Linear and nonlinear sloshing in a circular conical tank

Transcription

1 Fluid Dynamics Research 37 (5) Linear and nonlinear sloshing in a circular conical tank I.P. Gavrilyuk a,, I.A. Lukovsky b,, A.N. Timokha c,, a Berufakademie Thüringen-Staatliche Studienakademie, Am Wartenberg, Eisenach, 9987, Germany b Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska 3, Kiev, 6, Ukraine c Centre for Ships and Ocean Structures, NTNU, Trondheim, 749, Norway Received 8 March 4; received in revised form 6 April 5; accepted August 5 Communicated by M. Oberlack Abstract Linear and nonlinear fluid sloshing problems in a circular conical tank are studied in a curvilinear coordinate system. The linear sloshing modes are approximated by a series of the solid spheric harmonics. These modes are used to derive a new nonlinear modal theory based on the Moiseyev asymptotics. The theory makes it possible to both classify steady-state waves occurring due to horizontal resonant excitation and visualise nonlinear wave patterns. Secondary (internal) resonances and shallow fluid sloshing (predicted for the semi-apex angles α > 6 ) are extensively discussed. 5 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. Keywords: Sloshing; Variational method; Nonlinear modal system. Introduction A fluid partly occupying a moving tank undergoes wave motions (sloshing). These motions generate severe hydrodynamic loads that can be dangerous for structural integrity and stability of rockets, The paper is dedicated to the memory of Maurizio Landrini. Corresponding author. address: alextim@marin.ntnu.no (A.N. Timokha). Authors are grateful for support made by the German Research Council (DFG). Author acknowledges partial support of the Alexander-von-Humboldt Foundation /$3. 5 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. doi:.6/j.fluiddyn.5.8.4

2 4 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) satellites, ships, trucks and even stationary petroleum containers. If the tank interior is equipped with slosh-suppressing devices, the fluid motions can be accurately modelled by linear theories. Mathematical and numerical aspects of these theories have been documented in NASA Space Vehicle Design Criteria (968, 969) and by Abramson (966), Feschenko et al. (969) and Rabinovich (975). Environmental concerns require a new design for tanks of ships, roads and storage systems (Minowa, 994; Minowa et al., 994; Faltinsen and Rognebakke, ). This suggests double walls, bottom and roof and may as a result increase the total structural weight. A way to avoid the weight penalty consists of removing the slosh-suppressing structures. However, this invalidates linear theories. Moan and Berge (997), Cariou and Casella (999) and Frandsen (4) reported comparative surveys of the computational fluid dynamics (CFD) methods, which are applicable to nonlinear sloshing problems. By utilising among other programs the commercial FLOW3D-software developed by Flow Science, Inc., Solaas (995) and Landrini et al. (999) showed that, if sufficient care is not shown, many of these methods can numerically lose or generate fluid mass and kinetic energy on a long-time scale. An exception is the smoothed particle hydrodynamics (Landrini et al., 3) method. However, this method is CPU-time inefficient and runs into serious difficulties to identify steady-state fluid wave motions appearing after transients. An alternative to the CFD-methods consists of analytically oriented, modal approaches. Their fundamental aspects have been elaborated by Lukovsky (99), Faltinsen et al. (, 3), Faltinsen and Timokha (b) and La Rocca et al. (997, ). The modal approaches are based on the Fourier representation of the free surface x = f(y,z,t)= β i (t)f i (y, z) () in the Oxyz-coordinate system rigidly fixed with a moving tank, where {F i (y, z)} coincides with the linear surface modes. When substituting () into the original free boundary problem and using a variational scheme (Lukovsky, 976, 99; Miles, 976; La Rocca et al., ; Faltinsen et al., ; Shankar and Kidambi, ), one can derive an infinite-dimensional system of ordinary differential equations in β i (t) (modal system). Besides, assuming asymptotic relationships between β i (t) truncates this system to a finite-dimensional form. Such an asymptotic truncation may utilise the so-called Moiseyev asymptotics (Narimanov, 957; Moiseyev, 958; Ockendon and Ockendon, 973; Faltinsen, 974; Dodge et al., 965; Miles, 984a,b; Lukovsky, 99; Solaas and Faltinsen, 997; Gavrilyuk et al., ; Ibrahim et al., ; La Rocca et al., 997; Faltinsen et al.,, 3), for which applicability has been justified in the case of finite fluid depths. Asymptotic relationships of shallow sloshing were reported by Chester (968), Chester and Bones (968), Ockendon and Ockendon (973, ) and Faltinsen and Timokha (a). An analytical modal basis {F i (y, z)} exists for a very limited set of tank shapes.among these are tanks of two- or three-dimensional rectangular and vertical circular cylindrical geometry. The modal representation () admits the use of approximated {F i (y, z)} (Feschenko et al., 969; Lukovsky, 99; Solaas and Faltinsen, 997), but only if the tank walls are vertical in vicinity of the mean (hydrostatic) free surface. Otherwise, F i (y, z), i, have time-dependent domains of definition. The present paper is devoted to the case of strongly non-vertical (conical) walls. The study is based on a spatial transformation technique proposed by Lukovsky (975) and developed by Lukovsky and Timokha (). This assumes that the original tank cavity can be transformed to an artificial cylindrical domain (in curvilinear coordinates

3 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) (x,x,x 3 )), where the free surface allows for the normal parametrisation and ξ (x,x,x 3,t)= x f (x,x 3,t)= () f (x,x 3,t)= const + β (t) + β i (t) F i (x,x 3 ). (3) i= After the theoretical preliminaries in Section, we define admissible transformations of conical domains. In Section 3, we consider the linear sloshing problem in a curvilinear coordinate system and an analytically oriented variational method for approximating the linear sloshing modes (natural modes). The numerical results are in good agreement with experimental data by Bauer (98) and Mikishev and Dorozhkin (96). In Section 4, we derive a nonlinear modal system. This system is based on the Moiseyev asymptotics and generalises analogous modal systems by Dodge et al. (965), Lukovsky (99), Gavrilyuk et al. () and Faltinsen et al. (3) dealing with vertical cylindrical tanks. Shallow fluid flows (quantified for the semi-apex angles α > 6 ) and secondary resonances (predicted at α 6 and ), which lead to failure of the Moiseyev asymptotics, are not evaluated, but rather discussed. Applicability of the nonlinear modal system is justified in the range 5 < α < 6. Analysing periodic solutions of the system, which are associated with resonant steady-state sloshing due to horizontal harmonic excitations with frequencies close to the lowest natural frequency, reveals planar and swirling regimes as well as chaotic waves (both planar and swirling are not stable). Response curves corresponding to planar and swirling are studied. While response curves of the planar regime keep the soft-spring behaviour for α < 6, swirling demonstrates transition from the hard-spring to soft-spring behaviour as α passes through 4. (to the authors knowledge, this critical angle has never been estimated in the scientific literature). These and other results on steady-state resonant motions are compared with analogous results established for sloshing in a circular cylindrical tank (Abramson, 966; Miles, 984a,b; Lukovsky, 99; Gavrilyuk et al., ).. Theoretical preliminaries.. Free boundary problem We consider wave motions of an incompressible perfect fluid partly occupying a rigid mobile tank Q (Fig. ). The motions are described in a non-inertial Cartesian system Oxyz rigidly fixed with the tank. Motions of the tank are described in an absolute Cartesian coordinate system O x y z by the pair of time-dependent vectors v O (t) and ω(t). These vectors denote translatory and angular velocities of the Oxyz-frame, respectively. Since any absolute position vector r (t) = (x,y,z ) can be decomposed into the sum of r O (t) = O O and the relative position vector r = (x,y,z), the gravity potential U depends on (t,x,y,z), i.e. U(x,y,z,t)= g r, r =r O +r, where g is the gravity acceleration vector. Furthermore, we assume irrotational potential fluid flows and introduce the velocity potential Φ(x,y,z,t). The following free boundary problem (derived for instance by Narimanov et al., 977; Lukovsky, 99, 4) couples

