Natural Sloshing Frequencies in Truncated Conical · PDF fileLINEAR SLOSHING IN A TRUNCATED...

Natural Sloshing Frequencies in Truncated Conical Tanks

I. Gavrilyuk1, M. Hermann2, I. Lukovsky3, O. Solodun3, A. Timokha3

1Berufsakademie Thuringen-Staatliche Studienakademie, Am Wartenberg 2, D-99817, Eisenach, Germany2Friedrich-Schiller-Universitat Jena, Ernst-Abbe-Platz 1-2, Jena, D-07745, Germany,

3Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska 3, 01601, Kiev,Ukraine

SUMMARY

Conical-shaped or conical-bottom reservoirs are widely used as water containment for elevated tanks.After earthquakes water tanks play an important role, by making the water available needed forextinguishing fires which arise with such catastrophic events frequently. Therefore special care mustbe practiced with the construction of the tanks in order to assure their safety and functionality duringa seismic event. The occurrence of resonant vibrations caused by the closeness of the lowest naturalsloshing frequency of the contained water to the effective frequency domain of seismic excitations isvery dangerously. Robust and CPU-efficient numerical methods are therefore required for quantifyingthe natural sloshing frequencies and modes. The present paper employs the Treftz variational schemeand two distinct harmonic basic systems of functions to develop two numerical-analytical methods.Extensive numerical experiments establish the limits of their applicability in terms of semi-apex angleand position of the truncating plane. c© Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

key words: Linear sloshing frequencies; the Treftz method.

1. Introduction

Linear liquid sloshing in conical tanks was studied in the 50-60’s of the last century to evaluatethe hydrodynamic loads in rocket tanks. Selected principal results and design criteria areoutlined in Feschenko, et al. [12] (numerical method of Dokuchaev [6]) and in the NASAreport [1]. Experimental measurements of the lowest natural sloshing frequency were reportedby Mikishev and Dorozhkin [22] and Bauer [2]. In the 70-80’s the linear liquid sloshing dynamicsin ∨-shaped conical tanks was considered as a test problem for debugging the ComputationalFluid Dynamics (CFD) methods, see e.g. Bauer and Eidel [3], Lukovsky and Bilyk [17],Schiffner [24] and Lukovsky [16, 18].

The international standards concerning the megaliter elevated water tanks [9], which havebeen issued in the 90’s, represent a typical design for the concrete and steel reservoirs of conicaland conical-bottom shape. Two typical examples are illustrated in Fig. 1. The concrete tank(a) has the form of a truncated cone (the conical walls have a height of 15.2 m with the semi-apex angle 45◦, the thickness of the conical walls varies from 61 cm at the bottom up to 40 cmat the top). Its capacity is approximately 8000 tons. It is one of the first American tanks madein concrete instead of steel. The photo (b) shows a steel tower tank in Sydney.

c© Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

2 I. P. GAVRILYUK ET AL.

(a) (b)

Figure 1. Typical conical tower tanks: photo (a) – US Water Tank Projects, Boynton Beach, Florida.The Boynton Beach water tower in Florida has been manufactured in concrete instead of steel. It hasa capacity of approximately 8.000 tons, and is one of the first tanks of this type in the USA. The coneincreases with an angle of 450 to 15.2 m height. The thickness of the conical section varies from 61 cmat the floor level to 40 cm at the top. The tank is closed with a 34.4 m, 10 cm thick free span concretedome supported by a 61 cm × 1.5 m dome ring on the top of the conical vessel section. This ringis prestressed and transfers weight and lateral thrust from the dome roof to the conical vessel wall.Photo (b): The development of elevated water storages has achieved a new level with the ”wineglass”steel water tank in Sydney. It can store 3.2 megalitres, a capacity previously unseen in Sydney. Thewell-known tank manufacturer Saunders International has built the tank at Quakers Hill in Sydney’s

west to serve some 15,000 households in stage two of the vast Rouse Hill development.

Since seismic events are always possible, the effect of hydrodynamic loads that are causedby liquid motions in a water tank on the supporting tower must be studied. In the modellingof these events equivalent mechanical models (see, e.g. Dutta et al. [8], Shrimali and Jangid[25], Damatty and Sweedan [5]) can be used, which relate liquid dynamics to oscillations ofpendulum or spring-mass systems. The eigenfrequencies of those mechanical systems shouldcoincide with the lower natural sloshing frequencies. A reason for it is that the lower naturalsloshing modes are characterised by a relatively low damping and, therefore, give a dominatingcontribution to the hydrodynamic force and moment applied to the tank walls. An accurateprediction of these natural sloshing frequencies and modes is absolutely necessary (Damattyet

al. [4], Tanga [26], Dutta and Laha [7]). This can be done by various Computational FluidDynamics (CFD) methods or using semi-empirical approximation formulas (Damatty et al.

[4], Gavrilyuk et al. [13]). Although CFD-methods demand lots of CPU power, these methodsare often used since they provide a precise approximation of the natural sloshing spectrum.When concentrating on linear and nonlinear sloshing in ∨-shaped non-truncated conical tanks,Gavrilyuk et al. [13] showed that an alternative may consist in a semi-analytical solutionmethod. The method keeps the accuracy of the CFD-methods, but remains CPU-efficient andsimple in use. The last, but not the least argument is that the constructed semi-analyticalsolutions may facilitate development of nonlinear sloshing theories. These theories are ofprimary importance for studying a resonant coupled vibration of tower and contained liquid.The present paper proposes two different semi-analytical approximation techniques for thecomputation of lower natural sloshing frequencies and modes in truncated conical tanks. Themethods are quite accurate in a wide range of semi-apex angles, liquid fillings and positions


LINEAR SLOSHING IN A TRUNCATED CONICAL TANK 3

of the secant plane.

Because the approximate natural sloshing modes are smoothly expandable over the non-perturbed water plane, they may be adopted by nonlinear multi-modal sloshing theories(Lukovsky [19, 20], Lukovsky and Bilyk [17], Gavrilyuk et al. [13, 14], Bauer and Eidel [3],Faltinsen et al. [10, 11]).

In § 2, we consider a time-dependent boundary value problem, which describes linearliquid sloshing in a motionless tank (Lukovsky et al. [18], Ibrahim [15]). In order to findthe natural sloshing frequencies and modes, a spectral problem has to be solved. This spectralproblem admits a variational formulation (see Feschenko et al. [12]). We use this variationalformulation in a Treftz scheme, whose implementation needs an analytical harmonic basis. Twodifferent harmonic bases are proposed. The first Treftz basis is of polynomial type (so-calledharmonic polynomials). The second one employs the Legendre functions of first kind. Extensivenumerical experiments have been done to identify the geometrical characteristics (semi-apexangle, position of secant plane and liquid depth), for which the proposed functional basesguaranty a sufficient number of significant digits for the lower natural frequencies. In § 3.1,the main focus is on harmonic polynomials (see Lukovsky et al. [18]). For the ∨-shaped tanks,we show that 11-17 harmonic polynomials provide 4-6 significant digits of the lower naturalfrequencies for semi-apex angles that are smaller than 75◦ and larger than 10◦. For the ∨- and∧-shaped tanks which are characterised by semi-apex angles larger than 75◦, the same numberof harmonic polynomials guarantees only 3-4 significant digits. For the ∧-shaped tanks withsemi-apex angles smaller than 75◦, the convergence to the natural frequencies depends on theratio between the radixes of the mean liquid plane and the circular bottom. The method is notvery efficient (only about 2-3 significant digits can be obtained with 17-20 basic polynomials)when the semi-apex angle is smaller than 60◦ and the ratio between the specified radixes issmaller than 1/2. The slow convergence for the ∧-shaped tanks can be attributed partially tothe singular asymptotic behaviour of the natural sloshing modes in the neighbourhood of thecontact line formed by the mean free surface and the conical walls (Lukovsky et al. [18]).

