Nonlinear semi-analytical modeling of liquid sloshing in rectangular container with horizontal baffles

doi:10.1007/s10483-023-3054-8

Abstract

Abstract: A nonlinear semi-analytical scheme is proposed for investigating the finite-amplitude nonlinear sloshing in a horizontally baffled rectangular liquid container under the seismic excitation. The sub-domain method is developed to analytically derive the modal behaviors of the baffled linear sloshing. The viscosity dissipation effects from the interior liquid and boundary layers are considered. With the introduction of the generalized time-dependent coordinates, the surface wave elevation and velocity potential are represented by a series of linear modal eigenfunctions. The infinite-dimensional modal system of the nonlinear sloshing is formulated based on the Bateman-Luke variational principle, which is further reduced to the finite-dimensional modal system by using the Narimanov-Moiseev asymptotic ordering. The base force and overturning moment induced by the nonlinear sloshing are derived as the functions of the generalized time-dependent coordinates. The present results match well with the available analytical, numerical, and experimental results. The paper examines the surface wave elevation, base force, and overturning moment versus the baffle parameters and excitation amplitude in detail.

Key words: rectangular container, nonlinear sloshing, baffle, sub-domain, multi-modal method

2010 MSC Number:

76B07

Xun MENG, Ying SUN, Jiadong WANG, Ruili HUO, Ding ZHOU. Nonlinear semi-analytical modeling of liquid sloshing in rectangular container with horizontal baffles. Applied Mathematics and Mechanics (English Edition), 2023, 44(11): 1973-2004.

