Fully non-linear simulation of second-order resonance in a three-dimensional tank using the PSME method

Ming-Jyh Chern, Nima Vaziri*, Alistair G. L. Borthwick

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Sloshing occurs when a partially filled tank is subject to external excitation. If the excitation frequency is equal to half of the second-order natural frequency obtained from the linear theory, then second-order resonance may occur. But it gradually modifies to standing waves. Moreover, violent second-order resonant free surface motions may be induced when the sum (or difference) of the excitation frequency and any one of the natural frequencies is equal to another natural frequency. In this study, fully non-linear second-order resonance waves in a three-dimensional shallow water rectangular tank are simulated using a pseudospectral if-transformation model. The model is validated against results from three benchmark tests for which there are published analytical and numerical solutions available. A detailed analysis is presented of sloshing in a shallow water tank, with the main excitation and response frequencies identified from the power spectra. Large amplitude free surface motions are observed whenever second-order resonance occurs. In certain cases, the wave pattern in the tank is different to that predicted from linear analysis of second-order resonance, due to the effect of nonlinearity. Results are presented from a parameter study examining the influences of water depth, base aspect ratio, and excitation amplitude on the wave motions and patterns. It is found that the wave pattern is highly dependent on the water depth in the tank, but relatively insensitive to excitation amplitude. Also, the decay patterns are seen when the second-order resonance excitations are applied vertically. The study demonstrates that second-order resonance can be very pronounced in shallow water. (C) 2012 Elsevier Ltd. All rights reserved.

Original languageEnglish
Pages (from-to)22-32
Number of pages11
JournalApplied Ocean Research
Publication statusPublished - Aug 2012


  • Second-order resonance
  • PSME method
  • Three-dimensional tank
  • Shallow water


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