Lukovsky [19] has proposed a non-conformal mapping technique to develop the multimodalmethod for the nonlinear sloshing problem in a non-cylindrical tank. Lukovsky andTimokha [21] and Gavrilyuk et al. [13] have realised this technique for the case of a non-truncated ∨-tank. The natural sloshing modes were approximated by a series of adjoinedspheroidal harmonics of first kind. Their usage in variational methods was earlier justified byDokuchayev [12]. In § 3.2, we use the curvilinear coordinate system proposed by Gavrilyuk et

al. [13] and generalise the results on natural sloshing frequencies for the case of truncated ∨-and ∧-shaped conical tanks. The convergence of this method to the lower natural frequenciesis slower than of a method which uses the harmonic polynomials considered in § 3.1. Moreover,for the typical geometrical configuration of water tanks (a ∨-shaped cone with semi-apex anglebetween 30◦ and 60◦ and a small relative radius of the bottom), the method shows a fasterconvergence behavior than in the case of § 3.1. Six significant digits of the lowest sloshingfrequency can be found by using only 6-10 basis functions. A comparative analysis of the twoTreftz methods from § 3.1 and § 3.2 is presented in § 3.3.

In section 4, we discuss the dependence of the lowest natural sloshing frequency on thegeometrical shape of truncated conical tanks. Bearing in mind that the actual water tanks are∨-shaped, we present the lowest spectral parameter (with a accuracy of five significant digits)versus the semi-apex angle and the ratio between radixes of the mean water plane and thebottom. This should facilitate engineering computations of the lowest natural frequency.



2. Statement of the problem

2.1. Differential and variational formulations

We consider an ideal incompressible liquid with irrotational flow, which partly occupies anearth-fixed rigid conical tank with the semi-apex angle θ0. The mean (hydrostatic) liquidshape coincides with the domain Q0 as it is shown in Fig. 2.

Oy

zx

q00

S1S

r

r

0

1

O

yz

x

q0

0

0QS2

S r

r

0

1

g

Figure 2. Hydrostatic liquid domain in ∧- and ∨-shaped tanks. In the non-dimensional case we haver0 := 1, r1 := r1/r0 and h := h/r0, where h is the liquid depth.

The gravity acceleration is directed downwards along the symmetry axis Ox. The wettedconical walls are denoted by S1. The circle S2 is the tank bottom, S = S1 ∪ S2 and Σ0 isthe non-perturbed (hydrostatic) water-plane. The origin of the Oxyz-coordinate system issuperposed with the artificial apex of the conical surface. The time-dependent problem thatdescribes small relative liquid motions is considered in a non-dimensional statement assumingthat r0 (the bottom radius for ∧-shaped and the water plane radius for ∨-shaped tanks) ischosen as a characteristic geometrical dimension. In the following we use the same notationfor the unscaled and scaled quantities, e.g., r1 := r1/r0 (the ratio between the radixes of themean free surface and the bottom). The non-dimensional r1 and θ0 completely determine thegeometric proportions of Q0. The limit r1→1 implies that h := h/r0→0, i.e. the water becomesshallow. At a fixed r1, the non-dimensional depth h tends to zero as θ0→π/2.

Linear sloshing in a motionless conical tank is governed by the following boundary valueproblem (see Lukovsky et al. [18] and Ibrahim [15])

∆φ = 0 in Q0;∂φ

∂x=∂f

∂t,∂φ

∂t+ gf = 0 on Σ0;

∂φ

∂ν= 0 on S;

∫

Σ0

∂φ

∂xdS = 0, (1)

where φ(x, y, z, t) is the velocity potential, x = f(y, z, t) describes the free surface, ν is theouter normal to S and g is the gravity acceleration scaled by r0 (g := g/r0). Using the initialvalues

f(y, z, t0) = F0(y, z);∂f

∂t(y, z, t0) = F1(y, z), (2)

at an initial time t = t0, the linear boundary value problem (1) has a unique solution. Thefunctions F0 and F1 define initial displacements of the free surface and its velocity, respectively.



2.2. Fundamental solution and natural sloshing modes

Natural oscillations of the contained liquid are associated with the following solutions of (1)

φ(x, y, z, t) = ψ(x, y, z) exp(iσt), i2 = −1, (3)

where σ is the natural frequency and ψ(x, y, z) is the so-called natural mode. Inserting (3) into(1) leads to the spectral boundary problem

∆ψ = 0 in Q0;∂ψ

∂x= κψ on Σ0;

∂ψ

∂ν= 0 on S;

∫

Σ0

∂ψ

∂xdS = 0, (4)

where each natural frequency σ can be restored from an eigenvalue κ by the formula

σ =√gκ. (5)

The spectral problem (4) has a real positive pointwise spectrum (see Morand and Ohayon[23])

0 < κ1 ≤ κ2 ≤ . . . ≤ κn ≤ . . .

with a unique limit point at infinity, i.e. κn → ∞, n→ ∞. The projections of the eigenfunctionsψn onto Σ0, i.e. fn(y, z) = ψn|Σ0

, constitute an orthogonal basis in L2(Σ0). This means that{φn = ψn(x, y, z) exp(iσnt)} is the fundamental solution of (1). Using the known {κn} and{ψn}, we can present the solution of (1) and (2) as a Fourier series in φn, i.e. as a superpositionof the natural standing waves.

Problem (4) admits a minimax variational formulation (see Feschenko et al. [12], Chapter 6],Morand and Ohayon [23]), which is based on the functional

K(ψ) =

∫

Q0

(∇ψ)2dQ

∫

Σ0

ψ2dS(6)

defined on ψ ∈ W 12 (Q). The absolute minimum of the positive definite functional (6) coincides

with the lowest eigenvalue of the spectral problem (4).

3. Projective scheme

The Treftz projective scheme approximates the exact solutions of (4) by a finite sum ofharmonic functions, which constitute the corresponding functional basis. Substituting this suminto (6) and using the necessary condition for an extrema reduces (4) to the spectral matrixproblem (A− κB)a = 0, where A and B are two nonnegative symmetric matrices and a is thecorresponding eigenvector. By increasing the dimension of the sum, the non-zero eigenvaluesof the matrix problem converge (from above) to the lower eigenvalues of (4). Approximationsof the eigenfunctions can be obtained when the eigenvectors of the matrix problem are usedas the coefficients of the Treftz scheme.

A principal difficulty of the Treftz scheme consists in the determination of a suitablefunctional basis. The completeness of the set of harmonic functions depends significantly onthe actual shape of Q0. For star-shaped domains, such a functional basis may be associatedwith the so-called harmonic polynomials (Lukovsky et al. [18]).



x

x

O

q0

G

G

1

1

1

r 0

0x x

O

q0

h

h

1r

2

2

L

L

L

LL

L

1

1

Figure 3. Meridional plane of ∧- and ∨-shaped tanks.

3.1. The harmonic polynomials as a functional basis

Preliminary notes. We use the cylindrical coordinate system (X, ξ, η) linked with theoriginal Cartesian coordinates by

x = X +X0, y = ξ cos η, z = ξ sin η. (7)

Here, the lag X0 along the vertical axis is introduced in such a way to superpose the origin ofthe cylindrical coordinate system with the water-plane. Furthermore, we represent the solutionof (4) in the following form

ψ(X, ξ, η) = ϕm(X, ξ)

(

sinmηcosmη

)

, m = 0, 1, 2 . . . . (8)

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

∂

∂X

(

ξ∂ϕm

∂X

)

+∂

∂ξ

(

ξ∂ϕm

∂ξ

)

−m2

ξϕm = 0 in G;

∂ϕm

∂X= κm ϕm on L0;

∂ϕm

∂ν= 0 on L; (9)

|ϕm(X, 0)| <∞;

∫

L0

ξ∂ϕ0

∂Xdξ = 0.

