Multimodal Method in Sloshing Analysis: Analytical Mechanics Concept

Transcription

1 Multimodal Method in Sloshing Analysis: Analytical Mechanics Concept by Alexander Timokha CeSOS/AMOS, NTNU, Trondheim, NORWAY & Institute of Mathematics, National Academy of Sciences of Ukraine, UKRAINE 1 Trondheim, 28. May, 2013

2 Multimodal Method in Sloshing Analysis: Analytical Mechanics Concept Etymology comes from CONCEPT which is the most cited paper on sloshing of the last two decades 2

3 Multimodal Method in Sloshing Analysis: Analytical Mechanics Concept CONCEPT The concept originally appeared in XIX century but generalized in by the author together with Prof. Odd M. Faltinsen Liquid Sloshing Dynamics is treated as a conservative mechanical system with infinite degrees of freedom so that the Lagrange formalism can adopt the generalized coordinates and velocities responsible for the global liquid modes instead of working with typicallyaccepted hydrodynamic characteristics (velocity field, pressure, etc.) 3 Historical aspects & Ideas: Linear & Nonlinear

4 HISTORY & IDEAS 4 IDEAS come from aircraft and spacecraft applications: when the task consists of describing the coupled dynamics of a body with cavities filled by a liquid N.E. Joukowski (1885) paper `On the motion of a rigid body with cavities filled by a homogeneous fluid Joukowski theorem for a completely filled tank: `The rigid body-ideal irrotational incompressible fluid mechanical system can modelled as a rigid body with a specifically-modified inertia tensor Considering the fluid as frozen is a wrong way (boiled and fresh eggs!!!). The velocity field is described by the so-called Stokes-Joukowski potentials.

5 FULLY FILLED TANK by the liquid: ideal, incompressible, irrotational flow HISTORY & IDEAS Translatory velocity: v () t ( (), t (), t ()) t O Instant angular velocity: () t ( (), t (), t ()) t The liquid velocity potential (,,,) xyzt v () t r () t (,, xy z), r ( xyz,, ) O the Stokes-Joukowski potential 5

6 HISTORY & IDEAS Having known the time-independent Stokes- Joukowski potentials makes it possible to find the fluid flow for any time instant so that the velocity field the pressure, the resulting hydrodynamic force and moments, etc. are explicit functions of the six input generalized coordinates What about liquid sloshing (free surface)? 6

7 LINEAR SLOSHING, s of XX century Liquid: ideal incompressible irrotational flow v () t ( (), t (), t ()) t O HISTORY & IDEAS 7 Free surface () t ( (), t (), t ()) t O () () 0 N N N ( xyzt,,, ) v t r t (,, xyz) R () t (,, xy z) z (,,) xyt ( t) (,, x y 0) N Joukowski solution N generalized velocities N sloshing modes interpreted as generalized coordinates (infinite set!!!)

8 HISTORY & IDEAS As long as we know the time-independent Stokes-Joukowski potentials (Neumann problem in the unperturbed liquid domain) 0 (,, xyz) the natural sloshing modes (the spectral boundary problem in the unperturbed liquid domain) N (,, xyz ) then the free-surface elevations the hydrodynamic forces and moments (provided by the corresponding Lukovsky formulas) are functions of the input and the generalized coordinates where the latters are the solution of the linear oscillator problem: m 0( k3) m ( g ) ( g ), m 1,2,... m 2 1m 2m m m m 6 k k4 m 8 and the hydrodynamic coefficients are integrals over and xyz 0 (,, ) N (,, xyz )

9 NONLINEAR MULTIMODAL METHOD: new life in 00 s Liquid: ideal incompressible irrotational flow v () t ( (), t (), t ()) t O () t ( (), t (), t ()) t HISTORY & IDEAS 9 (,,,) xyzt v r (,,,{ xyz N ()} t ) R N () t N (,, xyz) O 0 N free surface ( t): Z xyz,,, ( t) 0 subject to volume conservation Generalized coordinates N Qt () The free surface elevations, the hydrodynamic forces and moments also remain functions of the six input and infinite set of the freesurface generalized coordinates dq 0 Generalized velocities

10 HISTORY & IDEAS THERE ARE APPLIED MATHEMATICAL & PHYSICAL PROBLEMS TO BE SOLVED TO IMPLEMENT THE NONLINEAR MULTIMODAL METHOD 10