4 4 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) ω(t) x Q x y r O Σ(t) z r o Q(t) S(t) r O z y Fig.. Sketch of a moving conical tank. Φ(x,y,z,t)and the fluid shape Q(t) (defined by the equation ξ(x,y,z,t)= ) ΔΦ = in Q(t), (4a) Φ ν = v ν + ω [r ν] on S(t), (4b) Φ ν = v ν + ω [r ν] ξ / ξ t on Σ(t), (4c) Φ + t ( Φ) Φ (v + ω r) + U = on Σ(t), (4d) dq = const, (4e) Q(t) where S(t) is the wetted walls and bottom, Σ(t) is the free surface and ν is the outer normal. The evolutional free boundary problem (4) should be completed by initial conditions. They define the initial fluid shape and velocity as follows: ξ(x,y,z,t ) = ξ (x,y,z); where ξ and ξ are known... Transformations ξ t (x,y,z,t ) = ξ (x,y,z), (5) Let us consider an open artificial cylindrical domain Q = (,d) D in the Ox x x 3 -coordinate system and let Q be the interior of a rigid tank in the Oxyz-system. We define smooth transformations that map Q to Q and back as follows: x = x, x = x (x, y, z), x 3 = x 3 (x,y,z); x = x, y = y(x,x,x 3 ), z = z(x,x,x 3 ). (6)

5 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) d x d x Q Σ Q * * Σ Q h Q * h y O x O z x 3 Fig.. Sketch of an admissible transformation. Here, the Oyz-plane has to be tangent to S = Q and x> for (x,y,z) Q and the Ox x 3 -plane should be superposed with the bottom of Q (Fig. ). Transformations (6) are also obligated to have the positive Jacobian J (x,x,x 3 ) = D(x,y,z) D(x,x,x 3 ), (J(x,y,z) = /J ) (7) inside of Q and, except a limited set of isolated points, on the boundary S = Q. Such a single point with J = invertible appears for conical, parabolic, etc. domains, because (6) maps the bottom of Q to the origin O (the situation is schematically depicted in Fig. ).... Linear natural modes If v = ω =, the free boundary problem (4) can be linearised relative to the trivial solution ξ = z + const, Φ = const, which determines a hydrostatic fluid shape Q (Fig. ). The linearisation implies the smallness of Φ and f (Σ(t) : x = f(t,y,z)), and, apparently, becomes mathematically justified only in a curvilinear coordinate system. The procedure includes transformation (6) (admitting () (3)), evaluates (4) in the (x x,x 3 )-coordinates, assumes Φ f h Φ f =O(ε)> and, finally, neglects the o(ε)-term. After linearising (4), we set up Φ = i gκ (x,x,x 3 )exp(i gκt); f =exp(i gκt)f(x,x ) and obtain a spectral boundary problem in Q =(,h) D. In accordance with theorems proved by Lukovsky and Timokha (), this spectral problem is isomorphically equivalent to Δ = in Q ; ν = on S ; z = κ on Σ ; which is considered in the (x,y,z)-coordinates (Feschenko et al., 969). Σ z dy dz =, (8)

6 44 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) z L x x L * G G L * * L O z L * x x Fig. 3. Transformations of the meridional cross-section. If Q is of an axial-symmetric shape, admissible transformations of Q to Q can be combined with separation of spatial variables. This leads to an infinite series of two-dimensional spectral problems in a rectangular domain. Reduction to these two-dimensional problems includes two steps. The first step implies the substitution x = z, y = z cos z 3, z= z sin z 3 (9) together with expression for : m (z,z,z 3 ) = ψ m (z,z ) cos sin (mz 3), m =,,,.... () Inserting (9) () into (8) leads to the following family of spectral problems in the meridional plane Oz z : ( ) ψ z m + ( ) ψ z m m ψ z z z z m = in G; z ψ m ν = on L ; ψ m (z, ) <, m=,,,..., ψ m = κψ z m on L, L ψ z dz =, () where L and L are the boundaries of G (see Fig. 3); ν is the outer normal to L (theory of () is given by Lukovsky et al., 984). The second step assumes z = x, z = ζ(x,x ), (z 3 = x 3 ), () which maps G to G as shown in Fig. 3 (L L,L L ). In the Ox x x 3 -coordinate system, the spectral problems () take the form ( p ψ m + q ψ m + s ψ m p + + q ) ψm x x x x x x + ( s x + q x x ) ψm x cm ψ m = in G,

7 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) where p ψ m + q ψ m = κpψ x x m on L ; s ψ m + q ψ m = on L x x, x ψ ζ ζ x dx =, m=,,,..., (3) p(x,x ) = ζ ζ ; q(x,x ) = pa; s(x,x ) = p(a + b ), x a(x,x ) = x ζ ; b(x,x ) = x ζ ; c(x,x ) = ζ. (4) ζ x As shown by Lukovsky and Timokha (), problem (3) does not need any boundary conditions on the artificial bottom L and along x =. However, ψ m should be bounded at x = and x =, simultaneously.... Nonlinear modal system We employ the Bateman Luke variational formulation of (4): t δw(φ, ξ) = δ L dt = ; t L = (p p ) dq = ρ Q(t) Q(t) [ Φ t + ] ( Φ) Φ (v + ω r) + U dq, (5a) δφ t,t = ; δξ t,t =. (5b) Here, the nonlinear functional (5a) is based on the pressure-integral Lagrangian. Lukovsky and Timokha () have mathematically established that the Lagrangian (5a) is invariant relative to transformations (6), namely, [ L L Φ = ρ + ] t ( Φ ) Φ (v + ω r) + U J dq, (6) Q (t) where Q (t) is the transformed domain, U = U(x(x,x,x 3 ), y(x,x,x 3 ), z(x,x,x 3 ), t), Φ = Φ(x(x,x,x 3 ), y(x,x,x 3 ), z(x,x,x 3 ), t), ) Φ = Φ,j Φ,j Φ 3,j Φ = (g,g,g, x j x j x j and g i,j = r x i r x j, i,j =,, 3, is the metric tensor (stars in (v +ω r) denote projections on the unit vectors of a curvilinear coordinate system).

