Natural Sloshing Frequencies in Truncated Conical Tanks

Natural Sloshing Frequencies in Truncated Conical Tanks I. Gavrilyuk1 , M. Hermann2 , I. Lukovsky3 , O. Solodun3 , A. Timokha3 1 Berufsakademie

Th¨ uringen-Staatliche Studienakademie, Am Wartenberg 2, D-99817, Eisenach, Germany Jena, Ernst-Abbe-Platz 1-2, Jena, D-07745, Germany, 3 Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska 3, 01601, Kiev, Ukraine 2 Friedrich-Schiller-Universit¨ at

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 for extinguishing fires which arise with such catastrophic events frequently. Therefore special care must be practiced with the construction of the tanks in order to assure their safety and functionality during a seismic event. The occurrence of resonant vibrations caused by the closeness of the lowest natural sloshing frequency of the contained water to the effective frequency domain of seismic excitations is very dangerously. Robust and CPU-efficient numerical methods are therefore required for quantifying the natural sloshing frequencies and modes. The present paper employs the Treftz variational scheme and 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 angle c Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha and position of the truncating plane. 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 evaluate the hydrodynamic loads in rocket tanks. Selected principal results and design criteria are outlined in Feschenko, et al. [12] (numerical method of Dokuchaev [6]) and in the NASA report [1]. Experimental measurements of the lowest natural sloshing frequency were reported by Mikishev and Dorozhkin [22] and Bauer [2]. In the 70-80’s the linear liquid sloshing dynamics in ∨-shaped conical tanks was considered as a test problem for debugging the Computational Fluid 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 have been issued in the 90’s, represent a typical design for the concrete and steel reservoirs of conical and 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 semiapex angle 45◦ , the thickness of the conical walls varies from 61 cm at the bottom up to 40 cm at the top). Its capacity is approximately 8000 tons. It is one of the first American tanks made in 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 has a capacity of approximately 8.000 tons, and is one of the first tanks of this type in the USA. The cone increases with an angle of 450 to 15.2 m height. The thickness of the conical section varies from 61 cm at the floor level to 40 cm at the top. The tank is closed with a 34.4 m, 10 cm thick free span concrete dome supported by a 61 cm × 1.5 m dome ring on the top of the conical vessel section. This ring is 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. The well-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 caused by liquid motions in a water tank on the supporting tower must be studied. In the modelling of 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 of pendulum or spring-mass systems. The eigenfrequencies of those mechanical systems should coincide with the lower natural sloshing frequencies. A reason for it is that the lower natural sloshing modes are characterised by a relatively low damping and, therefore, give a dominating contribution to the hydrodynamic force and moment applied to the tank walls. An accurate prediction 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 Fluid Dynamics (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 methods are 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 solution method. The method keeps the accuracy of the CFD-methods, but remains CPU-efficient and simple in use. The last, but not the least argument is that the constructed semi-analytical solutions may facilitate development of nonlinear sloshing theories. These theories are of primary importance for studying a resonant coupled vibration of tower and contained liquid. The present paper proposes two different semi-analytical approximation techniques for the computation of lower natural sloshing frequencies and modes in truncated conical tanks. The methods are quite accurate in a wide range of semi-apex angles, liquid fillings and positions c Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

LINEAR SLOSHING IN A TRUNCATED CONICAL TANK

3

of the secant plane. Because the approximate natural sloshing modes are smoothly expandable over the nonperturbed 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 linear liquid sloshing in a motionless tank (Lukovsky et al. [18], Ibrahim [15]). In order to find the natural sloshing frequencies and modes, a spectral problem has to be solved. This spectral problem admits a variational formulation (see Feschenko et al. [12]). We use this variational formulation in a Treftz scheme, whose implementation needs an analytical harmonic basis. Two different harmonic bases are proposed. The first Treftz basis is of polynomial type (so-called harmonic polynomials). The second one employs the Legendre functions of first kind. Extensive numerical experiments have been done to identify the geometrical characteristics (semi-apex angle, position of secant plane and liquid depth), for which the proposed functional bases guaranty 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 natural frequencies 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 number of harmonic polynomials guarantees only 3-4 significant digits. For the ∧-shaped tanks with semi-apex angles smaller than 75◦ , the convergence to the natural frequencies depends on the ratio between the radixes 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 60◦ and the ratio between the specified radixes is smaller than 1/2. The slow convergence for the ∧-shaped tanks can be attributed partially to the singular asymptotic behaviour of the natural sloshing modes in the neighbourhood of the contact 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 multimodal method for the nonlinear sloshing problem in a non-cylindrical tank. Lukovsky and Timokha [21] and Gavrilyuk et al. [13] have realised this technique for the case of a nontruncated ∨-tank. The natural sloshing modes were approximated by a series of adjoined spheroidal harmonics of first kind. Their usage in variational methods was earlier justified by Dokuchayev [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 frequencies is 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 angle between 30◦ and 60◦ and a small relative radius of the bottom), the method shows a faster convergence behavior than in the case of § 3.1. Six significant digits of the lowest sloshing frequency can be found by using only 6-10 basis functions. A comparative analysis of the two Treftz 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 the geometrical 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 the bottom. This should facilitate engineering computations of the lowest natural frequency. c Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

4


2. Statement of the problem 2.1. Differential and variational formulations We consider an ideal incompressible liquid with irrotational flow, which partly occupies an earth-fixed rigid conical tank with the semi-apex angle θ0 . The mean (hydrostatic) liquid shape coincides with the domain Q0 as it is shown in Fig. 2.