11 The multimodal method The nonlinear multimodal method 1. The method is of analytical nature. What are analytical limitations? 2. The Euler-Lagrange equation for liquid sloshing dynamics 3. Physical and mathematical arguments for choosing the generalized coordinates 4. How does it work? 11

12 MATHEMATICAL LIMITATIONS The following problems must be solved analytically: A. For any instant N (,, xyz ) must be defined in the time-varied liquid domain Qt () and satisfy the tank surface condition B. Derivation of the Euler-Lagrange equations with respect to the generalized coordinates and velocities Assuming A (we know analytical modes): hand-made product for each tank shape e.g., the Bateman-Luke variational principle Bateman-Luke variational principle derives the Euler- Largange equation 12

13 EULER-LAGRANGE EQUATION 13 K (Kinematic Eq.): (Dynamic Eq.): da N dt K A N K A F K NK K K N KL N N N N, changing AK 1 AKL l1 l2 l3 F K FKFL changing N 2 where A d Q; A d Q, N N Qt () N NK N Qt () * N * K N N N l x d Q; l y d Q; l z dq 1 Qt () * 2 Qt () * 3 Qt () * Application needs a finite-dimensional form in application, so how to use the Euler-Lagrange equation, e.g., for Steady-state and transient response? Wave elevations? Forces and moments? Coupling? Realistic clean tanks? Dissipation? Internal structures effect? N;

14 PHYSICAL AGRUMENTS 14 Problems to be solved analytically: A. For any instant N (,, xyz ) must be defined in the time-varied liquid domain Qt () and satisfy the tank surface condition B. Derivation of the Euler-Lagrange equations with respect to the generalized coordinates and velocities C. Reduction of the infinite-dimensional Euler-Lagrange equations to a finitedimensional form D. Accounting for specific phenomena neglected by the physical model (damping, inner structures, wall/roof impact, perforated bulkhead, and so on) hand-made product for each tank shape e.g., the Bateman-Luke variational principle Normally, asymptotic approaches results of the last decade

15 How does it work? 15 HOW does it work? Let s consider devils examples for: 1. For 2D flows 2. For 3D rectangular tanks with upright walls 3. For complex tank shapes What should be solved? Choosing the leading and negligible generalized coordinates Modifying the modal equation due to damping, specific phenomena and internal structures

16 2D in rectangular tank 16 When forcing the lowest natural sloshing frequency: Finite liquid depth (liquid depth/tank length>0.2) A small forcing magnitude Moiseev s asymptotics Increasing forcing magnitude, or the critical depth ratio (h/l= ) secondary resonances and adaptive asymptotics Intermediate and small depths (liquid depth/tank length < 0.2) Multiple secondary resonance Boussinesq ordering Internal structures (e.g., bulkheads) Small forcing amplitude quasilinear theory Increasing forcing amplitude secondary resonance Nonlinearity implies an energy transfer from the lowest, primary excited, to a set of higher modes. The latter modes can be resonantly excited due to the so-called secondary resonances. What does it mean in terms of leading generalized coordinates?

17 2D in rectangular tank 17 When forcing the lowest natural frequency: Finite liquid depth (liquid depth/tank length>0.2) A small forcing magnitude Moiseev s asymptotics Increasing forcing magnitude, or the critical depth ratio (h/l= ) secondary resonances and adaptive asymptotics Intermediate and small depths (liquid depth/tank length < 0.2) Multiple secondary resonance Boussinesq ordering Internal structures (e.g., bulkheads) Small forcing amplitude quasilinear theory Increasing forcing amplitude secondary resonance Nonlinearity implies an energy transfer from the lowest, primary excited, to a set of higher modes. The latter modes can be resonantly excited due to the so-called secondary resonances. What does it mean in terms of leading generalized coordinates?