8 46 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) By adopting derivations (based on (5)) of nonlinear modal systems by Lukovsky (99) and Faltinsen et al. () and the modal solutions f = x + β (t) + β i (t) F i (x,x 3 ), (7a) Φ = v r + i= Z n (t) n (x,x,x 3 ), n= (7b) where x = h, { F i (x,x 3 )}, { n (x,x,x 3 )} are the basic systems of functions on Σ and in Q, respectively, and, for brevity, ω =, we get the following infinite-dimensional nonlinear modal system: d dt A n A nk Z k =, n=,,..., (8a) k n Ż n A n β i + nk A nk β i Z n Z k + ( v g ) l β i + ( v g ) l + ( v 3 g 3 ) l 3 =, i =,,.... (8b) β i β i This system couples the generalised coordinates Z n (t), β i (t) and A n (β i ), A nk (β i ), l k (β i ) defined by the integrals ( ) f ( ) f A n = ρ n J dx dx dx 3 ; A nk = ρ ( n, k )J dx dx dx 3, (9) D l = ρ β i l = ρ β i l 3 = ρ β i Σ Σ Σ F i [x J (x,x,x 3 )] x =f dx dx 3, F i [y(x,x,x 3 )J (x,x,x 3 )] x =f dx dx 3, D F i [z(x,x,x 3 )J (x,x,x 3 )] x =f dx dx 3 () (the upper limit f in integrals (9) depends on β i (t) due to (7a)). 3. Linear sloshing in a circular conical tank Numerical solutions of the spectral problem (8) in a conical domain Q can be found by diverse methods based on spatial discretisation (Solaas, 995; Solaas and Faltinsen, 997). However, because these discrete solutions { n } are not expandable over the mean free surface Σ, their usage in (9) is generally impossible. To the authors knowledge, the current scientific literature contains only two numerical approaches to (8) which give satisfactory approximations of eigenfunctions { n } to be used as a basis in the variational scheme of Section... The first approach is based on the Treftz method with the harmonic polynomials as a functional basis. Results by Feschenko et al. (969) showed applicability

9 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) x z x Σ r L * L y h O α S Q z h α G O L r z x * L G * x x L* Fig. 4. Sketch of a conical tank, its meridional cross-section G and the transformed domain G ; x = h, x = tan α. of this approach for computing both eigenvalues and eigenfunctions. The approximate eigenfunctions are harmonic in space and, therefore, expandable over Σ. However, because the approximate { n } does not satisfy zero-neumann condition on conical walls over Σ, inserting them into (9) can lead to a numerical error for certain evolutions of Q(t). Physically, the error may be treated as inlet/outlet through the rigid walls over Σ. Another appropriate analytically oriented approach to (8) in a conical domain was reported by Dokuchaev (964), Bauer (98), Lukovsky and Bilyk (985) and Bauer and Eidel (988). It consists of replacing the planar surface Σ by an artificial spheric segment. In that case, problem (8) has analytical solutions which coincide with the solid spheric harmonics. Bauer (98) has shown that an error caused by the replacement is small for relatively small α, his numerical examples agreed well with model tests for α < π/(rad) = 5. Combining these two approaches, we will adopt the solid spheric harmonics appearing in the papers by Bauer (98) and Dokuchaev (964) as a functional basis in the variational method by Feschenko et al. (969) (instead of the harmonic polynomials). The method will be elaborated in the (x,x,x 3 )- coordinates, so that the linear natural modes (eigenfunctions) can be substituted into (7), (9) and (). 3.. Natural modes In accordance with definitions in Fig. 4, we superpose the origin O with the apex of an inverted cone and direct the Ox-axis upwards. In that case, the cone is determined by the equation x = cot α y + z and the mean free surface Σ (x = h) is a circle of radius r = h tan α, where h is the fluid depth. By substituting x := x/r,y := y/r,z := z/r, we get a non-dimensional formulation of the spectral problem (8). Finally, the resulting transformation (9) + () is proposed, x = x ; y = x x cos x 3 ; z = x x sin x 3, (a) x = x y ; x = + z r x ; x 3 = arctan z y, (b)

10 48 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) so that (3) takes the following form: x x ψ m x x x ψ m x x + x ( + x ) ψ m x + ( + x ) ψ m m ψ x x m = in G, (a) x x ψ m x x ψ m = κ m x x x x ψ m on L, (b) x (x + ) ψ m x x ψ m = x x on L, m=,,,..., (c) x ψ x dx = as m =, (d) where G ={(x,x ) : x x, x x }, x = tan α,x = h/r = /x and a = x ; b = ; p = x x x x ; q = x x ; s = x (x + ); c =. x Each nth eigenvalue of (), κ mn, computes the natural circular frequency gκmn gκmn σ mn = = r h tan α. (3) The surface sloshing modes are defined by ψ m as follows: where x = F mn (x,x 3 ) = mn (x,x,x 3 ); x x, x 3 π, (4) mn (x,x,x 3 ) = ψ mn (x,x ) cos sin mx 3 (5) is the nth eigenfunction of () corresponding to κ mn. 3.. Approximate natural modes 3... Separation of spatial variables As proved by Eisenhart (934), the Laplace equation allows for separation of spatial variables only in 7 inequivalent coordinate systems. The Ox x x 3 -coordinate system defined by (a) belongs to the admissible set. The separability of (a) and (c) leads to the following particular solutions ψ m [ν](x,x )= w (m) ν (x,x ) = (x /x ) ν v (m) (x ), where x ( + x )v(m) ν v (m) v ν (m) ν [ + ( + x νx )v(m) + ν ν(ν )x m x ] v (m) ν =, ν () <, (6a) x (x ) = ν + x v ν (m) (x ), m =,,...,ν m. (6b)