y S0

O

x q0

S0

z

r0

r0

r1

S1 Q0 S2

r1

x

g

q0

y z

O

Figure 2. Hydrostatic liquid domain in ∧- and ∨-shaped tanks. In the non-dimensional case we have r0 := 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 wetted conical walls are denoted by S1 . The circle S2 is the tank bottom, S = S1 ∪ S2 and Σ0 is the non-perturbed (hydrostatic) water-plane. The origin of the Oxyz-coordinate system is superposed with the artificial apex of the conical surface. The time-dependent problem that describes small relative liquid motions is considered in a non-dimensional statement assuming that r0 (the bottom radius for ∧-shaped and the water plane radius for ∨-shaped tanks) is chosen as a characteristic geometrical dimension. In the following we use the same notation for the unscaled and scaled quantities, e.g., r1 := r1 /r0 (the ratio between the radixes of the mean free surface and the bottom). The non-dimensional r1 and θ0 completely determine the geometric proportions of Q0 . The limit r1 →1 implies that h := h/r0 →0, i.e. the water becomes shallow. 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 value problem (see Lukovsky et al. [18] and Ibrahim [15]) Z ∂f ∂φ ∂φ ∂φ ∂φ ∆φ = 0 in Q0 ; = , + gf = 0 on Σ0 ; = 0 on S; dS = 0, (1) ∂x ∂t ∂t ∂ν ∂x Σ0

where φ(x, y, z, t) is the velocity potential, x = f (y, z, t) describes the free surface, ν is the outer normal to S and g is the gravity acceleration scaled by r0 (g := g/r0 ). Using the initial values ∂f f (y, z, t0 ) = F0 (y, z); (y, z, t0 ) = F1 (y, z), (2) ∂t at an initial time t = t0 , the linear boundary value problem (1) has a unique solution. The functions F0 and F1 define initial displacements of the free surface and its velocity, respectively. c Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

5


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 Z ∂ψ ∂ψ ∂ψ = κψ on Σ0 ; = 0 on S; dS = 0, (4) ∆ψ = 0 in Q0 ; ∂x ∂ν Σ0 ∂x 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 eigenfunctions ψn 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(iσn t)} 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 superposition of 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 R (∇ψ)2 dQ Q0 K(ψ) = R 2 (6) ψ dS Σ0

defined on ψ ∈ W21 (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 of harmonic functions, which constitute the corresponding functional basis. Substituting this sum into (6) and using the necessary condition for an extrema reduces (4) to the spectral matrix problem (A − κB)a = 0, where A and B are two nonnegative symmetric matrices and a is the corresponding eigenvector. By increasing the dimension of the sum, the non-zero eigenvalues of the matrix problem converge (from above) to the lower eigenvalues of (4). Approximations of the eigenfunctions can be obtained when the eigenvectors of the matrix problem are used as the coefficients of the Treftz scheme. A principal difficulty of the Treftz scheme consists in the determination of a suitable functional basis. The completeness of the set of harmonic functions depends significantly on the actual shape of Q0 . For star-shaped domains, such a functional basis may be associated with the so-called harmonic polynomials (Lukovsky et al. [18]). c Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

6


x

xL

1

q0 r1 L 0 x O G L1 h L2 1

O G h r1 L2

x

0

L1

q0

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 the original 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 of the cylindrical coordinate system with the water-plane. Furthermore, we represent the solution of (4) in the following form sin mη ψ(X, ξ, η) = ϕm (X, ξ) , m = 0, 1, 2 . . . . (8) cos mη This makes it possible to separate the angular coordinate η and to reduce the three-dimensional boundary value problem (4) to the m-parametric family (m is a non-negative integer) of twodimensional spectral boundary value problems ∂ ∂ϕm ∂ ∂ϕm m2 ξ + ξ − ϕm = 0 in G; ∂X ∂X ∂ξ ∂ξ ξ ∂ϕm ∂ϕm = κm ϕm on L0 ; = 0 on L; (9) ∂X ∂ν Z ∂ϕ0 |ϕm (X, 0)| < ∞; ξ dξ = 0. L0 ∂X Problem (9) is defined in the meridional plane of Q0 and L = L1 + L2 (see Fig. 3). This means that the eigenvalues of the original three-dimensional problem constitute a two-parametric set κ = κmi (m = 0, 1, . . . ; i = 1, 2, . . .), where i ≥ 1 enumerates the eigenvalues of (9) in ascending order. 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 Z 2 .Z ∂ ϕm ∂ 2 ϕm m2 2 Jm (ϕm )= ξ + + ϕ dXdξ ξϕ2m dξ ∂X 2 ∂ξ 2 ξ m G


L0

(10)

7


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 condition for an extrema of Jm . Harmonic polynomials. By definition, the harmonic polynomials are polynomial solutions of the first equation of (9). According to Feschenko et al. [12] and Lukovsky [19], these solutions can be specified as p 2(k − m)! k (m) (m) wk (X, ξ) = R Pk (µ), k ≥ m, R = X 2 + ξ 2 , and µ = cos η, (11) (k + m)! (m)

(m)

where Pk (µ) are Legendre‘s functions of first kind. It can be shown that wk has indeed a polynomial structure in terms of X and ξ. The first functions of the set (11) are of the form ξ2 , ... 2 ξ3 (1) (1) (1) w1 = ξ, w2 = Xξ, w3 = X 2 ξ − , ... 4 4 ξ (2) (2) (2) w2 = ξ 2 , w3 = Xξ 2 , w4 = X 2 , ξ 2 − , ... 6 ξ5 (3) (3) (3) 3 3 2 3 w3 = ξ , w4 = Xξ , w5 = X ξ − , ... 8 The computation of polynomials of higher order can be realized relations (0)

w0 = 1,

(0)

(m)

∂ wk ∂X

(0)

w2 = X 2 −

w1 = X,

(m)

(m)

= (k − m)wk−1 ;

ξ

∂wk ∂ξ

(m)

(m)

= kwk

(m = 0), (m = 1), (m = 2), (m = 3). by the following recurrence (m)

− (k − m)Xwk−1 ,

(m)

(m)

(k − m + 1)wk+1 = (2k + 1)Xwk − (k − m)(X 2 + ξ 2 )wk−1 , (m+1) (m) (m) (k − m + 1)ξwk = 2(m + 1) (X 2 + ξ 2 )wk−1 − Xwk . Variational method. We represent the solution of (9) in the form ϕm (X, ξ) =

q X

(m)

ak

(m)

wk+m−1 (X, ξ).