Problem (9) is defined in the meridional plane of Q0 and L = L1 + L2 (see Fig. 3). This meansthat the eigenvalues of the original three-dimensional problem constitute a two-parametric setκ = κmi (m = 0, 1, . . . ; i = 1, 2, . . .), where i ≥ 1 enumerates the eigenvalues of (9) in ascendingorder. The corresponding eigenfunctions of (4) are of the form (8) with ϕm = ϕmi(X, ξ).

The lowest eigenvalue κm1 of (9) yields the absolute minimum of the functional

Jm(ϕm)=

∫

G

[

ξ

(

∂2ϕm

∂X2+∂2ϕm

∂ξ2

)

+m2

ξϕ2

m

]

dXdξ/

∫

L0

ξϕ2mdξ (10)



on a complete set of harmonic functions, which are bounded at ξ = 0 (see Lukovsky et

al. [18], § 10]). The eigenvalues κmi, i ≥ 1, can be determined from the necessary conditionfor an extrema of Jm.

Harmonic polynomials. By definition, the harmonic polynomials are polynomial solutionsof the first equation of (9). According to Feschenko et al. [12] and Lukovsky [19], these solutionscan be specified as

w(m)k (X, ξ) =

2(k −m)!

(k +m)!RkP

(m)k (µ), k ≥ m, R =

√

X2 + ξ2, and µ = cos η, (11)

where P(m)k (µ) are Legendre‘s functions of first kind. It can be shown that w

(m)k has indeed a

polynomial structure in terms of X and ξ. The first functions of the set (11) are of the form

w(0)0 = 1, w

(0)1 = X, w

(0)2 = X2 − ξ2

2, ... (m = 0),

w(1)1 = ξ, w

(1)2 = Xξ, w

(1)3 = X2ξ − ξ3

4, ... (m = 1),

w(2)2 = ξ2, w

(2)3 = Xξ2, w

(2)4 = X2, ξ2 − ξ4

6, ... (m = 2),

w(3)3 = ξ3, w

(3)4 = Xξ3, w

(3)5 = X2ξ3 − ξ5

8, ... (m = 3).

The computation of polynomials of higher order can be realized by the following recurrencerelations

∂ w(m)k

∂X= (k −m)w

(m)k−1; ξ

∂w(m)k

∂ξ= kw

(m)k − (k −m)Xw

(m)k−1,

(k −m+ 1)w(m)k+1 = (2k + 1)Xw

(m)k − (k −m)(X2 + ξ2)w

(m)k−1,

(k −m+ 1)ξw(m+1)k = 2(m+ 1)

(

(X2 + ξ2)w(m)k−1 −Xw

(m)k

)

.

Variational method. We represent the solution of (9) in the form

ϕm(X, ξ) =

q∑

k=1

a(m)k w

(m)k+m−1(X, ξ). (12)

After substitution of (12) into the functional (10), the necessary condition for an extrema

∂ Jm

∂ a(m)k

= 0, k = 1, 2, . . . , q,

leads to the spectral matrix problem

det(

{α(m)ij } − κm {β(m)

ij })

= 0, (13)

where the elements {α(m)ij } and {β(m)

ij } are determined for the ∧-cones by the formulas

α(m)ij =

r1∫

0

(

ξ∂ w

(m)i+m−1

∂ Xw

(m)j+m−1

)

X=0

dξ +

0∫

−h

(

ξ∂ w

(m)i+m−1

∂ ξw

(m)j+m−1

)

ξ=tan θ0X−r1

dx



− tan θ0

0∫

−h

(

ξ∂ w

(m)i+m−1

∂ Xw

(m)j+m−1

)

ξ=tan θ0X−r1

dX −1∫

0

(

ξ∂ w

(m)i+m−1

∂ Xw

(m)j+m−1

)

X=−h

dξ,

β(m)ij =

r1∫

0

(

ξ w(m)i+m−1 w

(m)j+m−1

)

X=0dξ,

and for the ∨ cones by the formulas

α(m)ij =

1∫

0

(

ξ∂ w

(m)i+m−1

∂ Xw

(m)j+m−1

)

X=0

dξ +

0∫

−h

(

ξ∂ w

(m)i+m−1

∂ ξw

(m)j+m−1

)

ξ=tan θ0X+1

dX

− tan θ0

0∫

−h

(

ξ∂ w

(m)i+m−1

∂ Xw

(m)j+m−1

)

ξ=tan θ0X+1

dX −r1∫

0

(

ξ∂ w

(m)i+m−1

∂ Xw

(m)j+m−1

)

X=−h

dξ,

β(m)ij =

1∫

0

(

ξ w(m)i+m−1 w

(m)j+m−1

)

X=0dξ.

Since this variational method implies the minimization of the functional (10), the matrixproblem (13) gives the best approximation κm1 for the lowest eigenvalue. Moreover, theapproximation κm1 converges to the corresponding eigenvalue of (9) from above. This makesit possible to check the convergence by the number of significant digits which do not changewhen q is increased.

Convergence. Our numerical experiments were primary dedicated to the eigenvaluesκm1, m = 0, 1, 2, 3. These eigenvalues are responsible for the lowest natural modes, whichare weakly damped and, therefore, give a decisive contribution to hydrodynamic loads. Inthe case of ∨-tanks, the method shows a fast convergence to κm1 and provides satisfactoryaccuracy for the eigenvalues κm2 and κm3, too. It shows a slower convergence for the ∧-tanks.Furthermore, the convergence depends not only on the type of the tank (∨ or ∧-shaped), butalso on the semi-apex angle θ0 and the dimensionless parameter 0 < r1 < 1. The results inTable I (A) exhibit a typical convergence behavior for the ∨-tanks with 10◦ ≤ θ0 ≤ 75◦ and0.2≤r1≤0.9. The table shows that 5-6 digits are significant as q ≥ 14. The best accuracyis established for the lowest spectral parameter κ11. The accuracy grows with q for m 6= 0.However, computations of the value κ01, which is responsible for the axial-symmetric naturalmode, may become unstable for q > 17. This explains why we do not present numerical resultson this eigenvalue for q = 20. While m 6=1 and 0.2≤r1≤0.9, increasing θ0 > 75◦ leads to aslower convergence. In this case, q = 17, . . . , 20 guarantees only 3-4 significant digits (thenecessary accuracy in practice). The same number of harmonic polynomials keeps this numberof significant digits for the tanks with r1 < 0.2 and 10◦≤θ0≤75◦. When r1 tends to zero (thetruncated tank is close to a non-truncated one), the approximations κm1 were validated bynumerical results presented by Gavrilyuk et al. [13] and experimental data by Bauer [2].

In Table I (B) a typical convergence behavior for the ∧-shaped tanks with 10◦ ≤ θ0 ≤ 75◦

is presented. A comparative analysis of the parts (A) and (B) illustrates that the method is inthe latter case less efficient. In particular, computations of κ01 are not so precise. Moreover,



Table I. Convergence to κm1, m = 0, 1, 2, 3 for different values of the dimensionless parameter r1

versus the number of basic functions q in (12). The column (A) is for ∨-shaped tanks, (B) correspondsto ∧-shaped tanks. Here, θ0 = 30◦.

