A numerical study of three-dimensional liquid sloshing in tanks. (English) Zbl 1317.76061

Summary: A numerical model NEWTANK (Numerical Wave TANK) has been developed to study three-dimensional (3-D) nonlinear liquid sloshing with broken free surfaces. The numerical model solves the spatially averaged Navier-Stokes equations, which are constructed on a non-inertial reference frame having arbitrary six degree-of-freedom (DOF) of motions, for two-phase flows. The large-eddy-simulation (LES) approach is adopted to model the turbulence effect by using the Smagorinsky sub-grid scale (SGS) closure model. The two-step projection method is employed in the numerical solutions, aided by the Bi-CGSTAB technique to solve the pressure Poisson equation for the filtered pressure field. The second-order accurate volume-of-fluid (VOF) method is used to track the distorted and broken free surface. Laboratory experiments are conducted for both 2-D and 3-D nonlinear liquid sloshing in a rectangular tank. A linear analytical solution of 3-D liquid sloshing under the coupled surge and sway excitation is also developed in this study. The numerical model is first validated against the available analytical solution and experimental data for 2-D liquid sloshing of both inviscid and viscous fluids. The validation is further extended to 3-D liquid sloshing. The numerical results match with the analytical solution when the excitation amplitude is small. When the excitation amplitude is large where sloshing becomes highly nonlinear, large discrepancies are developed between the numerical results and the analytical solutions, the former of which, however, agree well with the experimental data. Finally, as a demonstration, a violent liquid sloshing with broken free surfaces under six DOF excitations is simulated and discussed.


76M25 Other numerical methods (fluid mechanics) (MSC2010)
76F65 Direct numerical and large eddy simulation of turbulence
76T10 Liquid-gas two-phase flows, bubbly flows
Full Text: DOI