11 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) Eqs. (6a), (6b) constitute the ν-parametric boundary value problem, which has non-trivial solutions only for certain values of ν. If we input a test ν in the differential (6a) and output v ν (m) (x ) and (x ), the boundary condition (6b) plays the role of a transcendental equation with respect to ν. This v (m) ν transcendental equation can be solved by means of an iterative algorithm. In order to get an approximate ψ mn (x,x ), we should fix m and calculate q lower roots {ν m < ν m < < ν mq } of this transcendental equation. The nth ( n q) approximate solution of (), which corresponds to κ mn, can then be posed as ψ mn (x,x ) = q k= where coefficients ā (m) nk ā (m) ā (m) nk ( x x ) νmk v (m) ν mk (x ), (7) have to satisfy the supplementary conditions {, m =, n = q nk c ν k, m=, k= ā(m) c νk = x x w () ν k (x,x )x dx (ν m =, v ν (m) m = ) to agree with (d). One should note that comparing v ν () with the Legendre function Pν deduces ( ) ν ( ) v ν () (x ) = + x Pν / + x. (8) In view of this point, P ν ( / ) + x = p= (ν p + )(ν p + ) (ν + p) (p!) + x p (9) (Bateman and Erdelyi, 953) and recurrence formulae for Pν m (Lukovsky et al., 984) re-written to the (x,x 3 )-coordinates (ν + m + )v (m) ν+ = (ν + )v(m) ν (ν m)( + x )v(m) ν, (ν + m + )x v ν (m+) = (m + )[( + x )v(m) ν v(m) ν ], dv (m) ν dx = x [νv (m) ν facilitate computing v (m) ν (ν m)v (m) ν ] (3) (x ) and v (m) (x ) for arbitrary x,mand ν m. ν 3... The Treftz method based on (7) Representation (7) and the Rayleigh Kelvin minimax principle for the spectral problems () (see, Feschenko et al., 969) make it possible to compute approximate eigenvalues κ mn. The numerical scheme implies J m ā (m) =, i =,,...,q; m =,,,..., (3a) ni

12 4 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) J m = G [ ( ) ψm p + q ψ ( ) m ψ m ψm + s x x x x ] dx dx κ pψ m dx, m=,,..., (3b) + m x ψ m and, as a consequence, leads to the spectral matrix problem L det {α (m) ij } κ{b (m) ij } =, i,j =,...,q, (3) where the symmetric positive matrices {α (m) ij } and {b (m) ij } are calculated by the following formulae: x b (m) ij = x x w ν (m) mi w ν (m) mj dx, i,j =,...,q, (33) α (m) ij {[ x = x x w ν (m) x mi x x w νmi x ] w (m) ν mj }x =x dx, i,j =,...,q. (34) The spectral problem (3) gives not only approximate eigenvalues, but also eigenvectors expressed in terms of coefficients {ā (m) nk,k=,...,q}, n=,...,q. Being substituted into (7), these eigenvectors determine eigenfunctions ψ mn (x,x ), n =,...,q Convergence Furthermore, positive eigenvalues {κ mn,n=,...,q} of (3) are posed in ascending order. Fixing m and evaluating some of the lower approximate κ mn versus q (see Table ), one displays convergence of the Treftz method. Our numerical experiments with κ m,m, showed that 5 6 basis functions (q = 5 or 6) guarantee 5 6 significant figures, in general, and significant figures for α 45, in particular. Weaker convergence for smaller α can be treated in terms of the mean fluid depth x = h and geometric proportions of Q versus α. Ifα then x and Q becomes geometrically similar to a fairly long circular cylinder. The linear sloshing modes (eigenfunctions of (8)) in a circular cylindrical tank are characterised by exponential decaying from Σ to the bottom (Abramson, 966; Lukovsky et al., 984; Gavrilyuk et al., ), but, in contrast, functions (7) have power asymptotics Natural frequencies versus α Fig. 5 shows that κ mn and, therefore, σ mn, which are defined by (3), decrease monotonically with increasing α. Besides, a general tendency consists of κ mn asα 9, but each eigenvalue κ mn has a proper decaying gradient. As a result, some of the natural frequencies may become equal at an isolated α. We demonstrate this fact by points A and B in Fig. 5. The point A corresponds to α 9., where σ 3 = σ, and B occurs at α 3, where σ = σ 5. In accordance with theorems by Feschenko et al. (969), problem (8) has a denumerable set of real positive eigenvalues and each κ mn continuously depends on smooth deformations of Q. Both appearance and split of multiple eigenvalues are consistent with these theorems. Moreover, the matrices {α (m) ij } and {b (m) ij } are symmetric and positive and, therefore, the crossings in Fig. 5 do not yield a numerical difficulty in solving (3) (Parlett, 998). On the other hand, Bauer et al. (975) and Bridges (987) showed that

13 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) Table κ mn versus q in (7) α = π/(rad) = 9,x = q κ κ κ κ 3 κ 4 κ κ α = π/8(rad) =,x = q κ κ κ κ 3 κ 4 κ κ α = π/(rad) = 5,x = q κ κ κ κ 3 κ 4 κ κ α = 7π/8(rad) = 7,x = q κ κ κ κ 3 κ 4 κ κ α = π/6(rad) = 3,x =.7358 q κ κ κ 3 κ κ 4 κ κ α = π/4(rad) = 45,x = q κ κ κ 3 κ κ 4 κ 5 κ

14 4 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) Table (continued) α = π/(rad) = 9,x = q κ κ κ κ 3 κ 4 κ κ 5 α = π/3(rad) = 6,x = q κ κ κ 3 κ κ 4 κ 5 κ κ ij A B α/ Fig. 5. Eigenvalues κ ij versus α. split of a multiple eigenvalue into simple eigenvalues may lead to secondary bifurcations of nonlinear sloshing regimes. In a forthcoming paper, we plan to conduct the corresponding nonlinear investigations Shapes of the natural surface modes Fig. 6 shows surface modes defined by (4). Normalisation of these modes, which is usually accepted in nonlinear analysis (to interpret each generalised coordinate in (7a) as a generalised amplitude of F mn ), requires ψ mn (x,x ) =. It revises (7) to the form where ψ mn (x,x ) = a (m) nk q k= a (m) nk ( x x = ā(m) nk ; N mn = ψ N mn (x,x ) = mn and {ā (m) nk } are eigenvectors of (3). ) νmk v (m) ν mk (x ), (35) q k= ā (m) nk v(m) ν mk (x ) (36)

15 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) Fig. 6. Natural surface modes for π/4(rad) = 45. The asymmetric modes (indexes,, 3, 4, ) have double multiplicity, their shapes differ from each other by azimuthal rotation of 9. (a) Standing wave with σ. (b) Standing wave with σ. (c) Standing wave with σ. (d) Standing wave with σ 3. (e) Standing wave with σ 4. (f) Standing wave with σ. σ (rad/sec) Bauer(98) π/6 Theory π/6 Bauer(98) π/ Theory π/ g/h 9 Fig. 7. The lowest natural frequency σ versus g/h. Experimental measurements by Bauer (98) are compared with our theoretical prediction Experimental validation Bauer (98) has performed experimental measurements of the lowest natural frequency σ for α = π/6(rad) (=3 ) and α = π/(rad) (=5 ) and distinct fluid fillings (depths h). In order to compare our results with his experimental data, we note that σ = κ x g/h and, therefore, κ x, which is an invariant for a fixed α, determines a proportionality coefficient between σ and g/h. Fig. 7 (σ versus g/h) shows good agreement between our theoretical prediction and experimental results by Bauer (98). A statistical experimental estimate of κ x =.63 for α = was also published by Mikishev and Dorozhkin (96). This value is consistent with our numerical prediction.67. Numerical results in Fig. 5 show that the natural frequencies σ m are approximately in proportion to α (rad) and to g/r, namely, κ m x /α is a constant. This deduces the following formula σ m = C m α g/r (our calculations give, for instance, C =.658). Engineering of conical tanks may also be based on formula (3) and numerical results in Table.