(12)

k=1

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

(m)

∂ ak

= 0,

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

leads to the spectral matrix problem (m) (m) det {αij } − κm {βij } = 0, (m)

(13)

(m)

where the elements {αij } and {βij } are determined for the ∧-cones by the formulas ! ! Zr1 Z0 (m) (m) ∂ wi+m−1 (m) ∂ wi+m−1 (m) (m) αij = ξ wj+m−1 dξ + ξ wj+m−1 dx ∂X ∂ξ 0

X=0


−h

ξ=tan θ0 X−r1

8


Z0

− tan θ0

(m)

∂ wi+m−1 (m) ξ wj+m−1 ∂X

−h (m)

βij

Zr1 (m) (m) = ξ wi+m−1 wj+m−1

X=0

!

dX −

ξ=tan θ0 X−r1

Z1

(m)

∂ wi+m−1 (m) ξ wj+m−1 ∂X

0

!

dξ, X=−h

dξ,

0

and for the ∨ cones by the formulas ! Z1 (m) ∂ wi+m−1 (m) (m) αij = ξ wj+m−1 ∂X

X=0

0

− tan θ0

Z0

(m)

∂ wi+m−1 (m) ξ wj+m−1 ∂X

−h (m)

βij

=

Z1 0

dξ +

(m)

(m)

ξ wi+m−1 wj+m−1

X=0

Z0

(m)

∂ wi+m−1 (m) ξ wj+m−1 ∂ξ

−h

!

dX −

ξ=tan θ0 X+1

Zr1 0

(m)

!

dX ξ=tan θ0 X+1

∂ wi+m−1 (m) ξ wj+m−1 ∂X

!

dξ, X=−h

dξ.

Since this variational method implies the minimization of the functional (10), the matrix problem (13) gives the best approximation κm1 for the lowest eigenvalue. Moreover, the approximation κm1 converges to the corresponding eigenvalue of (9) from above. This makes it possible to check the convergence by the number of significant digits which do not change when q is increased. Convergence. Our numerical experiments were primary dedicated to the eigenvalues κm1 , m = 0, 1, 2, 3. These eigenvalues are responsible for the lowest natural modes, which are weakly damped and, therefore, give a decisive contribution to hydrodynamic loads. In the case of ∨-tanks, the method shows a fast convergence to κm1 and provides satisfactory accuracy 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), but also on the semi-apex angle θ0 and the dimensionless parameter 0 < r1 < 1. The results in Table I (A) exhibit a typical convergence behavior for the ∨-tanks with 10◦ ≤ θ0 ≤ 75◦ and 0.2≤r1 ≤0.9. The table shows that 5-6 digits are significant as q ≥ 14. The best accuracy is 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 natural mode, may become unstable for q > 17. This explains why we do not present numerical results on this eigenvalue for q = 20. While m6=1 and 0.2≤r1 ≤0.9, increasing θ0 > 75◦ leads to a slower convergence. In this case, q = 17, . . . , 20 guarantees only 3-4 significant digits (the necessary accuracy in practice). The same number of harmonic polynomials keeps this number of significant digits for the tanks with r1 < 0.2 and 10◦ ≤θ0 ≤75◦ . When r1 tends to zero (the truncated tank is close to a non-truncated one), the approximations κm1 were validated by numerical 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 in the latter case less efficient. In particular, computations of κ01 are not so precise. Moreover, c Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

9


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) corresponds to ∧-shaped tanks. Here, θ0 = 30◦ . A q r1 = 0.2 r1 = 0.4 r1 = 0.6 r1 = 0.8 r1 = 0.9 2 5 8 11 14 17

5.032415 3.394779 3.385603 3.385600 3.385600 3.385600

4.577833 3.392286 3.385592 3.385590 3.385590 3.385590

4.152352 3.384865 3.381845 3.381827 3.381822 3.381819

3.805501 3.141171 3.138861 3.138718 3.138665 3.138644

2.961399 2.200342 2.197438 2.197238 2.197162 2.197138

2 5 8 11 14 17 20

1.343631 1.304413 1.304378 1.304377 1.304377 1.304377 1.304377

1.335723 1.301793 1.301694 1.301692 1.301687 1.301686 1.301686

1.299977 1.254157 1.254054 1.253982 1.253972 1.253969 1.253967

1.007261 0.934412 0.933885 0.933835 0.933823 0.933819 0.933817

0.607340 0.542503 0.542284 0.542257 0.542251 0.542249 0.542248

2 5 8 11 14 17 20

2.443739 2.263550 2.263151 2.263150 2.263150 2.263150 2.263150

2.424390 2.263351 2.263087 2.263087 2.263086 2.263086 2.263086

2.386633 2.255096 2.255004 2.254976 2.254972 2.254969 2.254968

2.191371 2.015602 2.014923 2.014838 2.014811 2.014801 2.014796

1.572048 1.361905 1.360928 1.360837 1.360805 1.360795 1.360790

2 5 8 11 14 17 20

3.533170 3.181530 3.180251 3.180249 3.180249 3.180249 3.180249

3.520882 3.181299 3.180249 3.180248 3.180247 3.180247 3.180247

3.459048 3.179541 3.179080 3.179077 3.179074 3.179073 3.179073

3.316198 3.047179 3.046742 3.046654 3.046620 3.046606 3.046599

2.697426 2.329022 2.327044 2.326879 2.326812 2.326790 2.326779

q κ01 8 10 12 14 16 18 κ11 8 10 12 14 16 18 20 κ21 8 10 12 14 16 18 20 κ31 8 10 12 14 16 18 20