A Bq r1 = 0.2 r1 = 0.4 r1 = 0.6 r1 = 0.8 r1 = 0.9 q r1 = 0.2 r1 = 0.4 r1 = 0.6 r1 = 0.8 r1 = 0.9

κ01

2 5.032415 4.577833 4.152352 3.805501 2.961399 8 70.063366 15.140938 6.899986 4.560991 2.6834115 3.394779 3.392286 3.384865 3.141171 2.200342 10 50.467624 11.124492 6.672266 4.551150 2.6831178 3.385603 3.385592 3.381845 3.138861 2.197438 12 39.061112 10.185632 6.666704 4.550877 2.68293211 3.385600 3.385590 3.381827 3.138718 2.197238 14 32.075719 10.017447 6.665678 4.550590 2.68279614 3.385600 3.385590 3.381822 3.138665 2.197162 16 27.649041 10.016124 6.665058 4.550422 2.68275117 3.385600 3.385590 3.381819 3.138644 2.197138 18 25.152987 10.014257 6.664886 4.550349 2.682730

κ11

2 1.343631 1.335723 1.299977 1.007261 0.607340 8 16.504854 5.694833 3.516782 1.661815 0.7265555 1.304413 1.301793 1.254157 0.934412 0.542503 10 13.947812 5.633730 3.515861 1.661684 0.7265078 1.304378 1.301694 1.254054 0.933885 0.542284 12 13.487667 5.631904 3.515547 1.661608 0.72648911 1.304377 1.301692 1.253982 0.933835 0.542257 14 12.247242 5.630560 3.515365 1.661559 0.72648414 1.304377 1.301687 1.253972 0.933823 0.542251 16 11.663497 5.629898 3.515269 1.661551 0.72648217 1.304377 1.301686 1.253969 0.933819 0.542249 18 11.450724 5.629777 3.515237 1.661537 0.72648020 1.304377 1.301686 1.253967 0.933817 0.542248 20 11.331833 5.629888 3.515214 1.661532 0.726478

κ21

2 2.443739 2.424390 2.386633 2.191371 1.572048 8 45.024108 9.797804 5.950925 3.724251 1.9232345 2.263550 2.263351 2.255096 2.015602 1.361905 10 28.652496 9.096131 5.941913 3.724049 1.9230198 2.263151 2.263087 2.255004 2.014923 1.360928 12 25.183942 8.975383 5.941107 3.723708 1.92295011 2.263150 2.263087 2.254976 2.014838 1.360837 14 24.771657 8.968027 5.941002 3.723640 1.92293014 2.263150 2.263086 2.254972 2.014811 1.360805 16 21.512726 8.966657 5.940651 3.723591 1.92291317 2.263150 2.263086 2.254969 2.014801 1.360795 18 19.738583 8.966531 5.940603 3.723549 1.92289920 2.263150 2.263086 2.254968 2.014796 1.360790 20 17.760234 8.966524 5.940589 3.723542 1.922895

κ31

2 3.533170 3.520882 3.459048 3.316198 2.697426 8 139.879750 15.670550 8.146078 5.634069 3.4040355 3.181530 3.181299 3.179541 3.047179 2.329022 10 83.822538 12.954922 8.046749 5.633996 3.4035688 3.180251 3.180249 3.179080 3.046742 2.327044 12 64.588532 12.229637 8.044916 5.633565 3.40346011 3.180249 3.180248 3.179077 3.046654 2.326879 14 53.410762 12.088554 8.044459 5.633498 3.40341114 3.180249 3.180247 3.179074 3.046620 2.326812 16 53.138732 12.079636 8.044234 5.633377 3.40336017 3.180249 3.180247 3.179073 3.046606 2.326790 18 38.855186 12.084001 8.044028 5.633343 3.40333620 3.180249 3.180247 3.179073 3.046599 2.326779 20 36.467491 12.084898 8.043988 5.633321 3.403328

18-20 harmonic polynomials lead to 4-5 significant digits of κm1 only for r1≥0.4. This is notthe case for lower values of r1. Furthermore, when r1≤0.2, q = 17, . . . , 20 can only guarantee2-3 significant digits for κ11. The slower convergence for the ∧-shaped tanks can in part beclarified by the occurrence of singular first derivatives of the eigenfunctions ψm at an innervertex between L0 and L1 (see the mathematical results of Lukovsky et al. [18]). The harmonicpolynomials are smooth in the (ξ,X)-plane, and, therefore, do not capture this singularbehaviour. The singularity disappears when the corner angle is less than 90◦. This occursonly for the ∧-shaped tanks. The ∨-shaped tanks are characterised by a similar singularityat the vertex formed by L1 and L2. However, because the natural modes (eigenfunctions ψm)should “decay” exponentially downward, the method may be sensitive with respect to thatsingularity only for shallow water. Our numerical experiments confirm this fact as r1 < 0.1.



10

x x

x

x

x

G

L

LL

00

20

1

1

2

2

*

*

*

*

Ox

y

G

O

L

L

L

q0 0

0

1

1

2

r

rO

x

y

G

L

L

L

q0

00

11

2

r

r

xx

xx

x

G

L

L

L

0

0

10

20

1

1

2

2

*

*

*

*

O

Figure 4. Meridional cross-sections of the original and transformed domains.

3.2. Specific harmonic basis in a curvilinear coordinate system

The nonlinear resonant sloshing is effectively studied by multi-modal methods. As it is shownby Lukovsky [16], Lukovsky and Timokha [21] and Gavrilyuk et al. [13], these methods requireanalytical expressions for the natural modes, which

(i) are analytically expandable over the waterplane,(ii) satisfy a zero-Neumann condition at the tank walls and, if the tank is non-cylindrical,(iii) can be transformed to a curvilinear coordinate system (x1, x2, x3), in which the free

surface is governed by the normal form x1 = f(x2, x3, t).

An example of such approximate natural modes for non-truncated ∨-tanks is given by Lukovsky[19] and Gavrilyuk et al. [13]. The present section generalizes these results.

Curvilinear coordinate system. The non-Cartesian parametrization proposed byLukovsky [16, 13] links the x, y, z coordinates with x1, x2, x3 as follows

x = x1, y = x1x2 cosx3, z = x1x2 sinx3. (14)

Thus the variable x3 = η is the polar angle in the Oyz-plane. In Fig. 4 it is demonstratedthat the hydrostatic liquid domain Q0 takes in the (x1, x2, x3)-system the form of an uprightcircular cylinder (x0 ≤ x1 ≤ x10, 0 ≤ x2 ≤ x20, 0 ≤ x3 ≤ 2π). The domain G∗ represents arectangle with sides h = x10 − x0 and x20 = tan θ0 in the meridional plane Ox2x1. Here, theradius of the undisturbed water plane is rt = 1 for the ∨-tanks and rt = r1 for the ∧-tanks.Having represented

ϕ(x1, x2, x3) = ψm(x1, x2)

(

sinmx3

cosmx3

)

, m = 0, 1, 2, . . . (15)

and following Gavrilyuk et al. [13], one obtains that the original three-dimensional problem (4)admits the separation of the spatial variables (x1, x2) and x3. Furthermore, the transformation(14) generates the following m-parametric family of spectral problems with respect toψm(x1, x2)

p∂2ψm

∂x21

+ 2q∂2ψm

∂x1∂x2+ s

∂2ψm

∂x22

+ d∂ψm

∂x2−m2cψm = 0 in G∗, (16)



p∂ψm

∂x1+ q

∂ψm

∂x2= κmpψm on L∗

0, (17)

s∂ψm

∂ x2+ q

∂ψm

∂ x1= 0 on L∗

1, (18)

p∂ψm

∂ x1+ q

∂ψm

∂ x2= 0 on L∗

2, (19)

|ψm(x1, 0)| <∞, m = 0, 1, 2, . . . , (20)

x20∫

0

ψ0x2dx2 = 0, (21)

where G∗ = {(x1, x2) : x0 ≤ x1 ≤ x10, 0 ≤ x2 ≤ x20}, p = x21x2, q = −x1x

22, s = x2(x

22 + 1),

d = 1 + 2x22, c = 1/x2 and the boundary of G∗ consists of the portions L∗

0,L∗

1 and L∗

2.It can be shown, that the solution of the spectral problem (16)–(21) coincides with the

extreme points of the functional

J (ψm) =

∫

G∗

[

p

(

∂ψm

∂x1

)2

+ 2q∂ψm

∂x1

∂ψm

∂x2+ s

(

∂ψm

∂x2

)2

+m2

x2ψ2

m

]

dx1dx2

/

∫

L∗

0

pψ2mdx2 (22)

on test functions satisfying (20).

