A two-dimensional computational hydrodynamic model is developed to investigate the propulsive performance of a flapping foil system in viscous incompressible flows, which consists of two anti-phase flapping foils in side-by-side arrangement. In the simulations, the gap between the two foils is varied from 1.0 to 4.0 times of the diameter of the semi-circular leading edge; the amplitude-based Strouhal number is changed from 0.06 to 0.55. The simulations therefore cover the flow regimes from negligible to strong interference in the wake flow. The generations of drag and thrust are investigated as well. The numerical results reveal that the counter-phase flapping motion significantly changes the hydrodynamic force generation and associated propulsive wake. Furthermore, the wake interference becomes important for the case with a smaller foil-foil gap and induces the inverted Bénard von Kármán vortex streets. The results show that the hydrodynamic performance of two anti-phase flapping foils can be significantly different from an isolated pitching foil. Findings of this study are expected to provide new insight for developing hydrodynamic propulsive systems by improving the performance based on the foil-foil interaction.

In this paper, instabilities in the flow over a circular cylinder of diameter $D$ with dual splitter plates attached to its rear surface are numerically investigated using the spectral element method. The key parameters are the splitter plate length $L$ , the attachment angle $\unicode[STIX]{x1D6FC}$ and the Reynolds number $Re$ . The presence of the plates was found to significantly modify the flow topology, leading to substantial changes in both the primary and secondary instabilities. The results showed that the three instability modes present in the bare circular cylinder wake still exist in the wake of the present configurations and that, in general, the occurrences of modes A and B are delayed, while the onset of mode QP is earlier in the presence of the splitter plates. Furthermore, two new synchronous modes, referred to as mode A $^{\prime }$ and mode B $^{\prime }$ , are found to develop in the wake. Mode A $^{\prime }$ is similar to mode A but with a quite long critical wavelength. Mode B $^{\prime }$ shares the same spatio-temporal symmetries as mode B but has a distinct spatial structure. With the exception of the case of $L/D=0.25$ , mode A $^{\prime }$ persists for all configurations investigated here and always precedes the transition through mode A. The onset of mode B $^{\prime }$ occurs for $\unicode[STIX]{x1D6FC}>20^{\circ }$ with $L/D=1.0$ and for $L/D>0.5$ with $\unicode[STIX]{x1D6FC}=60^{\circ }$ . The characteristics of all the transition modes are analysed, and their similarities and differences are discussed in detail in comparison with the existing modes. In addition, the physical mechanism responsible for the instability mode B $^{\prime }$ is proposed. The weakly nonlinear feature of mode B $^{\prime }$ , as well as that of mode A $^{\prime }$ , is assessed by employing the Landau model. Finally, selected three-dimensional simulations are performed to confirm the existence of these two new modes and to investigate the nonlinear evolution of the three-dimensional modes.

Three-dimensional numerical computations are carried out for a cylinder undergoing controlled sinusoidal oscillation perpendicular to the free stream. The results are examined for Re = 500 and an oscillation amplitude of $A/D=0.25$ , to allow comparison with the two-dimensional study by Blackburn and Henderson [J. Fluid Mech. 385, 255–286 (1999)]. The dynamic response of the flow is investigated over a wide oscillation frequency range from 0.5f_o to 1.5f_o, with f_o being the natural shedding frequency of the fixed cylinder. As the oscillation frequency passes through f_o, the ubiquitous abrupt shift in the phase of vortex shedding is again observed. On either side of this shift, the wakes exhibit Kármán mode of shedding, in the absence of the mode switch. The hypothesis for the mechanism of phase shift proposed by Blackburn and Henderson [J. Fluid Mech. 385, 255–286 (1999)] is further examined by quantitatively measuring the flux of vorticity generated on the base of the cylinder. Unlike in two-dimensional flow, where hysteresis is found to be associated with four branches, in three-dimensional flow only two longer branches K₁ and K₂ are identified, which bring out a remarkably simplified bifurcation structure. Dynamic mode decomposition (DMD) and its sparsity-promoting variant are used to characterize the coherent modes that govern the dynamics of the flow, as well as their corresponding frequencies. For the non-synchronized case, a multitude of DMD modes must be retained to accurately approximate the original flow, while for the synchronized case, only four DMD modes suffice to guarantee a same performance loss. In addition, the DMD modes that have the most profound impact on the hysteresis of lift and drag fluctuations are identified by DMD reconstruction.

Two-dimensional numerical computations are carried out for two rigidly connected cylinders of unequal sizes undergoing vortex-induced vibrations (VIV) perpendicular to the free stream. Results are examined for Re = 250 and a fixed diameter ratio of $d/D=0.2$ . The VIV response of the system is investigated for various positions of the small cylinder, covering a fine grid of wide radial (r) and azimuthal (θ) ranges, relative to the origin of the main cylinder. It is shown that the structural dynamics and hydrodynamic forces are strongly dependent on the arrangements. Regions of VIV reduction and amplification are distinguished, and the highest and lowest oscillation amplitudes are, respectively, acquired at configurations of $(r,\theta )=(0.7D,90°)$ and $(r,\theta )=(0.88D,130°)$ . A deeper analysis in terms of the wake topology and surface pressure is then provided for these two extreme cases, to figure out the underlying mechanisms that lead to such markedly distinct responses. For the former case, the shear layers from two cylinders intensely interact and amalgamate during the oscillation, setting off subsequent processes of shear layer reattachment and downflow that are responsible for the observed high-amplitude response, while for the latter case, the shear layers from the small cylinder are highly stretched and absent from direct interaction with that from the large cylinder, which is favorable for stabilizing the wake and maintaining the low-amplitude response. Proper orthogonal decomposition (POD) is further utilized to correlate the key features of the wake with the dominant coherent structures in the flow.

In this paper, the flow around a main circular cylinder with two small control cylinders is numerically investigated at a subcritical Reynolds number of 3900. Two small control cylinders with diameter ratio of d/D = 0.04 (d and D are the diameters of the control cylinders and the main cylinder, respectively) are symmetrically placed inside the separated shear layers emanating from the main cylinder. The computation is performed with highly resolved direct numerical simulations employing a high-order spectral/hp element method. For validation purposes, the results of the flow over a single cylinder are presented at first, and the current simulation is in good agreement with the existing experiments and large eddy simulations. For the main cylinder with the two control cylinders, the dynamics of shear layers is found to be modified strongly owing to the disturbance from the small control cylinders. As a consequence, the recirculation length is decreased by 89% as compared to that of a single cylinder. The presence of the small control cylinders further leads to a significant increase in the fluctuating lift and the mean drag exerting on the main cylinder. Furthermore, the statistical analysis and turbulent wake visualization imply that the turbulent intensity is amplified in the near wake region and becomes weaker further downstream. This study then suggests that the small control cylinders located in the near wake region of the main cylinder may work differently in the turbulent wake against that in the laminar wake.