B r1 = 0.6 r1 = 0.8 r1 = 0.9

r1 = 0.2

r1 = 0.4

70.063366 50.467624 39.061112 32.075719 27.649041 25.152987

15.140938 11.124492 10.185632 10.017447 10.016124 10.014257

6.899986 6.672266 6.666704 6.665678 6.665058 6.664886

4.560991 4.551150 4.550877 4.550590 4.550422 4.550349

2.683411 2.683117 2.682932 2.682796 2.682751 2.682730

16.504854 13.947812 13.487667 12.247242 11.663497 11.450724 11.331833

5.694833 5.633730 5.631904 5.630560 5.629898 5.629777 5.629888

3.516782 3.515861 3.515547 3.515365 3.515269 3.515237 3.515214

1.661815 1.661684 1.661608 1.661559 1.661551 1.661537 1.661532

0.726555 0.726507 0.726489 0.726484 0.726482 0.726480 0.726478

45.024108 28.652496 25.183942 24.771657 21.512726 19.738583 17.760234

9.797804 9.096131 8.975383 8.968027 8.966657 8.966531 8.966524

5.950925 5.941913 5.941107 5.941002 5.940651 5.940603 5.940589

3.724251 3.724049 3.723708 3.723640 3.723591 3.723549 3.723542

1.923234 1.923019 1.922950 1.922930 1.922913 1.922899 1.922895

139.879750 83.822538 64.588532 53.410762 53.138732 38.855186 36.467491

15.670550 12.954922 12.229637 12.088554 12.079636 12.084001 12.084898

8.146078 8.046749 8.044916 8.044459 8.044234 8.044028 8.043988

5.634069 5.633996 5.633565 5.633498 5.633377 5.633343 5.633321

3.404035 3.403568 3.403460 3.403411 3.403360 3.403336 3.403328

18-20 harmonic polynomials lead to 4-5 significant digits of κm1 only for r1 ≥0.4. This is not the case for lower values of r1 . Furthermore, when r1 ≤0.2, q = 17, . . . , 20 can only guarantee 2-3 significant digits for κ11 . The slower convergence for the ∧-shaped tanks can in part be clarified by the occurrence of singular first derivatives of the eigenfunctions ψm at an inner vertex between L0 and L1 (see the mathematical results of Lukovsky et al. [18]). The harmonic polynomials are smooth in the (ξ, X)-plane, and, therefore, do not capture this singular behaviour. The singularity disappears when the corner angle is less than 90◦ . This occurs only for the ∧-shaped tanks. The ∨-shaped tanks are characterised by a similar singularity at 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 that singularity only for shallow water. Our numerical experiments confirm this fact as r1 < 0.1. c Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

10


O

x y

x 1 L *0 x 1 x 20 x r0 L 0 O x 10 x2 * G * 0

L1 L q0 L 0 x0 r1 r1 * x0 G L * q0 L 2 G L1 1 * L L y r0 2 2 O O x10

G

L 1*

L *2

x2 x20

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 shown by Lukovsky [16], Lukovsky and Timokha [21] and Gavrilyuk et al. [13], these methods require analytical 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 by Lukovsky [16, 13] links the x, y, z coordinates with x1 , x2 , x3 as follows x = x1 ,

y = x1 x2 cos x3 ,

z = x1 x2 sin x3 .

(14)

Thus the variable x3 = η is the polar angle in the Oyz-plane. In Fig. 4 it is demonstrated that the hydrostatic liquid domain Q0 takes in the (x1 , x2 , x3 )-system the form of an upright circular cylinder (x0 ≤ x1 ≤ x10 , 0 ≤ x2 ≤ x20 , 0 ≤ x3 ≤ 2π). The domain G∗ represents a rectangle with sides h = x10 − x0 and x20 = tan θ0 in the meridional plane Ox2 x1 . Here, the radius of the undisturbed water plane is rt = 1 for the ∨-tanks and rt = r1 for the ∧-tanks. Having represented sin m x3 ϕ(x1 , x2 , x3 ) = ψm (x1 , x2 ) , m = 0, 1, 2, . . . (15) cos m x3

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 to ψm (x1 , x2 ) p

∂ 2 ψm ∂ 2 ψm ∂ 2 ψm ∂ψm + 2q + s +d − m2 cψm = 0 in 2 2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x2


G∗ ,

(16)

11


p

∂ψm ∂ψm +q = κm pψm ∂x1 ∂x2

on L∗0 ,

(17)

s

∂ψm ∂ψm +q =0 ∂ x2 ∂ x1

on L∗1 ,

(18)

p

∂ψm ∂ψm +q = 0 on L∗2 , ∂ x1 ∂ x2

(19)

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

(20)

Zx20

ψ0 x2 dx2 = 0,

(21)

0

where G = {(x1 , x2 ) : x0 ≤ x1 ≤ x10 , 0 ≤ x2 ≤ x20 }, p = x21 x2 , q = −x1 x22 , s = x2 (x22 + 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 # 2 2 Z" .Z ∂ψm ∂ψm ∂ψm ∂ψm m2 2 2 J (ψm ) = p + 2q +s + ψ dx1 dx2 pψm dx2 (22) ∂x1 ∂x1 ∂x2 ∂x2 x2 m ∗

G∗

L∗ 0

on test functions satisfying (20). Particular solutions of (16) + (18). Gavrilyuk et al. [13] operated with the a spectral problem 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, this separation gives the following particular solutions xν1 Tν(m) (x2 ) and

(m) T¯ν (x2 ) , ν ≥ 0. x1+ν 1

(23)

(m)

In order to find Tν , we get the following homogeneous boundary value problem with the real parameter ν in both the equation and boundary condition x22 (1 + x22 )Tν′′(m) + x2 (1 + 2x22 − 2νx22 )Tν′(m) + ν(ν − 1)x22 − m2 Tν(m) = 0, (24) Tν′(m) (x20 ) = ν

x20 T (m) (x20 ), 1 + x220 ν

|Tν(m) (0)| < ∞.