18 Example 1: Example 1: The Moiseev-type multimodal theory The modal solution (,,) yzt v r (, yz) R (, yz), O 0 n n n 1 z (,) yt () t (,0), y n 1 n cosh nz ( h) 1 n cos ny ( ) 2 cosh( nh) Moiseev proved for the steady-state (periodic) sloshing due to excitation of the lowest frequency when there are no secondary resonances / l / l O(); O() 1, 2 3 n 4 R O( ), R O( ), /3 2/3 2 2 R O( ); R O( ), n 4 n n 18

19 Example 1: Modal equations ( ) d ( ) d ( ) d K ( t), ( ) 0, ( ) q q () t d d q q q K ( ), 4,..., N 2 K t i i i i i Forcing term: () t g K t t t l l 2 () P S () () i i i 4 4 Coefficients are analytically found as functions of the liquid depth to the tank breadth ratio 19

20 Transients: Experiments Theory with different initial scenarios Example 1: Steady-state waves with the horizontal/angular harmonic forcing, (frequency, nondimensional amplitude ), the dominant amplitude parameter: h / l h / l

21 Example 2: 21 When forcing the lowest natural frequency: Finite liquid depth (liquid depth/tank length>0.2) A small forcing magnitude Moiseev s asymptotics Increasing forcing magnitude, or the critical depth ratio (h/l= ) secondary resonances and adaptive asymptotics Intermediate and small depths (liquid depth/tank length < 0.2) Multiple secondary resonance Boussinesq ordering Internal structures (e.g., bulkheads) Small forcing amplitude quasilinear theory Increasing forcing amplitude secondary resonance Nonlinearity implies an energy transfer from the lowest, primary excited, to a set of higher modes. The latter modes can be resonantly excited due to the so-called secondary resonances. What does it mean in terms of leading generalized coordinates?

22 Example 2: 22 Example 2: Adaptive modal systems for critical depth and increasing forcing amplitude Nonlinear multiple frequency effects excite higher natural frequencies Increased importance with decreasing depth and increasing forcing amplitude Reason for decreasing depth importance is that n n when the liquid depth goes to zero Implies that more then one mode is dominant

23 Different ordering of the generalized coordinates due to secondary resonances with a finite liquid depth Subharmonic regimes are predicted Example 2: Rectangular tank, 1x1m, nearly critical depth, h/l =

24 Increasing the forcing amplitude and accounting for roof impact, h/l= Example 2: Flow 3D impact neglected impact accounted for CFD: Symbols and represent numerical results by the viscous CFD-code FLOW-3D obtained for fresh water with different internal parameters of the code: for alpha=0.5, epsdj=0.01 and for alpha=1.0, epsdj= Experiments: The influence of viscosity:, fresh water;, reginoloil;, glycerol-water 63%,, glycerol-water 85%

25 Example 3: 25 When forcing the lowest natural frequency: Finite liquid depth (liquid depth/tank length>0.2) A small forcing magnitude Moiseev s asymptotics Increasing forcing magnitude, or the critical depth ratio (h/l= ) secondary resonances and adaptive asymptotics Intermediate and small depths (liquid depth/tank length < 0.2) Multiple secondary resonance Boussinesq ordering Internal structures (e.g., bulkheads) Small forcing amplitude quasilinear theory Increasing forcing amplitude secondary resonance Nonlinearity implies an energy transfer from the lowest, primary excited, to a set of higher modes. The latter modes can be resonantly excited due to the so-called secondary resonances. What does it mean in terms of leading generalized coordinates?

26 Intermediate and small depths normally cause amplification of higher modes and a series of local wave breaking & overturning Example 3: 26

27 As a consequence, a Boussinesq-type ordering can be proven being applicable to with R Oh l O i i i 1/4 ( / ) ( ), 1 Transients: Example 3: The measured and calculated wave elevations near the wall for horizontal forcing h / l 0.173, The solid and dashed lines correspond to experiments and the Boussinesq-type multimodal method, respectively. 27

28 Steady-state due to harmonic forcing = experiments Chester & Bones / l h Example 3: Multumodal theory Theory by Chester Agreement is almost ideal when no wave breaking occurs and, therefore, damping does not matter. Multi-peak response curves and damping are important when dealing with secondary resonances. 28

29 Examples vs. damping 29 Associated damping becomes important with Decreasing depth and increasing forcing amplitude Roof impact Internal structures Normally, viscous boundary layer effect is less important Damping terms can be incorporated into the modal equations following the strategy in Chapter 6 of «Sloshing» book. An open problem is damping due to the local freesurface phenomena: Overturning and impact on underlying fluid Breaking waves in the middle of the tank

30 Example 4: 30 When forcing the lowest natural frequency: Finite liquid depth (liquid depth/tank length>0.2) A small forcing magnitude Moiseev s asymptotics Increasing forcing magnitude, or the critical depth ratio (h/l= ) secondary resonances and adaptive asymptotics Intermediate and small depths (liquid depth/tank length < 0.2) Multiple secondary resonance Boussinesq ordering Internal structures (e.g., bulkheads) Small forcing amplitude quasilinear theory Increasing forcing amplitude secondary resonance Nonlinearity implies an energy transfer from the lowest, primary excited, to a set of higher modes. The latter modes can be resonantly excited due to the so-called secondary resonances. What does it mean in terms of leading generalized coordinates?