Particular solutions of (16) + (18). Gavrilyuk et al. [13] operated with the a spectralproblem similar (16)-(21). Following fundamental results by Eisenhart, they showed that (16)+ (18) allows for separation of the spatial variables x1 and x2. In the studied case, thisseparation gives the following particular solutions

xν1T

(m)ν (x2) and

T(m)ν (x2)

x1+ν1

, ν ≥ 0. (23)

In order to find T(m)ν , we get the following homogeneous boundary value problem with the real

parameter ν in both the equation and boundary condition

x22(1 + x2

2)T′′(m)ν + x2(1 + 2x2

2 − 2νx22)T

′(m)ν +

[

ν(ν − 1)x22 −m2

]

T (m)ν = 0, (24)

T ′(m)ν (x20) = ν

x20

1 + x220

T (m)ν (x20), |T (m)

ν (0)| <∞. (25)

It can be shown that the problem (24)-(25) has nontrivial solutions only for a countable set ofν = νmn > 0 (m = 0, 1, . . . ; n = 1, 2, . . .).

The second class of functions, T(m)ν , appears only in the case of x0 6= 0, i.e., when the conical

tank is truncated. Computing T(m)ν leads to the following ν-parametric problems

x22(1 + x2

2)T′′(m) + x2(1 + 4x2

2 + 2νx22)T

′(m) +[

(ν + 1)(ν + 2)x22 −m2

]

T (m) = 0, (26)

T ′(m)(x20) + (ν + 1)x20

1 + x220

T (m)(x20) = 0, (27)

which also have nontrivial solutions only for a countable set of nonnegative ν.



Let us show that the solutions of (24) and (26) can be expressed in terms of the spheroidalharmonics and the set {νmn} is the same for both (24)-(25) and (26)-(27). For this purpose,

we change the variables in (24) and (26) by µ = (1 + x22)

−1

2 and introduce the substitutionsy(µ) = µν T (µ) and y(µ) = µ−1−ν T (µ) for (24) and (26), respectively. This reduces bothequations to the same well-known differential equation

(1 − µ2)y′′(µ) − 2µy′(µ) +

[

ν(ν + 1) − m2

1 − µ2

]

y(µ) = 0,

whose solutions indeed coincide with the Legendre function of first kind, i.e. y(µ) = P(m)ν (µ).

Further, performing similar operations in the boundary conditions (25) and (27) andintroducing the substitution µ = cos θ, one obtains the same equation

∂P(m)ν (cos θ)

∂θ

∣

∣

∣

∣

∣

θ=θ0

= 0 (28)

instead of (25) and (27). Eq. (28) can be considered as a transcendental equation to compute{νmn}. Table II represents first 12 values of νmn (m = 0, 1, 2, 3) versus θ0.

In summary, we get the following nontrivial particular solutions

T (m)νmk

(x2) = (1 + x22)

νmk

2 P (m)νmk

(

1√

1 + x22

)

(29)

T (m)νmk

(x2) = (1 + x22)

−1−νmk

2 P (m)νmk

(

1√

1 + x22

)

. (30)

Particular solutions as a functional basis. The derived above particular solutions (23),(29)-(30) are re-written in the form

W(m)k (x1, x2) = N

(m)k xνmk

1 T (m)νmk

(x2), W(m)k (x1, x2) = N

(m)k x−1−νmk

1 T (m)νmk

(x2). (31)

Here, N(m)k and N

(m)k are multipliers chosen from the condition

1 = ||W (m)k ||2L∗

2+L∗

0

= ||W (m)k ||2L∗

2+L∗

0

=

∫ x20

0

x2[(W(m)k |x1=x10

)2 + (W(m)k |x1=x0

)2]dx2 =

=

∫ x20

0

x2[(W(m)k |x1=x10

)2 + (W(m)k |x1=x0

)2]dx2. (32)

Eq. (32) expresses the fact that W(m)k and W

(m)k have to have the unit norm (in the mean-

square metrics) on the boundary L∗

2 + L∗

0, where the conditions (17) and (19) should benumerically satisfied. Explicit formulas for these normalising multiplier has the following form

N(m)k =

1√

x2νmk

10 + x2νmk

0

1√

x20∫

0

(

1 + x22

)νmk

(

P(m)νmk

)2

dx2

,

N(m)k =

1√

x−2−2νmk

10 + x−2−2νmk

0

1√

x20∫

0

(

1 + x22

)

−1−νmk

(

P(m)νmk

)2

dx2

.



Table II. Values of νmn versus θ0.

θ0 = π/6n ν0n ν1n ν2n ν3n

1 6.83539808 3.11959709 5.49282500 7.752442352 12.90828411 9.71206871 12.37204261 14.918041773 18.93644560 15.82152796 18.58301633 21.246148494 24.95138112 21.87016680 24.68578005 27.416635675 30.96063428 27.89785663 30.74738346 33.522891356 36.96692917 33.91577437 36.78860888 39.595906807 42.97148871 39.92832912 42.81818932 45.649320018 48.97494349 45.93761963 48.84047013 51.690151189 54.97765160 51.94477437 54.85786626 57.7224054410 60.97983147 57.95045476 60.87183012 63.7485424611 66.98162391 63.95507442 66.88328881 69.7701594012 72.98312378 69.95890533 72.89286242 75.78833988

θ0 = π/41 4.40532918 2.00000000 3.63323872 5.201427062 8.44711262 6.33388964 8.13729380 9.874668473 12.46332875 10.39698483 12.25919783 14.064526734 16.47193967 14.42505010 16.31855929 18.163082935 20.47727740 18.44103192 20.35413803 22.224479776 24.48090964 22.45137475 24.37794511 26.266659237 28.48354098 26.45862229 28.39502631 30.297510298 32.48553497 30.46398566 32.40789182 34.321091769 36.48709810 34.46811614 36.41793652 38.3397185110 40.48835640 38.47139553 40.42599923 42.3548118511 44.48939109 42.47406256 44.43261536 46.3672945412 48.49025692 46.47627426 48.43814300 50.37779256

θ0 = π/31 3.19569115 1.46798738 2.75258821 4.000000002 6.21952915 4.65418866 6.04042231 7.388401393 9.22884937 7.69054134 9.11091186 10.498378284 12.23380906 10.70673365 12.14521751 13.555386375 15.23688626 13.71595755 15.16577182 16.590876366 18.23898124 16.72192772 18.17952282 19.615248997 21.24049935 19.72611149 21.18938780 22.633071798 24.24164994 22.72920771 24.19681751 25.646692979 27.24255203 25.73159226 27.20261792 28.6574511510 30.24327826 28.73348550 30.20727365 31.6661679311 33.24387547 31.73502524 33.21109395 34.6733766212 36.24437524 34.73630210 36.21428568 37.67943893

Note, that the case of m = 0 needs also the volume conservation condition (21). This

redefines the functions W(0)k and W

(0)k by W

(0)k := W

(0)k − c

(0)k , W

(0)k := W

(0)k − c

(0)k , where

c(0)k =

2

x220

x20∫

0

x2W(0)k (x10, x2) d x2, c

(0)k =

2

x220

x20∫

0

x2 W(0)k (x10, x2) d x2.