(25)

It can be shown that the problem (24)-(25) has nontrivial solutions only for a countable set of ν = νmn > 0 (m = 0, 1, . . . ; n = 1, 2, . . .). (m) The second class of functions, T¯ν , appears only in the case of x0 6= 0, i.e., when the conical (m) tank is truncated. Computing T¯ν leads to the following ν-parametric problems x22 (1 + x22 )T¯ ′′(m) + x2 (1 + 4x22 + 2νx22 )T¯ ′(m) + (ν + 1)(ν + 2)x22 − m2 T¯ (m) = 0, (26) x20 ¯ (m) T¯ ′(m) (x20 ) + (ν + 1) T (x20 ) = 0, 1 + x220

which also have nontrivial solutions only for a countable set of nonnegative ν. c Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

(27)

12


Let us show that the solutions of (24) and (26) can be expressed in terms of the spheroidal harmonics and the set {νmn } is the same for both (24)-(25) and (26)-(27). For this purpose, 1 we change the variables in (24) and (26) by µ = (1 + x22 )− 2 and introduce the substitutions ν −1−ν ¯ y(µ) = µ T (µ) and y(µ) = µ T (µ) for (24) and (26), respectively. This reduces both equations to the same well-known differential equation m2 y(µ) = 0, (1 − µ2 )y ′′ (µ) − 2µy ′ (µ) + ν(ν + 1) − 1 − µ2 (m)

whose solutions indeed coincide with the Legendre function of first kind, i.e. y(µ) = Pν (µ). Further, performing similar operations in the boundary conditions (25) and (27) and introducing the substitution µ = cos θ, one obtains the same equation (m) ∂Pν (cos θ) =0 (28) ∂θ θ=θ0

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 ! ν 1 (m) 2 mk (m) 2 Tνmk (x2 ) = (1 + x2 ) Pνmk p (29) 1 + x22 ! −1−νmk 1 (m) 2 (m) T¯νmk (x2 ) = (1 + x2 ) 2 Pνmk p . (30) 1 + x22

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

Wk

(m)

(x1 , x2 ) = Nk

xν1mk Tν(m) (x2 ), mk

¯ (m) (x1 , x2 ) = N ¯ (m) x−1−νmk T¯ (m) (x2 ). W νmk 1 k k

(31)

(m)

¯ (m) are multipliers chosen from the condition and N k Z x20 (m) (m) (m) 2 (m) 2 ¯ 1 = ||Wk ||L∗2 +L∗0 = ||Wk ||L∗2 +L∗0 = x2 [(Wk |x1 =x10 )2 + (Wk |x1 =x0 )2 ]dx2 = 0 Z x20 ¯ (m) |x1 =x10 )2 + (W ¯ (m) |x1 =x0 )2 ]dx2 . = x2 [(W (32) k k

Here, Nk

0

(m) ¯ (m) have to have the unit norm (in the meanEq. (32) expresses the fact that Wk and W k ∗ ∗ square metrics) on the boundary L2 + L0 , where the conditions (17) and (19) should be numerically satisfied. Explicit formulas for these normalising multiplier has the following form (m)

Nk

1 1 s =q , xR20 2νmk 2νmk νmk (m) 2 x10 + x0 2 1 + x2 Pνmk dx2 0

1

1 ¯ (m) = q s N . k xR20 −2−2νmk mk −1−νmk (m) 2 x−2−2ν + x 2 10 0 1 + x2 Pνmk dx2 0



13

Table II. Values of νmn versus θ0 . n 1 2 3 4 5 6 7 8 9 10 11 12

ν0n 6.83539808 12.90828411 18.93644560 24.95138112 30.96063428 36.96692917 42.97148871 48.97494349 54.97765160 60.97983147 66.98162391 72.98312378

1 2 3 4 5 6 7 8 9 10 11 12

4.40532918 8.44711262 12.46332875 16.47193967 20.47727740 24.48090964 28.48354098 32.48553497 36.48709810 40.48835640 44.48939109 48.49025692

1 2 3 4 5 6 7 8 9 10 11 12

3.19569115 6.21952915 9.22884937 12.23380906 15.23688626 18.23898124 21.24049935 24.24164994 27.24255203 30.24327826 33.24387547 36.24437524

θ0 = π/6 ν1n ν2n 3.11959709 5.49282500 9.71206871 12.37204261 15.82152796 18.58301633 21.87016680 24.68578005 27.89785663 30.74738346 33.91577437 36.78860888 39.92832912 42.81818932 45.93761963 48.84047013 51.94477437 54.85786626 57.95045476 60.87183012 63.95507442 66.88328881 69.95890533 72.89286242 θ0 = π/4 2.00000000 3.63323872 6.33388964 8.13729380 10.39698483 12.25919783 14.42505010 16.31855929 18.44103192 20.35413803 22.45137475 24.37794511 26.45862229 28.39502631 30.46398566 32.40789182 34.46811614 36.41793652 38.47139553 40.42599923 42.47406256 44.43261536 46.47627426 48.43814300 θ0 = π/3 1.46798738 2.75258821 4.65418866 6.04042231 7.69054134 9.11091186 10.70673365 12.14521751 13.71595755 15.16577182 16.72192772 18.17952282 19.72611149 21.18938780 22.72920771 24.19681751 25.73159226 27.20261792 28.73348550 30.20727365 31.73502524 33.21109395 34.73630210 36.21428568

ν3n 7.75244235 14.91804177 21.24614849 27.41663567 33.52289135 39.59590680 45.64932001 51.69015118 57.72240544 63.74854246 69.77015940 75.78833988 5.20142706 9.87466847 14.06452673 18.16308293 22.22447977 26.26665923 30.29751029 34.32109176 38.33971851 42.35481185 46.36729454 50.37779256 4.00000000 7.38840139 10.49837828 13.55538637 16.59087636 19.61524899 22.63307179 25.64669297 28.65745115 31.66616793 34.67337662 37.67943893

Note, that the case of m = 0 needs also the volume conservation condition (21). This (0) ¯ (0) by W (0) := W (0) − c(0) , W ¯ (0) := W ¯ (0) − c¯(0) , where redefines the functions Wk and W k k k k k k k (0) ck