31 Example 4: Example 4: Middle screen 31 The middle-screen causes: (a) migration of the resonances and supermultipeak response curves; (b) damping; (c) disappearance of the primary resonance with increasing the solidity ratios

32 Three-dimensional: nearly steady-state wave response 3D sloshing 32 Planar waves waves keeps symmetry with respect to the excitation plane: occur far from the primary resonance. Swirling exactly at the primary resonance Irregular (chaotic) waves. Weak chaos? For the rectangular shape, a diagonal-type (squares-like) waves waves with an angle to the excitation plane

33 3D sloshing Moiseev-type modal system for square base tank a ; a ; b ; b ; c, a ; c ; c ; b 1, 0 1 2, 0 2 0,1 1 0, 2 2 1,1 1 3, 0 3 2,1 21 1, , 3 3 g a a d( aa aa ) d( aa aa) daa P S dab b( dc dab) dcb d ba d abb d bc 0, , , 0 1,0 5 L1 L g b b d ( bb bb ) d ( bb b b ) dbb P S dba a ( dc dab) dca d a b d aba d ac 0, , ,1 0,1 4 L1 L a a daa da Modal equations with nine degrees of freedom 2 2 0; 2 2, b b dbb db 2 0; 2 0, c1 dab ,1 1 ˆ dba ˆ dab ˆ c 0, g 5 a3 3, 0a3 a1( qa 1 2 q2a1 ) q3a2a1 q4a1a1 q5aa 1 2 P 3, 0 S3,0 5 0, L1 L 1 c c a ( q c q ab ) b( q a q a ) q a b q ca q a b q aba q ac q a b 0, , c c b( qc qab) a ( qb qb ) q ba q cb q ba q abb q bc q ab 0, , g 4 b 3 0, 3b3 b 1( qb 1 2 q2b1 ) q3bb 2 1 q4b 1b1 q5bb 1 2 P 0, 3 S0, L1 L 1

34 For harmonic forcing, one can find analytically approximate steady-state solutions and study their stability (also analytically!!!). This makes it possible to classify the steady-state regimes and establish the frequency ranges where the regimes exist and stable. 3D sloshing 34 One can distinguish: the order (stable steady-state), the strong chaos (treated as no stable steadystate for leading generalized coordinates), the weak chaos (here, irregular for higherorder generalized coordinates)

35 Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = L 3D sloshing 35

36 Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = L 3D sloshing 36

37 Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = L 3D sloshing 37

38 Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = L 3D sloshing 38

39 Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = L 3D sloshing 39

40 Classification for longitudinal forcing: planar, diagonal (squares)-type, swirling and chaos excitation amplitude = L (relatively large!!!) 3D sloshing 40 Local phenomena for 3D, but the classification is Ok. Why? Why is the Moissev-type model still applicable?

41 3D sloshing Spherical tanks 41 The method was developed in 2012, after the book issued

42 ( and ) radius m ( and ) m ( and ο) h 0. 2 secondary resonance by (01) at / Classification of 3D sloshing 42

43 ( and ) radius m ( and ) m ( and ο) h 0.6 secondary resonance is far from the range Classification of 3D sloshing 43

44 ( and ) radius m ( and ) m ( and ο) h 1. 0 secondary resonance by (22) at / Classification of 3D sloshing 44

45 Swirling wave patterns taken for these input parameters from T.Hysing (1976) Det Norske Veritas, Høvik, Norway Specifically, Classification of 3D sloshing splashing, steep wave patterns, local breaking. 45

46 ( and ) radius m ( and ) m ( and ο) h 1.0 secondary resonance by (22) at / Classification of 3D sloshing 46

47 `planar splashing : ``...drops splashed from the tank wall and showered through the ullage...'' but wave patterns remain planar Classification of 3D sloshing for h 1 splashing (planar & swirling type) is all the frequency range 47

48 Swirling always causes secondary resonances and local phenomena when forcing amplitude increases, but classification remains correct. Again, why? Weak chaos in sloshing problems 48 This is explained by the concept of the weak chaos: Here, a clearly steady-state by a subset of [leading] generalized coordinates and irregular motions by other [infinite set] higher-order coordinates caused by higher resonances.

