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**Sloshing motions in excited tanks.**
*(English)*
Zbl 1115.76369

Summary: A fully nonlinear finite difference model has been developed based on inviscid flow equations. Numerical experiments of sloshing wave motion are undertaken in a 2-D tank which is moved both horizontally and vertically. Results of liquid sloshing induced by harmonic base excitations are presented for small to steep non-breaking waves. The simulations are limited to a single water depth above the critical depth corresponding to a tank aspect ratio of \(hs/b=0.5\). The numerical model is valid for any water depth except for small depth when viscous effects would become important. Solutions are limited to steep non-overturning waves. Good agreement for small horizontal forcing amplitude is achieved between the numerical model and second order small perturbation theory. For large horizontal forcing, nonlinear effects are captured by the third-order single modal solution and the fully non-linear numerical model. The agreement is in general good, both amplitude and phase. As expected, the third-order compared to the second-order solution is more accurate. This is especially true for resonance, high forcing frequency and mode interaction cases. However, it was found that multimodal approximate forms should be used for the cases in which detuning effects occur due to mode interaction. We present some test cases where detuning effects are evident both for single dominant modes and mode interaction cases. Furthermore, for very steep waves, just before the waves overturn, and for large forcing frequency, a discrepancy in amplitude and phase occurs between the approximate forms and the numerical model. The effects of the simultaneous vertical and horizontal excitations in comparison with the pure horizontal motion and pure vertical motion is examined. It is shown that vertical excitation causes the instability associated with parametric resonance of the combined motion for a certain set of frequencies and amplitudes of the vertical motion while the horizontal motion is related to classical resonance. It is also found that, in addition to the resonant frequency of the pure horizontal excitation, an infinite number of additional resonance frequencies exist due to the combined motion of the tank. The dependence of the non-linear behaviour of the solution on the wave steepness is discussed. It is found that for the present problem, non-linear effects become important when the steepness reaches about 0.1, in agreement with the physical experiments of Abramson [Rep. SP 106, NASA, 1966].

### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

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\textit{J. B. Frandsen}, J. Comput. Phys. 196, No. 1, 53--87 (2004; Zbl 1115.76369)

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