the separation of the spatial variablesx1andx2. For our problem, this separation leads to the following particular solutions:

xⁿ₁T^ðmÞ_n ðx₂Þ and T^ðmÞ_n ðx₂Þ

x^1þn₁ ; n$0: ð26Þ

In order to determineT^ðmÞ_n , we have to consider the following homogeneous boundary value problem, which depends on the real parametern:

x²₂ð1þx²₂ÞT⁰⁰_n^ðmÞþx2ð1þ2x²₂22nx²₂ÞT⁰⁰_n^ðmÞþ ½nðn21Þx²₂2m²T^ðmÞ_n ¼0; ð27Þ T⁰⁰_n^ðmÞðx₂₀Þ ¼n ^x20

1þx²₂₀ T^ðmÞ_n ðx₂₀Þ; jT^ðmÞ_n ð0Þj,1: ð28Þ It can be shown that the problem (27) and (28) has only nontrivial solutions for a countable set of valuesn¼nmn.0ðm¼0;1;. . .;n¼1;2;. . .Þ:

The second class of functions,T^ðmÞ_n , appears only in the case ofx0–0, i.e. when the conical tank is truncated. Computation of T^ðmÞ_n leads to the followingn-parametric problem:

x²₂ð1þx²₂ÞT

00ðmÞ

þx₂ð1þ4x²₂22nx²₂ÞT⁰_nþ ½ðnþ1Þðnþ2Þx²₂2m²T^ðmÞ ¼0; ð29Þ

T^0ðmÞðx₂₀Þ þ ðnþ1Þ x20

1þx²₂₀T^ðmÞðx₂₀Þ ¼0: ð30Þ Obviously, nontrivial solutions of equation (29) and equation (30) exist only for a countable set of nonnegative valuesn.

Let us now show that the solution of equations (27) and (29) can be expressed in terms 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 in equations (27) and (29) by m¼ ð1þx²₂Þ^2ð1=2Þ and substitute yðmÞ ¼mⁿTðmÞ and

Natural sloshing frequencies

529

yðmÞ ¼m²¹²ⁿTðmÞinto equation (27) and equation (29), respectively. This reduces the two equations to the same well-known differential equation:

ð12m²Þy⁰⁰ðmÞ22my⁰ðmÞ þ nðnþ1Þ2 m² 12m²

yðmÞ ¼0;

whose solutions coincide with the Legendre function of first kind, i.e.yðmÞ ¼P^ðmÞ_n ðmÞ.

Furthermore, treating the boundary conditions (28) and (30) in the same way and using the substitutionm ¼ cosu, the following common equation is obtained:

›P^ðmÞ_n ðcosuÞ

›u _u¼u

0

¼0: ð31Þ

This equation can be considered as a transcendental equation for the computation of the values {nmn}- Appendix (Figure A1) presents the first 12 values of {nmn} (m¼0,1,2,3) versusu0.

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

T^ðmÞ_nmkðx₂Þ ¼ ð1þx²₂Þ^nmk2 P^ðmÞ_nmk 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1þx²₂ q 0 B@

1

CA; ð32Þ

T^ðmÞ_nmkðx₂Þ ¼ ð1þx²₂Þ^212n2mkP^ðmÞ_nmk 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1þx²₂ q 0 B@

1

CA: ð33Þ

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

W^ðmÞ_k ðx₁;x2Þ ¼N^ðmÞ_k xⁿ₁^mkT^ðmÞ_nmkðx₂Þ; W^ðmÞ_k ðx₁;x2Þ ¼N^ðmÞ_k x²¹²ⁿ₁ ^mkT^ðmÞ_nmkðx₂Þ: ð34Þ Here,N^ðmÞ_k andN^ðmÞ_k are multipliers which are chosen to satisfy the following condition:

1¼ kW^ðmÞ_k jj²_L*

2þL*

0

¼ kW^ðmÞ_k jj²_L*

2þL*

0

¼ Z x20

0

x2½ðW^ðmÞ_k j_x1¼x10Þ²þ ðW^ðmÞ_k j_x1¼x0Þ²dx₂

¼ Z x20

0

x2½ðW^ðmÞ_k jx₁¼x10Þ²þ ðW^ðmÞ_k j_x1¼x0Þ²dx₂:

ð35Þ

equation (35) says that W^ðmÞ_k and W^ðmÞ_k . have the unit norm (in the mean-squares metrics) on the boundary L*

2þL*

0, where equations (21) and (23) should be approximately satisfied. Explicit formulae for these normalising multipliers have the form:

EC 25,6

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N^ðmÞ_k ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x²ⁿ₁₀^mkþx²ⁿ₀^mk

q 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rx20

0 ð1þx²₂Þ^nmk P^ðmÞ_n

mk

2

dx₂

r ;

N_k^ðmÞ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x²²²²ⁿ₁₀ ^mkþx²²²²ⁿ₀ ^mk

q 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rx20

0 ð1þx²₂Þ^212nmkP^ðmÞ_nmk 2

dx2

r :

The case m ¼ 0 requires, in addition, the volume conservation condition (25).