Variational method. In accordance with the Treftz scheme, we present the approximatesolution of (16)-(21) in the form

ψm(x1, x2) =

q1∑

k=1

a(m)k W

(m)k +

q2∑

l=1

a(m)l W

(m)l . (33)



Substituting (33) in the functional (22) and using the necessary extrema condition

∂ Jm

∂ a(m)k

= 0, k = 1, 2, . . . , q1;∂ Jm

∂ a(m)l

= 0, l = 1, 2, . . . , q2,

lead to the spectral matrix problem

det(

{α(m)ij } − κm {β(m)

ij })

= 0, i, j = 1, 2, ..., q1 + q2, (34)

with the spectral parameter κm. The spectral problem (34) has q1 + q2 eigenvalues. Because

the presentation (33) contains two types of functions, i.e. W(m)k and W

(m)l , there exist four

sub-matrices of {α(m)ij } and {β(m)

ij } that read as

α(m)ij =

(

α(m)ij1 α

(m)ij2

α(m)ij3 α

(m)ij4

)

, β(m)ij =

(

β(m)ij1 β

(m)ij2

β(m)ij3 β

(m)ij4

)

.

The elements {α(m)ijs } and {β(m)

ijs } s = 1, 4 are computed by the formulas

α(m)ij1 =

x20∫

0

(

x21x2

∂W(m)i

∂x1− x1x

22

∂W(m)i

∂x2

)

x1=ht

W(m)j dx2

−x20∫

0

(

x21x2

∂W(m)i

∂x1− x1x

22

∂W(m)i

∂x2

)

x1=hb

W(m)j dx2,

α(m)ij2 =

x20∫

0

(

x21x2

∂W(m)i

∂x1− x1x

22

∂W(m)i

∂x2

)

x1=ht

W(m)j dx2

−x20∫

0

(

x21x2

∂W(m)i

∂x1− x1x

22

∂W(m)i

∂x2

)

x1=hb

W(m)j dx2,

α(m)ij3 =

x20∫

0

(

x21x2

∂W(m)i

∂x1− x1x

22

∂W(m)i

∂x2

)

x1=ht

W(m)j dx2

−x20∫

0

(

x21x2

∂W(m)i

∂x1− x1x

22

∂W(m)i

∂x2

)

x1=hb

W(m)j dx2,

α(m)ij4 =

x20∫

0

(

x21x2

∂W(m)i

∂x1− x1x

22

∂W(m)i

∂x2

)

x1=ht

W(m)j dx2

−x20∫

0

(

x21x2

∂W(m)i

∂x1− x1x

22

∂W(m)i

∂x2

)

x1=hb

W(m)j dx2,

β(m)ij1 = h2

t

x20∫

0

x2

(

W(m)i W

(m)j

)

x1=ht

dx2, β(m)ij2 = h2

t

x20∫

0

x2

(

W(m)i W

(m)j

)

x1=ht

dx2,

β(m)ij3 = h2

t

x20∫

0

x2

(

W(m)i W

(m)j

)

x1=ht

dx2, β(m)ij4 = h2

t

x20∫

0

x2

(

W(m)i W

(m)j

)

x1=ht

dx2.



In the case of ∧- and ∨-tanks, ht = r1/ tan θ0; hb = 1/ tan θ0 and ht = 1/ tan θ0; hb = r1/ tan θ0,respectively.

Convergence. Column A in Table III demonstrates a typical convergence for the case of∨-tanks with 10◦ ≤ θ0 ≤ 75◦ and 0.2≤r1≤0.9. When 0.2≤r1≤0.55, the method shows that4-5 significant figures of κm1 may be stabilised within q = q1 = q2 = 7÷10 (14÷20 basicfunctions). This is consistent with the convergence results of § 3.1. However, in contrast to§ 3.1, our new harmonic basis keeps also a fast convergence to κm1 for r1 < 0.2. This includesthe case of κ01, which has not been satisfactory handled by the harmonic polynomials. Further,when 0 < r1≤0.4, the number of stabilised significant figures is larger (within equal numberof basic functions) for 15◦ ≤ θ0 < 30◦, but it is a little bit lower for 15◦ ≤ θ0. Gavrilyuk et al.

[13] related such a slower convergence of an analogous method for smaller semi-apex angleswith the asymptotic behaviour of the exact solution along vertical axis. The eigenfunctionsψm should exponentially decay downward Ox for a circular cylindrical tank, to which the

conical domain tends as θ0 decreases. However, W(m)k and W

(m)k do not capture this decaying.

Further, decreasing the non-dimensional liquid depth h (r1 → 1 or θ0 → 90◦) may cause alower accuracy (3-4 significant figures within 18-24 basic functions).

Column B in Table III shows convergence for the ∧-tanks with the same r1 and θ0 as inthe column A. It is seen that the numerical results may be of lower accuracy than those in§ 3.1. For instance, the same number of basic functions gives only 2-3 significant figures when0.2≤r1≤0.9. However, in contrast to the harmonic polynomials, the new basis provides stablecomputations for the case of the axial-symmetric mode, κ01. In addition, whereas 0.05≤r1≤0.4,the lowest eigenvalue κ11 is calculated with a better accuracy. For the same r1, increasing thesemi-apex angle may lead to a slower convergence. The number of significant figures withinq1 = q2 = 12 decreases also in the limit r1→1. These “shallow water” cases are handled with2-3 significant figures as q1 = q2 = 12÷14.

The presence of the two types of basic functions in (33) makes it possible to vary q1 and q2to obtain a better approximation with the same total number of basic functions Q = q1 + q2.Speculative variations of q1 and q2 with fixed Q≥16 showed that better accuracy of κm1 isexpected as q2 > q1. It is especially true for smaller liquid depths. For example, when the∨-shaped tank is characterised by θ0 = 30◦ and r1 = 0.9, the approximate κ11 = 0.54233738can be obtained with either q1 = q2 = 12, (Q = 24) or q1 = 7, q2 = 12, (Q = 19).

3.3. Comparative analysis of the two methods

Comparing the numerical experiments performed with the two different functional bases shows,that the method of § 3.1 has better accuracy for smaller liquid depths (0.6≤r1). However, largerdepths (r1≤0.4) are best handled by the second method. This is clearly seen for the ∧-shapedtanks: the calculations by the method of § 3.2 keep robustness with increasing the number ofbasic functions, while the first method invalidates for large dimensions. The accuracy of bothmethods is generally speaking similar for the ∨-tanks with 0.2≤r1≤0.55.

Even if the number of basic functions is not large, the two developed methods give accurateapproximations of the lowest eigenvalue κ11. Because this eigenvalue computes the lowestnatural frequency, i.e. σ11 =

√gκ11, which is of primary importance for modelling vibrations

of towers with ∨-shaped tanks, we placed special emphasis on comparing the numerical resultsobtained by our two methods fro κ11. This is illustrated by Figs. 5 (a) and (b). The figures



Table III. Convergence to κm1, m = 0, 1, 2, 3, for different values of the non-dimensional parameter r1

versus the number of basic functions q = q1 = q2 in (33). The column (A) is for ∨-shaped tanks, (B)corresponds to ∧-shaped tanks. The case of θ0 = 30◦.

