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f = −μV, μ the damping coefficient and V the local averaging flow velocity) are employed in this work to investigate the phenomenon of fluid resonance in narrow gaps between multi-bodies in close proximity under water waves. The numerical results are compared with experimental data available in the literature. The comparison demonstrates that both the viscous fluid model and the potential flow model are able to predict the resonant frequency reasonably well. However the conventional potential flow model (without artificial damping term) significantly over-predicts the wave height in narrow gaps around the resonant frequency. In order to calibrate the appropriate damping coefficient used for the potential model and make it work as well as the viscous fluid model in predicting the resonant wave height in narrow gaps but with little computational efforts, the dependence of damping coefficient μ on the body geometric dimensions is examined considering the parameters of gap width B g , body draft D, body breadth ratio B r and body number n (n = 2, 3), where B r = B B /B A for the case of two bodies (Body A and Body B) with different breadths of B A and B B , respectively. It was confirmed that the damping coefficient used for the potential flow model is not sensitive to the geometric dimensions and spatial arrangement. It was found that μ ∈ [0.4, 0.5] may guarantee the variation of H g /H 0 with kh to be generally in good agreement with the experimental data and the results of viscous fluid model, where H g is the excited wave height in narrow gaps under various dimensionless incident wave frequencies kh, H 0 is the incident wave height, k = 2π/L is the wave number and h is the water depth. narrow gap, fluid resonance, water wave, viscous fluid model, potential flow model, finite element method, boundary element method

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A numerical analysis of flow around a circular cylinder oscillating in-line with a steady flow is carried out over a range of driving frequencies $(f_{d})$ at relatively low amplitudes $(A)$ and a constant Reynolds number of 175 (based on the free-stream velocity). The vortex shedding is investigated, especially when the shedding frequency $(f_{s})$ synchronises with the driving frequency. A series of modes of synchronisation are presented, which are referred to as the $p/q$ modes, where $p$ and $q$ are natural numbers. When a $p/q$ mode occurs, $f_{s}$ is detuned to $(p/q)f_{d}$, representing the shedding of $p$ pairs of vortices over $q$ cycles of cylinder oscillation. The $p/q$ modes are further characterised by the periodicity of the transverse force over every $q$ cycles of oscillation and a spatial–temporal symmetry possessed by the global wake. The synchronisation modes $(p/q)$ with relatively small natural numbers are less sensitive to the change of external control parameters than those with large natural numbers, while the latter is featured with a narrow space of occurrence. Although the mode of synchronisation can be almost any rational ratio (as shown for $p$ and $q$ smaller than 10), the probability of occurrence of synchronisation modes with $q$ being an even number is much higher than $q$ being an odd number, which is believed to be influenced by the natural even distribution of vortices in the wake of a stationary cylinder.

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