16 44 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) Table κ, κ and κ versus α α κ κ κ Nonlinear sloshing in a conical tank The nonlinear modal method, which is presented in this section, generalises results by Narimanov (957), Lukovsky (99) and Faltinsen et al. (, 3). The method is based on the Moiseyev asymptotics and, therefore, it fails for shallow fluid flows characterised by progressive amplification of higher modes (Ockendon and Ockendon, ; Faltinsen and Timokha, a). 4.. Nonlinear modal system We normalise the original free boundary problem (4) and its modal analogy (8) by r. This implies g := g r ; v := v r ; β i (t) := β i(t) ; R i (t) := R i(t), i. (37) r r

17 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) Furthermore, we consider the horizontal harmonic excitations g = g, g = g 3 = ; v (t) = v (t), v 3 = εσ sin σt, (38) where ε> is the non-dimensional forcing amplitude and σ σ. The Moiseyev asymptotics (Moiseyev, 958; Gavrilyuk et al., ; Faltinsen et al., ) assumes that the lowest natural modes F (x,x 3 ) = ψ (x,x ) sin cos x 3 are of dominating character. By analysing this asymptotics for tanks of revolution, Lukovsky (99) proposed and justified the five-dimensional approximate solutions f (x,x 3,t)= x + f(x,x 3,t)= x + β (t) + p (t)f (x ) +[r (t) sin x 3 + p (t) cos x 3 ]f (x ) +[r (t) sin x 3 + p (t) cos x 3 ]f (x ), (39) φ(x,x,x 3 ) = P (t)ψ (x,x ) +[R (t) sin x 3 + P (t) cos x 3 ]ψ (x,x ) +[R (t) sin x 3 + P (t) cos x 3 ]ψ (x,x ), (4) where f i (x ) = ψ i (x,x ), i =,, (ψ i are normalised, see Section 3..5) and r R p P ε /3 ; p P r R p P ε /3. (4) When inserting (39) and (4) into the infinite-dimensional modal system (8) and accounting for (4), we get (correctly to O(ε)) the following nonlinear modal system coupling p,r,p,r and p (see its derivation in Appendix A): r + σ r + D (r r + r ṙ + r p p + r ṗ ) + D (p r + p ṙ ṗ r p p r ṗ ) D 3(p r r p +ṙ ṗ ṗ ṙ ) + D 4 (r p p r ) + D 5 (p r +ṙ ṗ ) + D 6 r p + σ [G r (r + p ) + G (p r r p ) + G 3 r p ]+Λ v 3 =, (4a) p + σ p + D (p p + p ṗ + r p r + p ṙ ) + D (r p + r ṙ ṗ r p r p ṙ ) + D 3(p p + r r +ṙ ṙ +ṗ ṗ ) D 4 (p p + r r ) + D 5 (p p +ṗ ṗ ) + D 6 p p + σ [G p (r + p ) + G (r r + p p ) + G 3 p p ]=, (4b) r + σ r D 9 (p r + r p ) D 7 ṙ ṗ + σ [G 4r p ]=, p + σ p + D 9 (r r p p ) + D 7 (ṙ ṗ ) σ [G 4(r p )]=, (4c) (4d) p + σ p + D (r r + p p ) + D 8 (ṙ +ṗ ) + σ [G 5(r + p )]=. (4e) Here σ =σ, σ =σ and σ =σ are defined by (3) and Λ, D i and G i, which depend only on α, are calculated by formulae (A.7). In order to help readers we give approximate values of these coefficients in Table 3. The modal system (4) differs from systems by Gavrilyuk et al. () (circular cylindrical) and Faltinsen et al. (3) (square-base) tanks by terms in the square brackets. These terms appear due to non-vertical walls, the coefficients G i vanish as α. Since, to the authors knowledge, the nonlinear fluid sloshing in a conical tank has never been studied and, as matter of fact, the modal system (4) has no analogies in the scientific literature, we tried our best to quantify its applicability. The quantification can be based on results by Ockendon et al. (996), Ockendon

18 46 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) Table 3 Coefficients of the nonlinear modal system (4) versus α α D D D 3 D 4 D 5 D 6 D 7 D 8 D 9 D α G G G 3 G 4 G 5 Λ k and Ockendon () and Faltinsen and Timokha (a) that associate failure of the Moiseyev ordering (4) with the secondary (internal) resonance. The secondary resonance has also been discussed by Bryant (989) (circular basin), it was examined for large amplitude forcing by Faltinsen and Timokha () (rectangular tank) and La Rocca et al. (997, ) and Faltinsen et al. (3, 5a) (square base tank). Quantification of critical semi-apex angles α, which yield the secondary resonance phenomena, can be done by analysing the dispersion relationship and higher periodic harmonics of steady-state solutions as σ = σ. Because of dramatical growth of the damping for higher harmonics and modes (Cocciano et al., 99; Faltinsen and Timokha, a; Faltinsen et al., 5b), our analysis can be limited to the

19 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) R3 R3 R R..8.6 R R α (grad) Fig. 8. Ri and Ri versus α. second-order terms. In that case, influence of the secondary resonance is associated with the equalities Ri = κi κ =, Ri = κi κ =, i. (43) Fig. 8 shows the graphs of Ri and Ri versus α. Using the resonance conditions (43) we predict the secondary resonance about α = 6 (by mode ) and α = (by mode ). When assuming Ri, Ri close, but not equal to, for instance, Ri, Ri <., we deduce that the Moiseyev-based modal system (4) is applicable for 5 < α < Steady-state wave motions By using (4) and accounting for results by Gavrilyuk et al. () and Faltinsen et al. (3), we pose the dominating modal functions of (39) as follows: r (t) = A cos σt + Ā sin σt + o(ε /3 ), p (t) = B cos σt + B sin σt + o(ε /3 ), (44) where A, Ā, B and B are unknown constants (dominating amplitudes) and σ is the excitation frequency. Representation (44) defines steady-state sloshing. By substituting (44) into (4c) (4e) and gathering primary harmonics, the Fredholm alternative deduces p (t) = c + c cos σt + c sin σt + o(ε /3 ), p (t) = s + s cos σt + s sin σt + o(ε /3 ), r (t) = e + e cos σt + e sin σt + o(ε /3 ), (45)