49 The weak chaos concept for low-dimensional Hamiltonian (conservative) systems (see, e.g. Henning et al. (2013), Physica D, 253): Weak chaos in sloshing problems 49 ORDER WEAK CHAOS CHAOS Lyapunov exponent: <0 >0, but small >0, finite «in domains of weak chaos trajectories (by higher-order generalized coordinates) slowly diffuse into thin chaotic layers and wander through a complicated network of higher order resonances» For our almost-conservative mechanical system with an infinite set of generalized coordinates (g.c.), this implies: ORDER WEAK CHAOS CHAOS stable by all g.c. stable by dominant g.c., unstable by all g.c. but chaos in higher-order g.c. unless a strong damping occurs For 3D sloshing, the weak chaos is the reality well modelled by the multimodal method

50 Weak chaos in sloshing problems 50 order strong chaos Swirling (weak chaos) order order strong chaos Swirling (weak chaos) order From the order to weak chaos and, thereafter, strong chaos for swirling with increasing the forcing amplitude Longitudinal resonant excitation with h/l=0.5 and ε = for a square-base tank. Higher-order (secondary) resonances.

51 Open problems: intensive 51 Open problems within the framework of the same paradigm, i.e., six degrees of freedom for the rigid tank and generalized coordinates for the free surface motions: 1. Different internal structures. 2. Accounting for damping due to wave breaking, overturning, etc. 3. Complex tank shapes. 4. Order weak chaos chaos. 5. Importance of weak chaos for coupled motions CFD are normally unapplicable on the long-time scale!

52 Open problems: extensive Input has more than six (infinite) degrees of freedom: inflow-outflow & sloshing (damaged ship tank, wave energy, etc.); elastic/hyperelastic tank walls (fish farms, membrane tanks, etc.); ship collapse 52

53 REFERENCES Books: 1. Faltinsen, O.M., Timokha A.N. (2009): Sloshing. Cambridge University Press. 608pp. (ISBN-13: ) Chinese Version of the book issued in 2012: P.R.C.:National Defense Industry Press. 783pp. (ISBN-13: ) 2. Gavrilyuk, I.P., Lukovsky, I.A., Makarov, V.L., Timokha, A.N. (2006): Evolutional problems of the contained fluid. Kiev: Publishing House of the Institute of Mathematics of NASU. 233pp. (ISBN ) Selected papers in peer-reviewed journals: 1. Faltinsen, O.M., Timokha, A.N. (2013): Multimodal analysis of weakly nonlinear sloshing in a spherical tank. Journal of Fluid Mechanics, 719, Faltinsen, O.M., Timokha, A.N. (2012): Analytically approximate natural sloshing modes for a spherical tank shape. Journal of Fluid Mechanics, 703, Faltinsen, O.M., Timokha, A.N. (2012): On sloshing modes in a circular tank. Journal of Fluid Mechanics, 695, Gavrilyuk, I., Hermann, M., Lukovsky, I., Solodun, O., Timokha, A. (2012): Multimodal method for linear liquid sloshing in a rigid tapered conical tank. Engineering Computations, 29, No 2, Lukovsky, I.A., Ovchynnykov, D.V., Timokha, A.N. (2012): Asymptotic nonlinear multimodal method for liquid sloshing in an upright circular cylindrical tank. Part 1: Modal equations. Nonlinear Oscillations, 14, No 4, Lukovsky, I.A., Timokha, A.N. (2011): Combining Narimanov--Moiseev' and Lukovsky--Miles' schemes for nonlinear liquid sloshing. Journal of Numerical and Applied Mathematics, 105, No 2, Faltinsen, O.M., Firoozkoohi, R., Timokha, A.N. (2011): Effect of central slotted screen with a high solidity ratio on the secondary resonance phenomenon for liquid sloshing in a rectangular tank. Physics of Fluids, 23, Issue 6, Art. No , Faltinsen, O.M., Firoozkoohi, R., Timokha, A.N. (2011): Analytical modeling of liquid sloshing in a twodimensional rectangular tank with a slat screen. Journal of Engineering Mathematics, 70, 1-2,