TrendMD

References

[1] SEZEN, H. and WHITTAKER, A. S. Seismic performance of industrial facilities affected by the 1999 Turkey earthquake. Journal of Performance of Constructed Facilities, 20(1), 28–36(2006)
[2] AKYILDIZ, H. Liquid sloshing in a baffled rectangular tank under irregular excitations. Ocean Engineering, 278, 114472(2023)
[3] YING, L., MENG, X., ZHOU, D., XU, X. L., ZHANG, J. D., and LI, X. H. Sloshing of liquid in a baffled rectangular aqueduct considering soil-structure interaction. Soil Dynamics and Earthquake Engineering, 122, 132–147(2019)
[4] BELLEZI, C. A., CHENG, L. Y., OKADA, T., and ARAI, M. Optimized perforated bulkhead for sloshing mitigation and control. Ocean Engineering, 187, 106171(2019)
[5] NIMISHA, P., JAYALEKSHMI, R., and VENKATARAMANA, K. Slosh damping in rectangular liquid tank with additional blockage effects under pitch excitation. Journal of Fluids Engineering, 144(12), 121403(2022)
[6] CHO, I. H. and KIM, M. H. Effect of a bottom-hinged, top-tensioned porous baffle on sloshing reduction in a rectangular tank. Applied Ocean Research, 104, 102345(2020)
[7] NAYAK, S. K. and BISWAL, K. C. Nonlinear seismic response of a partially-filled rectangular liquid tank with a submerged block. Journal of Sound and Vibration, 368, 148–173(2016)
[8] MITRA, S. and SINHAMAHAPATRA, K. P. Slosh dynamics of liquid-filled containers with submerged components using pressure-based finite element method. Journal of Sound and Vibration, 304(1-2), 361–381(2007)
[9] ISAACSON, M. and PREMASIRI, S. Hydrodynamic damping due to baffles in a rectangular tank. Canadian Journal of Civil Engineering, 28, 608–616(2001)
[10] MENG, X., ZHOU, D., LIM, Y. M., WANG, J. D., and HUO, R. L. Seismic response of rectangular liquid container with dual horizontal baffles on deformable soil foundation. Journal of Earthquake Engineering, 27(7), 1943–1972(2023)
[11] CHO, I. H. Liquid sloshing in a swaying/rolling rectangular tank with a flexible porous elastic baffle. Marine Structures, 75, 102865(2021)
[12] MENG, X., ZHOU, D., KIM, M. K., and LIM, Y. M. Free vibration and dynamic response analysis of liquid in a rectangular rigid container with an elastic baffle. Ocean Engineering, 216, 108119(2020)
[13] MENG, X., ZHOU, D., and WANG, J. D. Effect of vertical elastic baffle on liquid sloshing in rectangular rigid container. International Journal of Structural Stability and Dynamics, 21(12), 2150167(2021)
[14] KOLAEI, A. and RAKHEJA, S. Free vibration analysis of coupled sloshing-flexible membrane system in a liquid container. Journal of Vibration and Control, 25(1), 84–97(2019)
[15] SYGULSKI, R. Boundary element analysis of liquid sloshing in baffled tanks. Engineering Analysis with Boundary Elements, 35(8), 978–983(2011)
[16] HU, Z., ZHANG, X. Y., LI, X. W., and LI, Y. On natural frequencies of liquid sloshing in 2-D tanks using boundary element method. Ocean Engineering, 153, 88–103(2018)
[17] GAO, H. S., YIN, Z., LIU, J., ZANG, Q., and LIN, G. Finite element method for analyzing effects of porous baffle on liquid sloshing in the two-dimensional tanks. Engineering Computations, 38, 2105–2136(2021)
[18] LIU, D. G. and LIN, P. Z. A numerical study of three-dimensional liquid sloshing in tanks. Journal of Computational Physics, 227(8), 3921–3939(2008)
[19] AMANO, Y., ISHIKAWA, S., YOSHITAKE, T., and KONDOU, T. Modeling and design of a tuned liquid damper using triangular-bottom tank by a concentrated mass model. Nonlinear Dynamics, 104(3), 1917–1935(2021)
[20] BARABADI, S., IRANMANESH, A., PASSANDIDEH-FARD, M., and BARABADI, A. A numerical study on mitigation of sloshing in a rectangular tank using floating foams. Ocean Engineering, 277, 114267(2023)
[21] XUE, M. A., CHEN, Y. C., ZHENG, J. H., QIAN, L., and YUAN, X. L. Liquid dynamics analysis of sloshing pressure distribution in storage vessels of different shapes. Ocean Engineering, 192, 106582(2019)
[22] LUO, Z. Q. and CHEN, Z. M. Sloshing simulation of standing wave with time-independent finite difference method for Euler equations. Applied Mathematics and Mechanics (English Edition), 32(11), 1475–1488(2011) https://doi.org/10.1007/s10483-011-1516-6
[23] BELLEZI, C. A., CHENG, L. Y., and NISHIMOTO, K. A numerical study on sloshing mitigation by vertical floating rigid baffle. Journal of Fluids and Structures, 109, 103456(2022)