20 48 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) where with c = l (A + Ā + B + B ); c = h (A Ā B + B ), c = h (AĀ + B B); s = l (A + Ā B B ), s = h (A Ā + B B ); s = h (AĀ B B), e = l (A B + BĀ); e = h (ĀB A B); e = h (AB + Ā B), (46) h = D + D 8 σ G 5 ( σ 4), l = D D 8 σ G 5 σ, h = D 9 + D 7 + σ G 4 ( σ 4), l = D 9 D 7 + σ G 4 σ ; σ m = σ m, σ m=,,. By inserting (44), (45) into (4a), (4b) and using the Fredholm alternative, we derive the following system of nonlinear algebraic equations coupling A, Ā, B and B: A[ σ m (A + Ā + B ) m B ]+m 3 ĀB B = H Λ, Ā[ σ m (A + Ā + B ) m B ]+m 3 AB B =, B[ σ m (B + Ā + B ) m A ]+m 3 BAĀ =, B[ σ m (A + B + B ) m Ā ]+m 3 ĀAB =, (47) where ( m = D ) ( 5 h l + ) D3 h l + D6 h + D 4 h + D σ [ 34 ( G G l + h ) ( + G3 l + h )], ( m = D 3 l + 3 h ( ) + D5 l + h ) D6 h + 6D 4 h D + D σ [ 4 G + G (l 3 h ) + G 3 (l h ) ], (48) m 3 = m m. System (47) is similar to Eqs. (4) by Gavrilyuk et al. () (sloshing in circular cylindrical tanks); its resolvability condition is m 3 =. Solutions of (47) depend on the actual values of m i which are functions of σ and α (m i = m i (σ, α)). Taking into account that σ σ, we can consider m i (σ, α) (see the graphs in Fig. 9). These graphs establish that m 3 > for α < 6 and σ σ. Using derivations by Gavrilyuk et al. () and Faltinsen et al. (3) system (47) can be re-written to the equivalent form A( σ m A m B ) = H Λ, B( σ m B m A ) =, Ā = B =. (49) Vanishing Ā and B makes it possible to treat A and B as dominating longitudinal (along oscillations of the tank) and transversal (perpendicular to the oscillations) amplitudes of steady-state waves, respectively. System (49) has only two classes of solutions. The first class suggests B = and describes the so-called planar regime. Eqs. (49) then take the following form: A( σ m A ) = ΛH ; B =. (5)

21 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) shallow 6 4 m 3 m m 4 - m 5 m α (grad) Fig. 9. m i (σ, α) as functions of α. The second class (B > ) describes the so-called swirling regime (wave patterns imitate a rotation of the fluid volume around axis Ox). The algebraic system (49) then falls into the single equation with respect to A, A( σ m 4A ) = m 5 ΛH, m 5 = m, m 4 = m + m, (5) m 3 and the auxiliary formula for computing B B = ( σ m m A ) = ( ) ΛH m 5 m A + m A >. (5) Lukovsky (99), Gavrilyuk et al. () and Faltinsen et al. (3) showed that response curves of planar and swirling depend on m and m 4, respectively. Zeros of m and m 4 at isolated semi-apex angles α imply a passage from the hard-spring to soft-spring behaviour. If α < 6, response curves of the planar regime are always characterised by the soft-spring behaviour (similar to the case of circular cylindrical tanks, Lukovsky, 99). However, response curves of swirling change their behaviour at α 4. (x = h/r =.4...). Figs. and show the typical branching for α = 3 and α = 45, respectively. For the stability analysis the technique by Faltinsen et al. (3) was used. System (4) is linear in r i, p i. By inverting the matrix + D r + D p D 3p + D 5 p (D D )r p + D 3 r D 4 p D 4 r D 6 r (D D )r p + D 3 r + D p A = + D r + D 3p + D 5 p D 4 r D 4 p D 6 p D 9 p D 9 r D 9 r D 9 p D r D p it can be re-written to the normal form d p dt = A U(t, p, ṗ), (53)

22 4 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) A P P DD D D 3 3 D P+ D+.6 P.6 D.5 D+ D R C.3 F F.. P P. D 4. C σ /σ σ /σ B Fig.. Longitudinal (A) and transversal (B) amplitudes of steady-state resonant motions versus σ/σ. The results are given for α = 3 and H =.. Branches P P and P 3 P 4 imply planar, D D denotes swirling waves. R is the turning point, which divides the branch P + into stable (P R) and unstable RP subbranches. C is the Poincaré-bifurcation point. Here P 4 C denotes the stable planar solutions and CP 3 presents unstable ones. Swirling is associated with the branches D and D+. The Hopf-bifurcation point F divides D+ into D F and FD, where D F implies unstable solutions, but FD denotes a stable swirling. There are no stable steady-state solutions for σ/σ between abscissas of R and F. A P P 3 D D 3 P+ P D+ D.3 R C. F P P. 4 D B D D D 3 F D+ D. C σ / σ σ / σ Fig.. The same as in Fig., but for α = 45. where p = (r,p,r,p,p ) T and U = σ (r + G r (r + p ) + G (p r r p ) + G 3 r p ) D r (ṙ +ṗ ) D ṗ (p ṙ r ṗ ) + D 3 (ṙ ṗ ṗ ṙ ) D 5 ṙ ṗ + ΛH σ cos σt; U = σ (p + G p (r + p ) + G (r r + p p ) + G 3 p p ) D p (ṗ +ṙ ) D ṙ (r ṗ p ṙ ) D 3 (ṙ ṙ +ṗ ṗ ) D 5 ṗ ṗ ; U 3 = σ (r + G 4 r p ) + D 7 ṙ ṗ ; U 4 = σ (p G 4 (r p )) D 7(ṙ ṗ ); U 5 = σ (p + G 5 (r + p )) D 8(ṙ +ṗ ).