This implies a re-definition of the functions W^ð0Þ_k andW_k^ð0Þ by: W^ð0Þ_k UW^ð0Þ_k 2c^ð0Þ_k , W_k^ð0ÞUW_k^ð0Þ2c^ð0Þ_k ;where:

c^ð0Þ_k ¼ 2 x²₂₀

Z x20

0

x₂W^ð0Þ_k ðx₁₀;x₂Þdx₂; c^ð0Þ_k ¼ 2 x²₂₀

Z x20

0

x₂W_k^ð0Þðx₁₀;x₂Þdx₂:

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

cmðx₁;x₂Þ ¼X^q1

k¼1

a^ðmÞ_k W^ðmÞ_k þX^q2

l¼1

a_l^ðmÞW^ðmÞ_l : ð36Þ By separating the x3 coordinate in variational formulation (7) (after substitution of equation (19)), representation (36) leads to the m-parametric family of spectral problems:

X^Q

k¼1

a^ðmÞ_ik

2kmb^ðmÞ_ik

a_k¼0 ði¼1;. . .;QÞ;

detⁿa~^ðmÞ_ij ^o2kmⁿb~^ðmÞ_ik ^o ¼0:

ð37Þ

The spectral problem (37) hasQ¼q1þq2 eigenvalues. Because the representation (36) contains two types of functions, namely W^ðmÞ_k and, W_l^ðmÞthere exist four sub-matrices ofⁿa~^ðmÞ_ij ^oandⁿb~^ðmÞ_ij ^osuch that:

a~^ðmÞ_ij ¼

a^ðmÞ_ij1 a^ðmÞ_ij2 a^ðmÞ_ij3 a^ðmÞ_ij4 0

@

1

A; b~^ðmÞ_ij ¼

b^ðmÞ_ij1 b^ðmÞ_ij2 b^ðmÞ_ij3 b^ðmÞ_ij4 0

@

1 A:

The elementsⁿa^ðmÞ_ijs ^oandⁿb^ðmÞ_ijs ^o,s¼1,. . .,4, are computed by the formulae:

a^ðmÞ_ij1 ¼ Z x20

0

x²₁x₂›W^ðmÞ_i

›x₁ 2x₁x²₂›W^ðmÞ_i

›x₂

!

x1¼ht

W^ðmÞ_j dx₂

2 Z x20

0

x²₁x₂›W^ðmÞ_i

›x₁ 2x₁x²₂›W^ðmÞ_i

›x₂

!

x1¼hb

W^ðmÞ_j dx₂;

Natural sloshing frequencies

531

a^ðmÞ_ij2 ¼ Z x20

0

x²₁x2

›W^ðmÞ_i

›x1

2x1x²₂›W^ðmÞ_i

›x2

!

x1¼ht

W_j^ðmÞdx2

2 Z x20

0

x²₁x₂›W^ðmÞ_i

›x1

2x₁x²₂›W^ðmÞ_i

›x2

!

x1¼h_b

W_j^ðmÞdx₂;

a^ðmÞ_ij3 ¼ Z x20

0

x²₁x2

›W_i^ðmÞ

›x₁ 2x1x²₂›W_i^ðmÞ

›x₂

!

x1¼ht

W^ðmÞ_j dx2

2 Z x20

0

x²₁x2

›W_i^ðmÞ

›x₁ 2x1x²₂›W_i^ðmÞ

›x₂

!

x1¼hb

W^ðmÞ_j dx2;

a^ðmÞ_ij4 ¼ Z x20

0

x²₁x₂›W_i^ðmÞ

›x1

2x₁x²₂›W_i^ðmÞ

›x2

!

x1¼h_t

W_j^ðmÞdx₂

2 Z x20

0

x²₁x₂›W_i^ðmÞ

›x₁ 2x₁x²₂›W_i^ðmÞ

›x₂

!

x1¼h_b

W_j^ðmÞdx₂;

b^ðmÞ_ij1 ¼h²_t Z x20

0

x₂W^ðmÞ_i W^ðmÞ_j

x1¼ht

dx₂; b^ðmÞ_ij2 ¼h²_t Z x20

0

x₂W^ðmÞ_i W_j^ðmÞ

x1¼ht

dx₂;

b^ðmÞ_ij3 ¼h²_t Z x20

0

x2W_i^ðmÞW^ðmÞ_j

x1¼ht

dx2; b^ðmÞ_ij4 ¼h²_t Z x20

0

x2W_i^ðmÞW_j^ðmÞ

x1¼ht

dx2: In the case ofL- and V-tanks, we haveh_t ¼r₁=tanu0,h_b¼1=tanu0andh_t¼1=tanu0, h_b¼r₁=tanu0, respectively.

Convergence.Column A in Table II shows a typical convergence behaviour in the case of V-tanks with 108 #u0#758 and 0.2#r1#0.9. When 0.2#r1#0.55, the method generates 4-5 significant digits of km1for q¼q1¼q2¼7,. . .,10 (14,. . .,20 basic functions). This is consistent with the convergence results in Section 3. However, the SFB keeps also a fast convergence tokm1forr1,0.2. This includes the case ofk01, 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) for 158 #u0,308, but it is only marginally smaller for 158 #u0. Gavrilyuket al.(2005) related such a slower convergence of an analogous method for smaller semi-apex angles to the asymptotic behaviour of the exact solution along the vertical axis. Their conclusion is that the eigenfunctions cmshould exponentially decay downward Ox for a circular cylindrical tank, to which the conical domain tends as u0 decreases.

However, W^ðmÞ_k and W_k^ðmÞ do not capture this decaying. Furthermore, decreasing the dimensionless liquid depth h(r1!1 oru0!908) may cause a lower accuracy (3-4 significant digits for 18-24 basic functions).

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