[24] PIZZOLI, M., SALTARI, F., MASTRODDI, F., MARTINEZ-CARRASCAL, J., and GONZÁLEZ-GUTIÉRREZ, L. M. Nonlinear reduced-order model for vertical sloshing by employing neural networks. Nonlinear Dynamics, 107, 1469–1478(2022)
[25] AKYILDIZ, H., ÜNAl, N. E., and AKSOY, H. An experimental investigation of the effects of the ring baffles on liquid sloshing in a rigid cylindrical tank. Ocean Engineering, 59, 190–197(2013)
[26] SANAPALA, V. S., RAJKUMAR, M., VELUSAMY, K., and PATNAIK, B. S. V. Numerical simulations of parametric liquid sloshing in a horizontally baffled rectangular container. Journal of Fluids and Structures, 76, 229–250(2018)
[27] NAYAK, S. K. and BISWAL, K. C. Liquid damping in rectangular tank fitted with various internal objects — an experimental investigation. Ocean Engineering, 108, 552–562(2015)
[28] WU, C. H., FALTINSEN, O. M., and CHEN, B. F. Numerical study of sloshing liquid in tanks with baffles by time-independent finite difference and fictitious cell method. Computer & Fluids, 63, 9–26(2012)
[29] LU, L., JIANG, S. C., ZHAO, M., and TANG, G. Q. Two-dimensional viscous numerical simulation of liquid sloshing in rectangular tank with/without baffles and comparison with potential flow solutions. Ocean Engineering, 108, 662–677(2015)
[30] HWANG, S. C., PARK, J. C., GOTOH, H., KHAYYER, A., and KANG, K. J. Numerical simulations of sloshing flows with elastic baffles by using a particle-based liquid-structure interaction analysis method. Ocean Engineering, 118, 227–241(2016)
[31] HERMANGE, C., OGER, G., CHENADEC, Y. L., and TOUZÉ, D. L. A 3D SPH-FE coupling for FSI problems and its application to tire hydroplaning simulations on rough ground. Computer Methods in Applied Mechanics and Engineering, 355, 558–590(2019)
[32] YANG, Q., JONES, V., and MCCUE, L. Free-surface flow interactions with deformable structures using an SPH-FEM model. Ocean Engineering, 55, 136–147(2012)
[33] BELAKROUM, R., KADJA, M., MAI, T. H., and MAALOUF, C. An efficient passive technique for reducing sloshing in rectangular tanks partially filled with liquid. Mechanics Research Communications, 37(3), 341–346(2010)
[34] NAKAYAMA, T. and WASHIZU, K. The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems. International Journal for Numerical Methods in Engineering, 17(11), 1631–1646(1981)
[35] ZHANG, C. W. Nonlinear simulation of resonant sloshing in wedged tanks using boundary element method. Engineering Analysis with Boundary Elements, 69, 1–20(2016)
[36] ZHANG, Z. L., KHALID, M. S. U., LONG, T., CHANG, J. Z., and LIU, M. B. Investigation on sloshing mitigation using elastic baffles by coupling smoothed finite element method and decoupled finite particle method. Journal of Fluids and Structures, 94, 102942(2020)
[37] JIN, X., ZHENG, H. Y., LIU, M. M., ZHANG, F. G., YANG, Y. Z., and REN, L. Damping effects of dual vertical baffles on coupled surge-pitch sloshing in three-dimensional tanks: a numerical investigation. Ocean Engineering, 261, 112130(2022)
[38] FALTINSEN, O. M. A numerical nonlinear method of sloshing in tanks with two-dimensional flow. Journal of Ship Research, 22(3), 193–202(1978)
[39] SAGHI, R., HIRDARIS, S., and SAGHI, H. The influence of flexible liquid structure interaction on sway induced tank sloshing dynamics. Engineering Analysis with Boundary Elements, 131, 206–217(2021)
[40] BISWAL, K. C., BHATTACHARYYA, S. K., and SINHA, P. K. Non-linear sloshing in partially liquid filled containers with baffles. International Journal for Numerical Methods in Engineering, 68(3), 317–337(2006)
[41] ESWARAN, M. and REDDY, G. R. Numerical simulation of tuned liquid tank-structure systems through σ-transformation based liquid-structure coupled solver. Wind and Structures, 23(5), 421– 447(2016)
[42] GOUDARZI, M. A. and DANESH, P. N. Numerical investigation of a vertically baffled rectangular tank under seismic excitation. Journal of Fluid Mechanics, 61, 450–460(2016)
[43] FALTINSEN, O. M., ROGNEBAKKE, O. F., LUKOVSKY, I. A., and TIMOKHA, A. N. Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. Journal of Fluid Mechanics, 407, 201–234(2000)
[44] FALTINSEN, O. M. and TIMOKHA, A. N. An adaptive multimodal approach to nonlinear sloshing in a rectangular tank. Journal of Fluid Mechanics, 432, 167–200(2001)
[45] FALTINSEN, O. M. and TIMOKHA, A. N. A multimodal method for liquid sloshing in a two-dimensional circular tank. Journal of Fluid Mechanics, 665, 457–479(2010)