2 = 2 x20

Zx20 (0) x2 Wk (x10 , x2 ) d x2 ,

(0) c¯k

0

2 = 2 x20

Zx20 ¯ (0) (x10 , x2 ) d x2 . x2 W k 0

Variational method. In accordance with the Treftz scheme, we present the approximate solution of (16)-(21) in the form ψm (x1 , x2 ) =

q1 X

(m)

(m)

ak Wk

k=1


+

q2 X l=1

(m)

a ¯l

¯ (m) . W l

(33)

14


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

∂

(m) ak

= 0,

k = 1, 2, . . . , q1 ;

∂ Jm

(m)

∂a ¯l

lead to the spectral matrix problem (m) (m) det {α ˜ ij } − κm {β˜ij } = 0,

= 0,

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

i, j = 1, 2, ..., q1 + q2 ,

(34)

with the spectral parameter κm . The spectral problem (34) has q1 + q2 eigenvalues. Because (m) ¯ (m) , there exist four the presentation (33) contains two types of functions, i.e. Wk and W l (m) (m) sub-matrices of {α ˜ ij } and {β˜ij } that read as ! ! (m) (m) (m) (m) αij1 αij2 β β (m) (m) ij2 α ˜ ij = , β˜ij = ij1 . (m) (m) (m) (m) αij3 αij4 βij3 βij4 (m)

(m)

The elements {αijs } and {βijs } s = 1, 4 are computed by the formulas ! Zx20 (m) (m) ∂Wi (m) (m) 2 ∂Wi 2 αij1 = x1 x2 − x1 x2 Wj dx2 ∂x1 ∂x2 x1 =ht 0 ! Zx20 (m) (m) ∂Wi ∂Wi (m) − x21 x2 − x1 x22 Wj dx2 , ∂x1 ∂x2 x1 =hb 0 ! Zx20 (m) (m) (m) 2∂Wi 2 ∂Wi ¯ (m) dx2 αij2 = x1 x2 − x1 x2 W j ∂x1 ∂x2 x1 =ht 0 ! Zx20 (m) (m) ∂Wi ∂Wi ¯ (m) dx2 , − x21 x2 − x1 x22 W j ∂x1 ∂x2 x1 =hb

0

(m) αij3

−

Zx20 0

=

Zx20 0

¯ (m) ∂W i x21 x2 ∂x1

¯ (m) ∂W i x21 x2 ∂x1

−

−

¯ (m) ∂W i x1 x22 ∂x2

¯ (m) ∂W i x1 x22 ∂x2

!

!

(m)

Wj

dx2

x1 =ht (m)

Wj

dx2 ,

x1 =hb

Zx20

! ¯ (m) ¯ (m) ∂ W ∂ W i i ¯ (m) dx2 = x21 x2 − x1 x22 W j ∂x1 ∂x2 x1 =ht 0 ! Zx20 ¯ (m) ¯ (m) ∂ W ∂ W i i ¯ (m) dx2 , − x1 x22 W − x21 x2 j ∂x1 ∂x2

(m) αij4

x1 =hb

0

(m) βij1

(m)

Zx20 (m) (m) 2 = ht x2 Wi Wj

βij3 = h2t

dx2 ,

x1 =ht

0 Zx20 0

¯ (m) W (m) x2 W i j

(m) βij2

(m)

=

h2t

dx2 , βij4 = h2t

x1 =ht


Zx20 (m) ¯ (m) x2 Wi W j

dx2 ,

x1 =ht

0 Zx20 0

¯ (m) W ¯ (m) x2 W i j

dx2 .

x1 =ht


15

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 that 4-5 significant figures of κm1 may be stabilised within q = q1 = q2 = 7÷10 (14÷20 basic functions). 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 includes the 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 number of 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 angles with the asymptotic behaviour of the exact solution along vertical axis. The eigenfunctions ψm should exponentially decay downward Ox for a circular cylindrical tank, to which the (m) ¯ (m) do not capture this decaying. conical domain tends as θ0 decreases. However, Wk and W k Further, decreasing the non-dimensional liquid depth h (r1 → 1 or θ0 → 90◦ ) may cause a lower 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 in the 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 when 0.2≤r1 ≤0.9. However, in contrast to the harmonic polynomials, the new basis provides stable computations 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 the semi-apex angle may lead to a slower convergence. The number of significant figures within q1 = q2 = 12 decreases also in the limit r1 →1. These “shallow water” cases are handled with 2-3 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 q2 to 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 is expected 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 = 0.54233738 can 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, larger depths (r1 ≤0.4) are best handled by the second method. This is clearly seen for the ∧-shaped tanks: the calculations by the method of § 3.2 keep robustness with increasing the number of basic functions, while the first method invalidates for large dimensions. The accuracy of both methods 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 accurate approximations of the lowest eigenvalue κ11 . Because this eigenvalue computes the lowest √ natural 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 results obtained by our two methods fro κ11 . This is illustrated by Figs. 5 (a) and (b). The figures c Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