54 54 9. Faltinsen, O.M., Firoozkoohi, R., Timokha, A.N. (2011): Steady-state liquid sloshing in a rectangular tank with a slat-type screen in the middle: Quasilinear modal analysis and experiments. Physics of Fluids, 23, Issue 4, Art. No , Faltinsen, O.M., Timokha, A.N. (2011): Natural sloshing frequencies and modes in a rectangular tank with a slat-type screen. Journal of Sound and Vibration, 330, Barnyak, M., Gavrilyuk, I., Hermann, M., Timokha, A. (2011): Analytical velocity potentials in cells with a rigid spherical wall. ZAMM, 91, No 1, Faltinsen, O.M., Timokha, A.N. (2010): A multimodal method for liquid sloshing in a two-dimensional circular tank. Journal of Fluid Mechanics, 665, Hermann, M., Timokha, A. (2008): Modal modelling of the nonlinear resonant fluid sloshing in a rectangular tank II: Secondary resonance. Mathematical Models and Methods in Applied Sciences, 18, N 11, Gavrilyuk, I., Hermann, M., Lukovsky, I., Solodun, O., Timokha, A. (2008): Natural sloshing frequencies in rigid truncated conical tanks. Engineering Computations, 25, Issue 6, Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2007): Two-dimensional resonant piston-like sloshing in a moonpool. Journal of Fluid Mechanics, 575, [Supplementary material] 16. Gavrilyuk, I., Lukovsky, I., Trotsenko, Yu., Timokha, A. (2007): Sloshing in a vertical circular cylindrical tank with an annular baffle. Part 2. Nonliear resonant waves. Journal of Engineering Mathematics, 57, Gavrilyuk, I., Lukovsky, I., Trotsenko, Yu., Timokha, A. (2006): Sloshing in a vertical circular cylindrical tank with an annular baffle. Part 1. Linear fundamental solutions. Journal of Engineering Mathematics, 54, Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2006): Resonant three-dimensional nonlinear sloshing in a square-base basin. Part 3. Base ratio perturbations. Journal of Fluid Mechanics, 551, Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2006): Transient and steady-state amplitudes of resonant three-dimensional sloshing in a square base tank with a finite fluid depth. Physics of Fluids, 18, Art. No , Gavrilyuk, I.P., Lukovsky, I.A., Timokha, A.N. (2005): Linear and nonlinear sloshing in a circular conical tank. Fluid Dynamics Research, 37, Hermann, M., Timokha, A. (2005): Modal modelling of the nonlinear resonant sloshing in a rectangular tank I: A single-dominant model. Mathematical Models and Methods in Applied Sciences, 15, N 9, Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2005): Classification of three-dimensional nonlinear sloshing in a square-base tank with finite depth. Journal of Fluids and Structures, 20, Issue 1,

55 Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2005): Resonant three-dimensional nonlinear sloshing in a square base basin. Part 2. Effect of higher modes. Journal of Fluid Mechanics, 523, Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N. (2003): Resonant three-dimensional nonlinear sloshing in a square base basin. Journal of Fluid Mechanics, 487, Faltinsen, O.M., Timokha, A.N. (2002): Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth. Journal of Fluid Mechanics, 470, Lukovsky, I.A., Timokha, A.N. (2002): Modal modeling of nonlinear sloshing in tanks with non-vertical walls. Non-conformal mapping technique. International Journal of Fluid Mechanics Research, 29, Issue 2, Gavrilyuk, I., Lukovsky, I.A., Timokha, A.N. (2001): Sloshing in a circular conical tank. Hybrid Methods in Engineering, 3, Issue 4, Faltinsen, O.M., Timokha, A.N. (2001): Adaptive multimodal approach to nonlinear sloshing in a rectangular tank. Journal of Fluid Mechanics, 432, Lukovsky, I.A., Timokha, A.N. (2001): Asymptotic and variational methods in nonlinear problems on interaction of surface waves with acoustic field. J. Applied Mathematics and Mechanics. 65, Issue 3, Faltinsen, O.M., Rognebakke, O.F., Lukovsky, I.A., Timokha, A.N. (2000): Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. Journal of Fluid Mechanics, 407, Gavrilyuk, I., Lukovsky, I.A., Timokha, A.N. (2000): A multimodal approach to nonlinear sloshing in a circular cylindrical tank. Hybrid Methods in Engineering, 2, Issue 4,

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