23 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) Fig.. Visualisation of swirling for α = 3, H =., σ/σ =.9967,A =.35,B =.49 with r () =.35,p () =.,p () =.469, ṗ () =.494, ṙ () =.36. The initial conditions should define r () = r ; p () = p ; p () = p ; p () = p ; r () = r, ṙ () =ṙ ; ṗ () =ṗ ; ṗ () =ṗ ; ṗ () =ṗ ; ṙ () =ṙ. (54) We solved the Cauchy problem (53), (54) by the fourth-order Runge Kutta method. The simulations were made by a Pentium-II 366 computer. The simulation time depended on parameters of excitation. It varied between to 3 of the real time-scale. Solutions (44), (45) made it possible to get initial conditions (54) to simulate steady-state regimes. These initial conditions were r () = A; p k () = A (l k + h k ), k =, ; p () = r () =, ṙ k () =ṗ k () = ; k =, ; ṗ = (55) for planar, and r () = A; p () = ; p () = A (l + h ) + B (h l ), p () = A (l + h ) + B (l h ), ṙ () = ; ṗ () = σb; ṙ () = 4σh AB; ṗ () = ; ṗ () = (56) for swirling regime, respectively. Typical three-dimensional wave patterns are presented in Figs. 4. In particular, Figs. 3 and 4 illustrate the travelling wave phenomenon (see the movements of the crest C), which is explainable by contributions of the second-order modal functions r,p and p. 5. Conclusions Linear and nonlinear sloshing of an incompressible fluid in a conical tank was analysed within the framework of inviscid potential theory. Using a domain transformation technique by Lukovsky (975)

24 4 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) Fig. 3. Visualisation of planar regime for α = 3, H =., σ/σ =.936,A=. with r () =.,p () =.7, p () =.34. Fig. 4. Visualisation of planar regime for α = 45, H =., σ/σ =.9463,A=. with r () =.,p () =., p () =.37. and a functional basis, which satisfies both the Laplace equation and the Neumann boundary condition on the tank walls, we found the approximate linear sloshing modes. The method guarantees 5 6 significant figures of the linear natural frequencies with only 5 6 basis functions. The numerical results were validated by experimental data. By utilising the approximate linear natural modes and results by Lukovsky (99) and Faltinsen et al. (), we derived a nonlinear finite-dimensional asymptotic modal system. The derived modal system is a novelty in the scientific literature. It couples five natural modes and makes it possible to analyse resonant sloshing due to a horizontal harmonic excitation with the forcing frequency close to the lowest natural frequency. Applicability of the modal system is limited by possible progressive activation of higher

25 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) modes caused by secondary resonance. This was predicted for the semi-apex angles 6 and. Besides, shallow fluid sloshing (Faltinsen and Timokha, a; Ockendon and Ockendon, ) was quantified for α > 6. Our nonlinear modal theory should be applicable in the range 5 < α < 6. Passage to α > 6 needs significant revisions of the present modal technique based on the Boussinesq asymptotics, in the manner of Faltinsen and Timokha (a). The present paper is the first attempt to classify steady-state waves in conical tanks. The analysis finds planar and swirling steady-state regimes as well as a frequency domain, where chaotic waves (there are no stable steady-state regimes) are realised. Advantages and possibilities of the modal system in engineering and for visualising realistic wave patterns were demonstrated. A further perspective can be a detailed study of chaotic waves. The papers by Funakoshi and Inoue (99, 99) are useful in this context. Appendix A. Derivation of the modal system Solutions (39), (4) deal with { F i (x,x 3 )}, { n (x,x,x 3 )} and modal functions β i (t), R i (t), i, of (7) as follows: where = ψ ; = ψ sin x 3 ; 3 = ψ cos x 3 ; 4 = ψ sin x 3 ; 5 = ψ cos x 3, F = f (x ) = ψ (x,x ); F = f (x ) sin x 3 = ψ (x,x ) sin x 3, F 3 = f (x ) cos x 3 = ψ (x,x ) cos x 3 ; F 4 = f (x ) sin x 3 = ψ (x,x ) sin x 3, F 5 = f (x ) cos x 3 = ψ (x,x ) cos x 3, P (t) = Z (t); R (t) = Z (t); P (t) = Z 3 (t); R (t) = Z 4 (t); P (t) = Z 5 (t), β (t) = p (t); β (t) = r (t); β 3 (t) = p (t); β 4 (t) = r (t); β 5 (t) = p (t), f m (x ) = a (m) + q b (m) k= k (x ); b (m) k (x ) = a (m) k v(m) νmk (x ), m =,,. The time-dependent β (t) = O(ε /3 ) is a function of p,r,p,r and p, i.e. f = x + k (r (t) + p (t)) + p (t)f (x ) +[r (t) sin x 3 + p (t) cos x 3 ]f (x ) +[r (t) sin x 3 + p (t) cos x 3 ]f (x ), (A.) where the coefficient k = x x x x f (x ) dx is derived from the volume conservation condition π x Q(t) Q = (x f + x f + 3 ) f 3 x dx dx 3 = considered correctly to O(ε /3 ).

26 44 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) The integrals A n,a nk and l n / β i are linearly incorporated in (8) and, moreover, these integrals are linear in ρ. The density ρ can therefore be omitted. The integrals can be expanded in series by p,r,p,r and p correctly to O(ε).Accounting for da n /dt = 5 i= ( A n / β i ) βi and pursuing O(ε) in ( A n / β i ) βi, ( A n / β i )Z n and A nk Z n, Z n ( A nk / β i )Z k we get A = a 4 (r + p ) + a 7p, A = a 5 r + a 6 r (r + p ) + a 8(p r r p ) + a 4 r p, A 3 = a 5 p + a 6 p (p + r ) + a 8(r r + p p ) + a 4 p p, A 4 = a 3 r a 7 r p ; A 5 = a 3 p + a 7 (r p ) (A.) and A = a ; A = A = a 5 r ; A 3 = A 3 = a 5 p, A = a + a r + a p + a 9p a 6 p, A 3 = A 3 = a 8 r p + a 6 r ; A 55 = A 44 = a, A 33 = a + a p + a r + a 6p + a 9 p, A 4 = A 4 = a 3 p ; A 5 = A 5 = a 3 r ; A 43 = A 34 = a 3 r, A 53 = A 35 = a 3 p ; A 4 = A 4 = A 5 = A 5 = A 54 = A 45 =, (A.3) where coefficients a,...,a 8 are given by the following integrals: a = π x x a 3 = π x a 4 = π a 5 = π a 6 = π a 7 = π x B () x x x a 8 = π a 9 = π x a = π x F (,) (x )x dx ; a = π x (F (,) (x ) + x B (,) (x ))f (x )x dx, (B () (x )f (x ) + k B () (x ))x dx, (x )f (x )x dx, ( ) 34 B (3) (x )f (x ) + k B () (x ) f (x )x dx, B () (x )f (x )x dx, ( ) F (,) (x ) x B (,) (x ) f (x )x dx, [ ] F (,) (x ) + x B (,) (x ) f (x )x dx, ( ) F (,) (x ) + x B (,) (x ) x dx, (F (,) (x ) + 4 x B (,) (x ))x dx,