16


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 q r1 = 0.2 r1 = 0.4 r1 = 0.6 r1 = 0.8 r1 = 0.9 r1 = 0.2 κ01 2 3.385676 3.385666 3.382064 3.148405 2.224870 20.829848 3 3.385606 3.385596 3.381920 3.143923 2.211744 20.292399 4 3.385601 3.385591 3.381878 3.141937 2.206179 20.151838 5 3.385600 3.385590 3.381859 3.140888 2.203283 20.097111 6 3.385600 3.385590 3.381847 3.140267 2.201583 20.070700 7 3.385600 3.385590 3.381840 3.139870 2.200501 20.056100 8 3.385600 3.385590 3.381835 3.139601 2.199769 20.047236 9 3.385600 3.385590 3.381832 3.139410 2.199252 20.041473 10 3.385600 3.385590 3.381829 3.139270 2.198872 20.037525 11 3.385600 3.385590 3.381827 3.139164 2.198585 20.034708 12 3.385600 3.385590 3.381826 3.139082 2.198364 20.032629 κ11 2 1.304378 1.301707 1.254338 0.935957 0.544861 11.335332 3 1.304377 1.301695 1.254148 0.934864 0.543483 11.315765 4 1.304377 1.301691 1.254073 0.934437 0.542976 11.308794 5 1.304377 1.301689 1.254036 0.934226 0.542729 11.305577 6 1.304377 1.301688 1.254016 0.934106 0.542589 11.303842 7 1.304377 1.301687 1.254003 0.934032 0.542502 11.302805 8 1.304377 1.301687 1.253994 0.933983 0.542445 11.302137 9 1.304377 1.301687 1.253988 0.933949 0.542405 11.301682 10 1.304377 1.301686 1.253984 0.933924 0.542376 11.301359 11 1.304377 1.301686 1.253981 0.933906 0.542354 11.301121 12 1.304377 1.301686 1.253978 0.933892 0.542337 11.300941 κ21 2 2.263162 2.263100 2.255147 2.019323 1.371212 18.039183 3 2.263151 2.263088 2.255060 2.017249 1.366295 17.982154 4 2.263150 2.263087 2.255026 2.016335 1.364234 17.960135 5 2.263150 2.263087 2.255007 2.015850 1.363151 17.949506 6 2.263150 2.263086 2.254996 2.015563 1.362510 17.943614 7 2.263150 2.263086 2.254989 2.015378 1.362099 17.940025 8 2.263150 2.263086 2.254985 2.015252 1.361819 17.937683 9 2.263150 2.263086 2.254981 2.015163 1.361620 17.936074 10 2.263150 2.263086 2.254979 2.015097 1.361473 17.934922 11 2.263150 2.263086 2.254977 2.015047 1.361362 17.934070 12 2.263150 2.263086 2.254976 2.015008 1.361276 17.933423 κ31 2 3.180280 3.180279 3.179147 3.051144 2.346969 24.361239 3 3.180251 3.180250 3.179101 3.049247 2.338329 24.254295 4 3.180249 3.180248 3.179090 3.048337 2.334334 24.210302 5 3.180249 3.180249 3.179085 3.047828 2.332117 24.188236 6 3.180249 3.180247 3.179082 3.047514 2.330753 24.175694 7 3.180249 3.180247 3.179080 3.047306 2.329853 24.167920 8 3.180249 3.180247 3.179078 3.047161 2.329228 24.162784 9 3.180249 3.180247 3.179077 3.047056 2.328776 24.159223 10 3.180249 3.180247 3.179077 3.046978 2.328438 24.156655 11 3.180249 3.180247 3.179076 3.046918 2.328179 24.154745 12 3.180249 3.180247 3.179076 3.046871 2.327976 24.153288


r1 = 0.4

B r1 = 0.6 r1 = 0.8 r1 = 0.9

10.414889 10.146162 10.075881 10.048517 10.035311 10.028011 10.023579 10.020697 10.018723 10.017315 10.016275

6.934470 6.754682 6.707583 6.689248 6.680396 6.675501 6.672529 6.670596 6.669272 6.668327 6.667629

4.781070 4.629653 4.588863 4.572788 4.564916 4.560515 4.557819 4.556053 4.554837 4.553964 4.553317

2.868894 2.748954 2.715911 2.702514 2.695817 2.692006 2.689637 2.688066 2.686973 2.686181 2.685590

5.647088 5.637221 5.633703 5.632078 5.631202 5.630678 5.630340 5.630110 5.629947 5.629826 5.629735

3.528677 3.521265 3.518590 3.517343 3.516665 3.516258 3.515995 3.515815 3.515687 3.515592 3.515520

1.672867 1.666870 1.664611 1.663527 1.662926 1.662559 1.662319 1.662153 1.662034 1.661945 1.661878

0.732410 0.729246 0.728093 0.727536 0.727224 0.727032 0.726906 0.726818 0.726754 0.726707 0.726671

9.019177 8.990657 8.979646 8.974330 8.971384 8.969588 8.968417 8.967612 8.967036 8.966610 8.966287

5.977616 5.958186 5.950674 5.947040 5.945023 5.943793 5.942990 5.942438 5.942042 5.941750 5.941527

3.761864 3.742630 3.734876 3.731012 3.728818 3.727457 3.726556 3.725930 3.725477 3.725140 3.724882

1.952759 1.938182 1.932215 1.929167 1.927400 1.926285 1.925537 1.925011 1.924627 1.924338 1.924115

12.180611 12.127139 12.105142 12.094109 12.087838 12.083951 12.081383 12.079602 12.078318 12.077363 12.076635

8.115651 8.079896 8.065186 8.057806 8.053610 8.051009 8.049290 8.048098 8.047239 8.046600 8.046112

5.700453 5.667935 5.654291 5.647308 5.643277 5.640748 5.639061 5.637882 5.637026 5.636386 5.635894

3.473876 3.441587 3.427352 3.419803 3.415319 3.412440 3.410481 3.409090 3.408066 3.407290 3.406689

LINEAR SLOSHING IN A TRUNCATED CONICAL TANK 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

80 70 60 50 40 30 20 10

80 70 60 50 40 30 20 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

r1

0

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(b)

80

80 70 60 50 40 30 20 10

70

60 50 40 30 20 10

q0

q0

0

17

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

r1

1

Figure 5. The number of significant figures of κ11 obtained for ∨-tanks with twenty basic functions by the 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 of significant figures with twenty basic functions. One can see that the accuracy of the first method (§ 3.1) may become low only for small θ0 and r1 , e.g. for large liquid depths. In the other 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 converges slowly 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 the eigenvalues. For the ∧-tanks, increasing θ0 increases κm1 . The dependence of κm1 on nondimensional r1 is illustrated in Figs. 6 (a) and (b). The natural sloshing frequency σ11 is of primary practical importance for design of water towers (Damatty and Sweedan [5]). Remembering that, we present in Table IV approximate values of κ11 versus θ0 and r1 . The dimensional natural sloshing frequency σ11 is computed by κ11 via the formula s σ11 =

g κ11 (θ0 , rr10 ) r0

,

(35)