[1] Akyildiz, H.; Ünal, E., Experimental investigation of pressure distribution on a rectangular tank due to the liquid sloshing, Ocean eng., 32, 1503-1516, (2005)
[2] Armenio, V.; La Rocca, M., On the analysis of sloshing of water in rectangular containers: numerical and experimental investigation, Ocean eng., 23, 705-739, (1996)
[3] Chen, W.; Haroun, M.A.; Liu, F., Large amplitude liquid sloshing in seismically excited tanks, Earthquake eng. struct. dyn., 25, 653-669, (1996)
[4] Chen, B.-F.; Chiang, H.-W., Complete 2D and fully nonlinear analysis of ideal fluid in tanks, J. eng. mech.-ASCE, 125, 70-78, (1999)
[5] Chen, B.-F., Viscous fluid in a tank under coupled surge, heave and pitch motions, J. waterw. port coast. Ocean eng.-ASCE, 131, 239-256, (2005)
[6] Chen, B.-F.; Nokes, R., Time-independent finite difference analysis of 2D and nonlinear viscous liquid sloshing in a rectangular tank, J. comput. phys., 209, 47-81, (2005) · Zbl 1329.76224
[7] Cho, J.R.; Lee, H.W., Non-linear finite element analysis of large amplitude sloshing flow in two-dimensional tank, Int. J. numer. method eng., 61, 514-531, (2004) · Zbl 1075.76568
[8] Chorin, A.J., Numerical solution of the navier – stokes equations, Math. comput., 22, 745-762, (1968) · Zbl 0198.50103
[9] Chorin, A.J., On the convergence of discrete approximations of the navier – stokes equations, Math. comput., 232, 341-353, (1969) · Zbl 0184.20103
[10] Faltinsen, O.M., A numerical nonlinear method of sloshing in tanks with two-dimensional flow, J. ship res., 22, 193-202, (1978)
[11] Faltinsen, O.M.; Rognebakke, O.F.; Lukovsky, I.A.; Timokha, A.N., Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth, J. fluid mech., 407, 201-234, (2000) · Zbl 0990.76006
[12] Faltinsen, O.M.; Timokha, A.N., Adaptive multimodal approach to nonlinear sloshing in a rectangular tank, J. fluid mech., 432, 167-200, (2001) · Zbl 0991.76009
[13] Frandsen, J.B.; Borthwick, A.G.L., Simulation of sloshing motions in fixed and vertically excited containers using a 2-D inviscid σ-transformed finite difference solver, J. fluid struct., 18, 197-214, (2003)
[14] Frandsen, J.B., Sloshing motions in excited tanks, J. comput. phys., 196, 53-87, (2004) · Zbl 1115.76369
[15] Gueyffier, D.; Li, J.; Nadim, A.; Scardovelli, R.; Zaleski, S., Volume-of-fluid interface tracking with smoothed surface stress methods for 3-D flows, J. comput. phys., 152, 423-456, (1999) · Zbl 0954.76063
[16] Hill, D.F., Transient and steady-state amplitudes of forced waves in rectangular basins, Phys. fluid, 15, 1576-1587, (2003) · Zbl 1186.76228
[17] Ibrahim, R.A.; Pilipchuk, V.N.; Ikeda, T., Recent advances in liquid sloshing dynamics, Appl. mech. rev., 54, 133-199, (2001)
[18] Ibrahim, R.A., Liquid sloshing dynamics: theory and applications, (2005), Cambridge University Press New York, USA · Zbl 1103.76002
[19] Kim, Y., Numerical simulation of sloshing flows with impact load, Appl. Ocean res., 23, 53-62, (2001)
[20] Kim, Y.; Shin, Y.-S.; Lee, K.H., Numerical study on slosh-induced impact pressures on 3-D prismatic tanks, Appl. Ocean res., 26, 213-226, (2004)
[21] Lin, P., Numerical modeling of water waves, (2008), Taylor & Francis Co.
[22] Lin, P.; Liu, P.L.-F., A numerical study of breaking waves in the surf zone, J. fluid mech., 359, 239-264, (1998) · Zbl 0916.76009
[23] Lin, P.; Li, C.-W., A σ-coordinate 3-D numerical model for surface wave propagation, Int. J. numer. method fluid, 38, 1045-1068, (2002) · Zbl 1094.76547
[24] Lin, P.; Li, C.-W., Wave – current interaction with a vertical square cylinder, Ocean eng., 30, 855-876, (2003)
[25] Lin, P.; Man, C., A staggered numerical algorithm for the extended Boussinesq equations, Appl. math. model., 31, 349-368, (2007) · Zbl 1169.76042
[26] D. Liu, Numerical modeling of three-dimensional water waves and their interaction with structures, Ph.D. thesis, National University of Singapore, 2007.
[27] Nakayama, T.; Washizu, K., Nonlinear analysis of liquid motion in a container subjected to forced pitching oscillation, Int. J. numer. method eng., 15, 1207-1220, (1980) · Zbl 0438.76012
[28] Nakayama, T.; Washizu, K., The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems, Int. J. numer. method eng., 17, 1631-1646, (1981) · Zbl 0474.76028
[29] Okamoto, T.; Kawahara, M., Two-dimensional sloshing analysis by Lagrangian finite element method, Int. J. numer. method fluid, 11, 453-477, (1990) · Zbl 0711.76008
[30] Okamoto, T.; Kawahara, M., 3-D sloshing analysis by an arbitrary lagrangian – eulerian finite element method, Int. J. comput. fluid dyn., 8, 129-146, (1997) · Zbl 0894.76039
[31] Rider, W.J.; Kothe, D.B., Reconstructing volume tracking, J. comput. phys., 141, 112-152, (1998) · Zbl 0933.76069
[32] Smagorinsky, J., General circulation experiments with the primitive equations, Mon. weather rev., 91, 99-164, (1963)
[33] Wang, C.Z.; Khoo, B.C., Finite element analysis of two-dimensional nonlinear sloshing problems in random excitations, Ocean eng., 32, 107-133, (2005)
[34] Wei, G.; Kirby, J.T., Time-dependent numerical code for extended Boussinesq equations, J. waterw. port coast. Ocean eng.-ASCE, 121, 251-261, (1995)
[35] Wu, G.X.; Ma, Q.A.; Taylor, R.E., Numerical simulation of sloshing waves in a 3D tank based on a finite element method, Appl. Ocean res., 20, 337-355, (1998)
[36] Wu, G.X.; Taylor, R.E.; Greaves, D.M., The effect of viscosity on the transient free-surface waves in a two-dimensional tank, J. eng. math., 40, 77-90, (2001) · Zbl 1006.76026
[37] Verhagen, H.G.; Wijingaarden, L., Non-linear oscillation of fluid in a container, J. fluid mech., 22, 737-751, (1965) · Zbl 0128.43101
[38] Van der Vorst, H.A., Iterative Krylov methods for large linear systems, (2003), Cambridge University Press New York, USA · Zbl 1023.65027
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