27 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) a = π [ ] x (k F (,) (x ) + x B (,) (x ) [ ] ) + 3 F (,) (x ) + 4 3x B (,) (x ) f (x ) x dx, a = π [ ] x (k F (,) (x ) + x B (,) (x ) [ ] ) F (,) (x ) + x B (,) (x ) f (x ) x dx, a 3 = π x x a 4 = π a 6 = π x 4 x a 7 = π B () (x )f (x )x dx, B () (x )f (x )f (x )x dx ; a 5 = π x [ ] F (,) (x ) x B (,) (x ) f (x )x dx, B () (x )f (x )x dx ; a 8 = π x F (,) (x )f (x )x dx, B () f (x )f (x )x dx. Here, we introduce the functions B (),B(),B(), B(),B(3),B(),B(), B(,), B (,), B (,),B (,) B (,) and F (,),F (,),F (,),F (,),F (,),F (,),F (,) depending on b (m) k (x ) and c (m) k (x ) = a (m) dv ν (m) mk k, m=,, ; k =,,...,q. dx These functions are expressed as follows: B () m (x ) = x B (3) m (x ) = 6 B (m,n) q k= b (m) k (x ); B () m (x ) = x q (ν mk + )b (m) k (x ), k= q (ν mk + )(ν mk + )b (m) k (x ), m =,, ; k= (x ) = x B (m,n) = x Π (m,n) ij q i,j= i,j= b (m) i (x )b (n) j (x ) ν mi + ν nj + ; q B(m,n) = i,j= q (ν mi + ν nj )b (m) i b (n) j, m,n=, ; (x ) = ν mi ν nj b (m) i b (n) j x (ν mi b (m) i c (n) j + ν ni b (n) i c (m) j b (m) i (x )b (n) j (x ); ) + ( + x )c(m) i c (n) j,,

28 46 I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) F (m,n) (x ) = x q i,j= Π (m,n) ij (x ) ν mi + ν nj + ; F (m,n) (x ) = q i,j= Π (m,n) ij (x ), F (m,n) (x ) = x q (ν mi + ν nj )Π (m,n) ij (x ), m, n =,,. i,j= Relationship (4) and g = g, g = g 3 = mean that the terms l / β i and l 3 / β i of (8) should be calculated correctly to O() as follows: { { l, βi / p =, l 3, βi / r =, x λ = x 3 β i λ, β i p, β i λ, β i r, π x f (x ) dx. (A.4) The scalar function l reads as l = l () +[l () (r + p ) + l() p + l(3) (r + p ) + l(4) (r + p ) + l (5) ( p p r p + r p r ) + l (6) p (r + p )], (A.5) where l () = π 4 x4 x ; l() = x G ; l () = x G ; l (3) = x G, l (4) = 6 3 G + 3 x k G ; l (5) = x G ; l (6) = x G and x G = π x G = π x f dx ; f f x dx ; G = π x G = π x x f dx ; x f f dx ; G = π x G = π x f dx, x x f 4 dx. Consider (8a) as a system of linear algebraic equations in Z k (t). Accounting for (A.) and (A.3) and solving (8a) correctly to O(ε) one obtains R (t) = Q ṙ + C r ṙ + D 3 p ṙ + C r p ṗ + D (r ṗ p ṙ ) + C 3 (p ṙ r ṗ ) + B p ṙ + B 3 r ṗ, P (t) = Q ṗ + C p ṗ + D 3 r ṗ + C p r ṙ + D (r ṙ + p ṗ ) + C 3 (r ṙ + p ṗ ) + B p ṗ + B 3 p ṗ, P (t) = C (r ṙ + p ṗ ) + D ṗ ; R (t) = Q ṙ D (r ṗ + p ṙ ), P (t) = Q ṗ + D (r ṙ p ṗ ), (A.6) where C = a 4 a a 5a 5 4a a ; D = a 7 a ; C = a ( a 6 a 4a 5 a 5a 8 + a ) 5a5, a 4a 8a a

29 Q = a 3 a ; C 3 = B 3 = a I.P. Gavrilyuk et al. / Fluid Dynamics Research 37 (5) a ( a 4 a 5a 7 a ( a 8 a 3a 3 a ( D 3 = a 6 a + Q 3 a + a 3a 7 a 4a a a a a 5 ) ; B = ( a 4 a 5a 9 a a ) ; D = a 7 a + a 3a 5 4a a ; D = ) ; C = D 3 + C. a ) ; Q = a 5, a ( a 8 a ) 5a 6, a Substituting (A.) (A.6) in (8b) and gathering the terms up to O(ε) lead to the modal system (4a) (4d), where D = d μ ; D = d μ ; D 3 = d 3 μ ; D 4 = d 4 μ ; D 5 = d 5 μ ; D 6 = d 6 μ, D 7 = d 7 μ ; D 8 = d 8 μ ; D 9 = d 4 μ ; D = d 6 μ ; Λ = λ μ, G = dk ; G = dk ; G 3 = dk 3 ; G 4 = dk ; G 5 = dk 3, μ κ μ κ μ κ μ κ μ κ where μ = a 7 D ; μ = a 5 Q ; μ = a 3 Q, d = a 4 C + a 7 D + a 5 C + 3a 6 Q ; d = a 5 D 3 + a 6 Q + a 7 D, d 3 = a 5 D + a 8 Q ; d 4 = a 7 Q a 5 C 3 ; d 5 = a 5 B + a 4 Q, d 6 = a 4 D + a 5 B 3 ; d 7 = d 4 + d 3; d 8 = d 6 d 5, d k = 4l(4) ; dk = l(5) ; dk 3 = l(6). (A.7) References Abramson, H.N., 966. The dynamics of liquids in moving containers. NASA Report, SP 6. Bateman, H., Erdelyi, A., 953. Higher Transcendental Functions. McGraw-Hill Book Company, Inc., New York, Toronto, London. Bauer, H.F., 98. Sloshing in conical tanks. Acta Mechanica 43 (3 4), 85. Bauer, H.F., Eidel, W., 988. Non-linear liquid motion in conical container. Acta Mech. 73 ( 4), 3. Bauer, L., Keller, H.B., Reiss, E.L., 975. Multiple eigenvalues lead to secondary bifurcation. SIAM Rev. 7,. Bridges, T.J., 987. Secondary bifurcation and change of type for three dimensional standing waves in finite depth. J. Fluid Mech. 79, Bryant, P.J., 989. Nonlinear progressive waves in a circular basin. J. Fluid Mech. 5, Cariou, A., Casella, G., 999. Liquid sloshing in ship tanks: a comparative study of numerical simulation. Mar. Struct., Chester, W., 968. Resonant oscillation of water waves. I. Theory. Proc. R. Soc. London 36, 5. Chester, W., Bones, J.A., 968. Resonant oscillation of water waves. II. Experiment. Proc. R. Soc. London 36, 3 3. Cocciano, B., Faetti, S., Nobili, M., 99. Capillary effects on surface gravity waves in a cylindrical container: wetting boundary conditions. J. Fluid Mech. 3, Dodge, F.T., Kana, D.D., Abramson, H.N., 965. Liquid surface oscillations in longitudinally excited rigid cylindrical containers. AIAA J. 3 (4),