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

5. Conclusions and future work Two efficient numerically-analytical methods are proposed to compute the natural sloshing frequencies and modes in truncated conical tanks. These methods are based on the Treftz variational scheme. As a functional basis, the first method uses harmonic functions of polynomial type. The second method employs a non-conformal mapping technique and derives c Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

18


100

3.5

κm1

90

θ0=π/6

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

80

3

κ

θ =π/6

m1

0

2.5

70 60

2

50 1.5

40 30

1

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

20 0.5 10

r1

0 0

0.2

0.4

0.6

0.8

1

40

r1

0 0

0.2

0.4

0.6

0.8

0.6

0.8

1

3

κ

θ =π/4

m1

35

κm1

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

0

θ =π/4 0

2.5

30 2 25 20

1.5

15 1 m=0 m=1 m=2 m=3

10 0.5 5

r1 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

40

2.5

κ 35

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

θ =π/3

m1

0

r1

0 0

0.2

0.4

κ

1

θ =π/3

m1

0

2

30 25

1.5

20 1

15 10

0.5 5

r1 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a)

0 0

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

r1 0.2

0.4

0.6

0.8

1

(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 domain of 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 contact line formed by water-plane and conical walls. This was discussed in § 3.1. From mathematical c Gavrilyuk, Hermann, Lukovsky, Solodun, Timokha

19


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

r1 \θ0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

10◦ 1.6743 1.6743 1.6743 1.6743 1.6743 1.6743 1.6743 1.6743 1.6743 1.6743 1.6743 1.6743 1.6740 1.6727 1.6674 1.6472 1.5781 1.3702 0.8631

15◦ 1.5862 1.5862 1.5862 1.5862 1.5862 1.5862 1.5862 1.5862 1.5862 1.5860 1.5856 1.5842 1.5799 1.5683 1.5391 1.4709 1.3251 1.0471 0.5955

20◦ 1.4950 1.4950 1.4950 1.4950 1.4950 1.4950 1.4950 1.4949 1.4945 1.4935 1.4908 1.4842 1.4697 1.4395 1.3808 1.2741 1.0945 0.8199 0.4461

25◦ 1.4011 1.4011 1.4011 1.4011 1.4010 1.4010 1.4008 1.4002 1.3987 1.3952 1.3877 1.3730 1.3456 1.2973 1.2172 1.0918 0.9084 0.6599 0.3511

30◦ 1.3044 1.3044 1.3044 1.3044 1.3043 1.3041 1.3034 1.3017 1.2980 1.2908 1.2774 1.2540 1.2153 1.1544 1.0632 0.9339 0.7606 0.5423 0.2849

35◦ 1.2052 1.2052 1.2052 1.2051 1.2049 1.2043 1.2027 1.1994 1.1930 1.1816 1.1625 1.1320 1.0856 1.0181 0.9242 0.7994 0.6416 0.4521 0.2356

40◦ 1.1037 1.1037 1.1037 1.1035 1.1030 1.1017 1.0990 1.0938 1.0846 1.0696 1.0461 1.0110 0.9607 0.8915 0.8002 0.6844 0.5438 0.3800 0.1970

45◦ 1.0000 1.0000 1.0000 0.9996 0.9986 0.9966 0.9927 0.9857 0.9743 0.9566 0.9304 0.8932 0.8423 0.7751 0.6897 0.5850 0.4614 0.3207 0.1657

50◦ 0.8943 0.8943 0.8941 0.8935 0.8922 0.8894 0.8845 0.8762 0.8633 0.8442 0.8171 0.7800 0.7309 0.6681 0.5905 0.4978 0.3906 0.2704 0.1394

55◦ 0.7868 0.7868 0.7865 0.7857 0.7840 0.7806 0.7750 0.7660 0.7525 0.7332 0.7067 0.6715 0.6261 0.5693 0.5007 0.4202 0.3284 0.2267 0.1167

60◦ 0.6777 0.6776 0.6772 0.6763 0.6742 0.6706 0.6647 0.6556 0.6425 0.6242 0.5996 0.5676 0.5271 0.4775 0.4183 0.3499 0.2727 0.1879 0.0966

65◦ 0.5671 0.5670 0.5665 0.5655 0.5634 0.5598 0.5541 0.5456 0.5335 0.5171 0.4954 0.4676 0.4330 0.3910 0.3417 0.2851 0.2218 0.1526 0.0784

70◦ 0.4553 0.4551 0.4547 0.4536 0.4517 0.4484 0.4433 0.4359 0.4256 0.4117 0.3936 0.3707 0.3425 0.3086 0.2691 0.2241 0.1741 0.1197 0.0615

point of view, augmenting the smooth bases by a harmonic function, which has the specified singular behaviour, may considerably improve convergence. Examples of such an augmenting for two-dimensional spectral sloshing problems are given by Lukovsky et al. [18]. However, to the authors’ knowledge, the literature does not exemplify those analytical harmonic function for the studied case. Special emphasis should also be placed on the shallow water case. This requires a dedicated study based on a nonlinear dissipative sloshing model. Nonlinear phenomena are also of importance for resonant sloshing (Gavrilyuk et al. [13]). Results of the present paper may be utilised to develop the multi-modal technique (Lukovsky [19], Faltinsen et al. [10], Gavrilyuk et al. [13]) and study the nonlinear sloshing in a truncated conical tank. This is main purpose of the forthcoming studies.

ACKNOWLEDGEMENTS

Authors thank the German Research Society (DFG) for financial support (project DFG 436UKR 113/33/00). The last author (A.T.) is grateful for the sponsorship presented by the Alexander von Humboldt Foundation.

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