[46] FALTINSEN, O. M. and TIMOKHA, A. N. Resonant three-dimensional nonlinear sloshing in a square-base basin, part 4: oblique forcing and linear viscous damping. Journal of Fluid Mechanics, 822, 139–169(2017)
[47] FALTINSEN, O. M., ROGNEBAKKE, O. F., and TIMOKHA, A. N. Resonant three-dimensional nonlinear sloshing in a square-base, part 2: effect of higher modes. Journal of Fluid Mechanics, 523, 199–218(2005)
[48] FALTINSEN, O. M., LAGODZINSKYI, O. E., and TIMOKHA, A. N. Resonant three-dimensional nonlinear sloshing in a square base basin, part 5: three-dimensional non-parametric tank forcing. Journal of Fluid Mechanics, 894, A10(2020)
[49] FALTINSEN, O. M. and TIMOKHA, A. N. Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth. Journal of Fluid Mechanics, 470, 319–357(2002)
[50] GAVRILYUK, I., LUKOVSKY, I., and TIMOKHA, A. N. Linear and nonlinear sloshing in a circular conical tank. Fluid Dynamics Research, 37(6), 399–429(2005)
[51] GAVRILYUK, I., HERMANN, M., LUKOVSKY, I. A., SOLODUM, O. V., and TIMOKHA, A. N. Weakly nonlinear sloshing in a truncated circular conical tank. Fluid Dynamics Research, 45(5), 055512(2013)
[52] YU, Y. S., MA, X. R., and WANG, B. L. Multidimensional modal analysis of liquid nonlinear sloshing in right circular cylindrical tank. Applied Mathematics and Mechanics (English Edition), 28(8), 1007–1018(2007) https://doi.org/10.1007/s10483-007-0803-y
[53] FALTINSEN, O. M. and TIMOKHA, A. N. Multimodal analysis of weakly nonlinear sloshing in a spherical tank. Journal of Fluid Mechanics, 719, 129–164(2013)
[54] FALTINSEN, O. M., FIROOZKOOTHI, R., and TIMOKHA, A. N. Analytical modeling of liquid sloshing in a two-dimensional rectangular tank with a slat screen. Journal of Engineering Mathematics, 70(1), 93–109(2011)
[55] FALTINSEN, O. M., FIROOZKOOTHI, R., and TIMOKHA, A. N. 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(6), 062106(2011)
[56] LOVE, J. S. and TAIL, M. J. Nonlinear simulation of a tuned liquid damper with damping screens using a modal expansion technique. Journal of Fluids and Structures, 26(7-8), 1058–1077(2010)
[57] LOVE, J. S. and TAIL, M. J. Non-linear multimodal model for tuned liquid dampers of arbitrary tank geometry. International Journal of Non-Linear Mechanics, 46(8), 1065–1075(2011)
[58] LOVE, J. S. and TAIL, M. J. Nonlinear multimodal model for TLD of irregular tank geometry and small liquid depth. Journal of Fluids and Structures, 43, 83–99(2013)
[59] LOVE, J. S. and HASKETT, T. C. Nonlinear modelling of tuned sloshing dampers with large internal obstructions: damping and frequency effects. Journal of Fluids and Structures, 79, 1–13(2018)
[60] GAVRILYUK, I., LUKOVSKY, I., TROTSENKO, Y., and TIMOKHA, A. N. Sloshing in a vertical circular cylindrical tank with an annular baffle, part 2: nonlinear resonant waves. Journal of Engineering Mathematics, 57(1), 57–78(2007)
[61] ZHOU, D., WANG, J. D., and LIU, W. Q. Nonlinear sloshing of liquid in rigid cylindrical container with a rigid annular baffle: free vibration. Nonlinear Dynamics, 78(4), 2557–2576(2014)
[62] WANG, J. D., LO, S. H., ZHOU, D., and DONG, Y. Nonlinear sloshing of liquid in a rigid cylindrical container with a rigid annular baffle under lateral excitation. Shock and Vibration, 2019, 5398038(2019)
[63] LAMB, H. Hydrodynamics, Cambridge University Press, Cambridge (1945)
[64] HENDERSON, D. M. and MILES, J. W. Surface-wave damping in a circular cylinder with a fixed contact line. Journal of Fluid Mechanics, 275, 285–299(1994)
[65] FALTINSEN, O. M. and TIMOKHA, A. Sloshing, Cambridge University Press, Cambridge (2009)
[66] BATEMAN, H. Partial Differential Equations of Mathematical Physics, Dover Publications, New York (1944)
[67] LUKE, J. C. A variational principle for a liquid with a free surface. Journal of Fluid Mechanics, 27, 395–397(1967)
[68] NARIMANOV, G. S. Movement of a tank partly filled by a liquid: the taking into account of non-smallness of amplitude. Journal of Applied Mathematics and Mechanics, 21, 513–524(1957)
[69] MOISEEV, N. N. To the theory of nonlinear oscillations of a limited liquid volume of a liquid. Journal of Applied Mathematics and Mechanics, 22, 612–621(1958)
[70] LUKOVSKY, I. A. Nonlinear Dynamics: Mathematical Models for Rigid Bodies with a Liquid, De Gruyter, Berlin (2015)