A Bq r1 = 0.2 r1 = 0.4 r1 = 0.6 r1 = 0.8 r1 = 0.9 r1 = 0.2 r1 = 0.4 r1 = 0.6 r1 = 0.8 r1 = 0.9

κ01

2 3.385676 3.385666 3.382064 3.148405 2.224870 20.829848 10.414889 6.934470 4.781070 2.8688943 3.385606 3.385596 3.381920 3.143923 2.211744 20.292399 10.146162 6.754682 4.629653 2.7489544 3.385601 3.385591 3.381878 3.141937 2.206179 20.151838 10.075881 6.707583 4.588863 2.7159115 3.385600 3.385590 3.381859 3.140888 2.203283 20.097111 10.048517 6.689248 4.572788 2.7025146 3.385600 3.385590 3.381847 3.140267 2.201583 20.070700 10.035311 6.680396 4.564916 2.6958177 3.385600 3.385590 3.381840 3.139870 2.200501 20.056100 10.028011 6.675501 4.560515 2.6920068 3.385600 3.385590 3.381835 3.139601 2.199769 20.047236 10.023579 6.672529 4.557819 2.6896379 3.385600 3.385590 3.381832 3.139410 2.199252 20.041473 10.020697 6.670596 4.556053 2.68806610 3.385600 3.385590 3.381829 3.139270 2.198872 20.037525 10.018723 6.669272 4.554837 2.68697311 3.385600 3.385590 3.381827 3.139164 2.198585 20.034708 10.017315 6.668327 4.553964 2.68618112 3.385600 3.385590 3.381826 3.139082 2.198364 20.032629 10.016275 6.667629 4.553317 2.685590

κ11

2 1.304378 1.301707 1.254338 0.935957 0.544861 11.335332 5.647088 3.528677 1.672867 0.7324103 1.304377 1.301695 1.254148 0.934864 0.543483 11.315765 5.637221 3.521265 1.666870 0.7292464 1.304377 1.301691 1.254073 0.934437 0.542976 11.308794 5.633703 3.518590 1.664611 0.7280935 1.304377 1.301689 1.254036 0.934226 0.542729 11.305577 5.632078 3.517343 1.663527 0.7275366 1.304377 1.301688 1.254016 0.934106 0.542589 11.303842 5.631202 3.516665 1.662926 0.7272247 1.304377 1.301687 1.254003 0.934032 0.542502 11.302805 5.630678 3.516258 1.662559 0.7270328 1.304377 1.301687 1.253994 0.933983 0.542445 11.302137 5.630340 3.515995 1.662319 0.7269069 1.304377 1.301687 1.253988 0.933949 0.542405 11.301682 5.630110 3.515815 1.662153 0.72681810 1.304377 1.301686 1.253984 0.933924 0.542376 11.301359 5.629947 3.515687 1.662034 0.72675411 1.304377 1.301686 1.253981 0.933906 0.542354 11.301121 5.629826 3.515592 1.661945 0.72670712 1.304377 1.301686 1.253978 0.933892 0.542337 11.300941 5.629735 3.515520 1.661878 0.726671

κ21

2 2.263162 2.263100 2.255147 2.019323 1.371212 18.039183 9.019177 5.977616 3.761864 1.9527593 2.263151 2.263088 2.255060 2.017249 1.366295 17.982154 8.990657 5.958186 3.742630 1.9381824 2.263150 2.263087 2.255026 2.016335 1.364234 17.960135 8.979646 5.950674 3.734876 1.9322155 2.263150 2.263087 2.255007 2.015850 1.363151 17.949506 8.974330 5.947040 3.731012 1.9291676 2.263150 2.263086 2.254996 2.015563 1.362510 17.943614 8.971384 5.945023 3.728818 1.9274007 2.263150 2.263086 2.254989 2.015378 1.362099 17.940025 8.969588 5.943793 3.727457 1.9262858 2.263150 2.263086 2.254985 2.015252 1.361819 17.937683 8.968417 5.942990 3.726556 1.9255379 2.263150 2.263086 2.254981 2.015163 1.361620 17.936074 8.967612 5.942438 3.725930 1.92501110 2.263150 2.263086 2.254979 2.015097 1.361473 17.934922 8.967036 5.942042 3.725477 1.92462711 2.263150 2.263086 2.254977 2.015047 1.361362 17.934070 8.966610 5.941750 3.725140 1.92433812 2.263150 2.263086 2.254976 2.015008 1.361276 17.933423 8.966287 5.941527 3.724882 1.924115

κ31

2 3.180280 3.180279 3.179147 3.051144 2.346969 24.361239 12.180611 8.115651 5.700453 3.4738763 3.180251 3.180250 3.179101 3.049247 2.338329 24.254295 12.127139 8.079896 5.667935 3.4415874 3.180249 3.180248 3.179090 3.048337 2.334334 24.210302 12.105142 8.065186 5.654291 3.4273525 3.180249 3.180249 3.179085 3.047828 2.332117 24.188236 12.094109 8.057806 5.647308 3.4198036 3.180249 3.180247 3.179082 3.047514 2.330753 24.175694 12.087838 8.053610 5.643277 3.4153197 3.180249 3.180247 3.179080 3.047306 2.329853 24.167920 12.083951 8.051009 5.640748 3.4124408 3.180249 3.180247 3.179078 3.047161 2.329228 24.162784 12.081383 8.049290 5.639061 3.4104819 3.180249 3.180247 3.179077 3.047056 2.328776 24.159223 12.079602 8.048098 5.637882 3.40909010 3.180249 3.180247 3.179077 3.046978 2.328438 24.156655 12.078318 8.047239 5.637026 3.40806611 3.180249 3.180247 3.179076 3.046918 2.328179 24.154745 12.077363 8.046600 5.636386 3.40729012 3.180249 3.180247 3.179076 3.046871 2.327976 24.153288 12.076635 8.046112 5.635894 3.406689



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1r1

10

20

30

40

50

60

70

80

q0

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 1r1

10

20

30

40

50

60

70

80

q0

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

Figure 5. The number of significant figures of κ11 obtained for ∨-tanks with twenty basic functions bythe methods of § 3.1 (Case a) and § 3.2 (Case b).

identify domains in the (r1, θ0)-plane, for which each method gives the same number ofsignificant figures with twenty basic functions. One can see that the accuracy of the firstmethod (§ 3.1) may become low only for small θ0 and r1, e.g. for large liquid depths. In theother cases, the method guarantees a fast convergence and high accuracy. At the same time,small θ0 and r1 are well handled by the second method (§ 3.2). However, this method convergesslowly as r1 → 1 and θ0 > 45◦, e.g. for small liquid depths.

4. The lowest natural sloshing frequency σ11

The eigenvalues κm1 are functions of liquid depth h, semi-apex angle θ0 and type of the tank,i.e., whether the tank is ∧-shaped or ∨-shaped. For the ∨-tanks, increasing θ0 decreases theeigenvalues. For the ∧-tanks, increasing θ0 increases κm1. The dependence of κm1 on non-dimensional r1 is illustrated in Figs. 6 (a) and (b).

The natural sloshing frequency σ11 is of primary practical importance for design of watertowers (Damatty and Sweedan [5]). Remembering that, we present in Table IV approximatevalues of κ11 versus θ0 and r1. The dimensional natural sloshing frequency σ11 is computedby κ11 via the formula

σ11 =

√

g κ11(θ0,r1

r0

)

r0, (35)

where g, r0 and r1 are dimensional. The numerical data can be used in actual engineeringcomputations or for validation of semi-empirical formulas.

