This study topologically describes near-wall flows around a surface-mounted cylinder at a high Reynolds number ( $Re$ ) of $5\times 10^4$ and in a very thick boundary layer, which were partially measured or technically approximated from the literature. For complete and rational flow construction, we use high-resolution simulations and critical-point theory. The large-scale near-wake vortex is composed of two connected segments rolled up from the sides of the cylinder and from the free end. Another large-scale side vortex clearly roots on two notable foci on the lower side wall. In the junction region, the side vortex moves upwards with a curved trajectory, which induces the formation of nodes on the ground surface. In the free-end region, the side vortex is compressed, which results in a smaller trailing-edge vortex and its downstream movement. Only tip vortices are observed in the far wake. The origin of the tip vortices and their distinction from the near-wake vortex are discussed. Further analyses suggest that $Re$ independence should be treated with high caution when $Re$ increases from 500 to ${O}(10^4)$ . The occurrence of upwash flow behind the cylinder strongly depends on the increase in $Re$ , the mechanism of which is also provided. The separation–reattachment process in the junction region and the trailing-edge vortices are discovered only at a high $Re$ . The former should significantly affect the strength of the side vortex in the junction region and the latter should cause a sharp drop in pressure near the trailing edge.

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.