5. Conclusions and future work

Two efficient numerically-analytical methods are proposed to compute the natural sloshingfrequencies and modes in truncated conical tanks. These methods are based on the Treftzvariational scheme. As a functional basis, the first method uses harmonic functions ofpolynomial type. The second method employs a non-conformal mapping technique and derives



0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

θ0=π/6

r1

κm1 m=0

m=1m=2m=3

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

θ0=π/6

r 1

κm1

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

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

θ0=π/4

r1

κm1

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

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

θ0=π/4

r 1

κm1

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

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

θ0=π/3

r1

κm1

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

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5θ

0=π/3

r 1

κm1

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

(a) (b)

Figure 6. Eigenvalues κm1 versus r1 for ∧-shaped (Case a) and ∧-shaped tanks (Case b).

special functional basis, which can be associated with the Legendre functions of first kind.Extensive numerical experiments showed that these methods have slightly different domainof the applicability in terms of semi-apex angle, liquid depth and type of the tank (∨- or∧-shaped).

The methods may demonstrate a slow convergence and even invalidate for ∧-shaped tanks.This point can be explained by the singular behaviour of the natural modes at the contactline formed by water-plane and conical walls. This was discussed in § 3.1. From mathematical



Table IV. κ11 versus θ0 and r1 for ∨-shaped tanks.

r1\θ0 10◦ 15◦ 20◦ 25◦ 30◦ 35◦ 40◦ 45◦ 50◦ 55◦ 60◦ 65◦ 70◦

0.05 1.6743 1.5862 1.4950 1.4011 1.3044 1.2052 1.1037 1.0000 0.8943 0.7868 0.6777 0.5671 0.45530.10 1.6743 1.5862 1.4950 1.4011 1.3044 1.2052 1.1037 1.0000 0.8943 0.7868 0.6776 0.5670 0.45510.15 1.6743 1.5862 1.4950 1.4011 1.3044 1.2052 1.1037 1.0000 0.8941 0.7865 0.6772 0.5665 0.45470.20 1.6743 1.5862 1.4950 1.4011 1.3044 1.2051 1.1035 0.9996 0.8935 0.7857 0.6763 0.5655 0.45360.25 1.6743 1.5862 1.4950 1.4010 1.3043 1.2049 1.1030 0.9986 0.8922 0.7840 0.6742 0.5634 0.45170.30 1.6743 1.5862 1.4950 1.4010 1.3041 1.2043 1.1017 0.9966 0.8894 0.7806 0.6706 0.5598 0.44840.35 1.6743 1.5862 1.4950 1.4008 1.3034 1.2027 1.0990 0.9927 0.8845 0.7750 0.6647 0.5541 0.44330.40 1.6743 1.5862 1.4949 1.4002 1.3017 1.1994 1.0938 0.9857 0.8762 0.7660 0.6556 0.5456 0.43590.45 1.6743 1.5862 1.4945 1.3987 1.2980 1.1930 1.0846 0.9743 0.8633 0.7525 0.6425 0.5335 0.42560.50 1.6743 1.5860 1.4935 1.3952 1.2908 1.1816 1.0696 0.9566 0.8442 0.7332 0.6242 0.5171 0.41170.55 1.6743 1.5856 1.4908 1.3877 1.2774 1.1625 1.0461 0.9304 0.8171 0.7067 0.5996 0.4954 0.39360.60 1.6743 1.5842 1.4842 1.3730 1.2540 1.1320 1.0110 0.8932 0.7800 0.6715 0.5676 0.4676 0.37070.65 1.6740 1.5799 1.4697 1.3456 1.2153 1.0856 0.9607 0.8423 0.7309 0.6261 0.5271 0.4330 0.34250.70 1.6727 1.5683 1.4395 1.2973 1.1544 1.0181 0.8915 0.7751 0.6681 0.5693 0.4775 0.3910 0.30860.75 1.6674 1.5391 1.3808 1.2172 1.0632 0.9242 0.8002 0.6897 0.5905 0.5007 0.4183 0.3417 0.26910.80 1.6472 1.4709 1.2741 1.0918 0.9339 0.7994 0.6844 0.5850 0.4978 0.4202 0.3499 0.2851 0.22410.85 1.5781 1.3251 1.0945 0.9084 0.7606 0.6416 0.5438 0.4614 0.3906 0.3284 0.2727 0.2218 0.17410.90 1.3702 1.0471 0.8199 0.6599 0.5423 0.4521 0.3800 0.3207 0.2704 0.2267 0.1879 0.1526 0.11970.95 0.8631 0.5955 0.4461 0.3511 0.2849 0.2356 0.1970 0.1657 0.1394 0.1167 0.0966 0.0784 0.0615

point of view, augmenting the smooth bases by a harmonic function, which has the specifiedsingular behaviour, may considerably improve convergence. Examples of such an augmentingfor two-dimensional spectral sloshing problems are given by Lukovsky et al. [18]. However, tothe authors’ knowledge, the literature does not exemplify those analytical harmonic functionfor the studied case.

Special emphasis should also be placed on the shallow water case. This requires a dedicatedstudy based on a nonlinear dissipative sloshing model. Nonlinear phenomena are also ofimportance for resonant sloshing (Gavrilyuk et al. [13]). Results of the present paper may beutilised to develop the multi-modal technique (Lukovsky [19], Faltinsen et al. [10], Gavrilyuket al. [13]) and study the nonlinear sloshing in a truncated conical tank. This is main purposeof the forthcoming studies.

ACKNOWLEDGEMENTS

Authors thank the German Research Society (DFG) for financial support (project DFG 436UKR113/33/00). The last author (A.T.) is grateful for the sponsorship presented by the Alexander vonHumboldt Foundation.

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4. Damatty El A., Korol R. M., Tang L. M. Analytcial and experimental investigation of the dynamic responseof liquid-filled conical tanks. Proceedings of the World Conference of Earthquake Engineering New Zelan:2000; Paper No. 966, Topic No. 7: 8.

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6. Dokuchaev L. V. On the solution of a boundary value problem on the sloshing of a liquid in conical cavities.Applied Mathematics and Mechanics (PMM). 1964; 28: 151–154.

7. Dutta S., Laha M. K. Analysis of the small amplitude sloshing of a liquid in a rigid container of arbitraryshape using a low-order boundary element method. International Journal for Numerical Methods inEngineering 2000; 47, Issue 9: 1633–1648.

8. Dutta S., Mandal A., Dutta S.C. Soilstructure interaction in dynamic behaviour of elevated tanks withalternate frame staging configurations. Journal of Sound and Vibration 2004; 277: 825–853.

9. European Committie for Standatization (CEN) Eurocode 8. Design Provisions of Earthquake Resistanceof Structures. Brussels: Silos, Tanks and Pipelines Part 4 1998; 56.

10. Faltinsen O. M., Rognebakke O. F., Lukovsky I. A., Timokha A. N. Multidimensional modal analysis ofnonlinear sloshing in a rectangular tank with finite water depth. Journal of Fluid Mechanics 2000; 407:201–234.

11. Faltinsen O. M., Rognebakke O. F., Timokha A. N. Resonant three-dimensional nonlinear sloshing in asquare base basin. Journal of Fluid Mechanics 2003; 487: 1–42.

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22. Mikishev G. N., Dorozhkin N. Y. An experimental investigation of free oscillations of a liquid in containers.Izvestiya Akademii Nauk SSSr, Otdelenie Tekhnicheskikh Nauk, Mekhanika: Mashinostroenie 1961; N 4:48–53.

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