This paper presents a numerical study on steady flow around two identical circular cylinders of various arrangements at a low subcritical Reynolds number ( $\mathit{Re}=10^{3}$ ). The ratio of centre-to-centre pitch distance ( $P$ ) to the diameter of the cylinder ( $D$ ) ranges from 1.5 to 4, and the alignment angle $({\it\alpha})$ between the two cylinders and the direction of the cross-flow varies from 0 to 90°. The detailed flow information obtained from direct numerical simulation allows a comprehensive interpretation of the underlying physics responsible for some interesting flow features observed around two staggered cylinders. Four distinct vortex shedding regimes are identified and it is demonstrated that accurate classification of vortex shedding regimes around two staggered cylinders should consider the combination of the flow visualization with the analyses of lift forces and velocity signal in the wake. It is revealed that the change in pressure distribution, as a result of different vortex shedding mechanisms, leads to a variety of characteristics of hydrodynamic forces on both cylinders, including negative drag force, attractive and repulsive lift forces. Two distinct vortex shedding frequencies are identified and are attributed to the space differences based on the flow structures observed in the wake of the cylinders. It is also found that the three-dimensionality of flow in the gap and the shared wake region is significantly weakened in almost two of the classified flow regimes; however, compared with the flow around a single cylinder, active wake interaction at large ${\it\alpha}$ does not clearly increase the three-dimensionality.

This paper presents the response and the wake modes of a freely vibrating D-section prism with varying angles of attack ( $\alpha = 0^\circ \text {--}180^\circ$ ) and reduced velocity ( $U^* = 2\text {--}20$ ) by a numerical investigation. The Reynolds number, based on the effective diameter, is fixed at 100. The vibration of the prism is allowed only in the transverse direction. We found six types of response with increasing angle of attack: typical vortex-induced vibration (VIV) at $\alpha = 0^\circ \text {--}35^\circ$ ; extended VIV at $\alpha = 40^\circ \text {--}65^\circ$ ; combined VIV and galloping at $\alpha = 70^\circ \text {--}80^\circ$ ; narrowed VIV at $\alpha = 85^\circ \text {--}150^\circ$ ; transition response, from narrowed VIV to pure galloping, at $\alpha = 155^\circ \text {--}160^\circ$ ; and pure galloping at $\alpha = 165^\circ \text {--}180^\circ$ . The typical and narrowed VIVs are characterized by linearly increasing normalized vibration frequency with increasing $U^*$ , which is attributed to the stationary separation points of the boundary layer. On the other hand, in the extended VIV, the vortex shedding frequency matches the natural frequency in a large $U^*$ range with increasing $\alpha$ generally. The galloping is characterized by monotonically increasing amplitude with enlarging $U^*$ , with the largest amplitude being $A^* = 3.2$ . For the combined VIV and galloping, the vibration amplitude is marginal in the VIV branch while it significantly increases with $U^*$ in the galloping branch. In the transition from narrowed VIV to pure galloping, the vibration frequency shows a galloping-like feature, but the amplitude does not monotonically increase with increasing $U^*$ . Moreover, a partition of the wake modes in the $U^*$ – $\alpha$ parametric plane is presented, and the flow physics is elucidated through time variations of the displacement, drag and lift coefficients and vortex dynamics. The angle-of-attack range of galloping is largely predicted by performing a quasi-steady analysis of the galloping instability. Finally, the effects of $m^*$ and ${\textit {Re}}$ , the roles of afterbody and the roles of separation point in determining vibration responses and vortex shedding frequency are further discussed.

This paper presents a numerical study of three-dimensional (3D) vortex shedding flow in the wake of four circular cylinders in a square configuration with a constant space-to-diameter ratio of 2. Numerical tests are carried out for Reynolds number (Re) in the range from 100 to 500. Four wake flow regimes are identified at this spacing ratio. Regime 1 ($100\le \mathrm{Re}\le 220$ ) is characterized by the inclination of weak spanwise vortices and weak streamwise vortices, where the wake behind four cylinder array shares similar features to that behind a single cylinder with a large equivalent diameter. It is observed that the onset of three-dimensionality in the wake behind four cylinder array occurs at lower Re than that behind a single cylinder. Regime 2 ($240\le \mathrm{Re}\le 300$ ) is characterized by the appearance of the regular wavy spanwise vortices and rib-shaped streamwise vortices. The wavelength of the spanwise vortices is about 1.2 and the wake flow is similar to the transition mode B of a single cylinder. Regime 3 ($320\le \mathrm{Re}\le 380$ ) is characterized by severe vortex dislocations in the wake of the cylinders and regime 4 ($400\le \mathrm{Re}\le 500$ ) is characterized by the absence of vortex dislocations and the strong streamwise vortices. The flow between the upstream and the downstream cylinders is predominantly two-dimensional in regimes 1, 2, and 3 and becomes 3D in regime 4. Physical mechanisms responsible for different flow regimes are proposed and discussed in details. Significant changes in the root-mean-square force coefficients, wake formation length, and phase angle of the lift coefficients on the downstream cylinders are observed when the flow transits from one regime to another.

Sinusoidally oscillatory flow around four circular cylinders in an in-line square arrangement is numerically investigated at Keulegan–Carpenter numbers ( $\mathit{KC}$ ) ranging from 1 to 12 and at Reynolds numbers ( $\mathit{Re}$ ) from 20 to 200. A set of flow patterns is observed and classified based on known oscillatory flow regimes around a single cylinder. These include six types of reflection symmetry regimes to the axis of flow oscillation, two types of spatio-temporal symmetry regimes and a series of symmetry-breaking flow patterns. In general, at small gap distances, the four structures behave more like a single body, and the flow fields therefore resemble those around a single cylinder with a large effective cylinder diameter. With increasing gap distance, flow structures around each individual cylinder in the array start to influence the overall flow patterns, and the flow field shows a variety of symmetry and asymmetry patterns as a result of vortex and shear layer interactions. The characteristics of hydrodynamic forces on individual cylinders as well as on the cylinder group are also examined. It is found that the hydrodynamic forces respond in a similar manner to the flow field to the cylinder proximity and wake interference.

A stable three-dimensional (3D) wake structure for flow past a circular cylinder has been discovered through 3D direct numerical simulations (DNS). The stable 3D wake structure occurs over a small range of Reynolds number (Re) below the critical Re for Mode A^∗ (i.e., Mode A with large-scale vortex dislocations, where Mode A is the first 3D instability mode which will evolve into Mode A^∗) instability. It is believed that the stable 3D wake structure discovered in this study is the stable state of Mode A wake structure inferred by Williamson [“Three-dimensional wake transition,” J. Fluid Mech. 328, 345–407 (1996)]. This confirms the wake transition sequence of 2D → A → A^∗ → B suggested by Williamson. Compared with conventional Mode A structure, the stable state of Mode A structure has much weaker amplitude and does not evolve into large-scale vortex dislocations. The stable state of Mode A structure is triggered by small-scale spanwise disturbance introduced upstream of the cylinder, due to energy amplification through convective instability of the flow. The stable state of Mode A is transient and is damped out eventually under a transient initial disturbance condition, but is sustained throughout under a persistent disturbance condition. The emergence of the stable state of Mode A structure is correlated with both Re and the disturbance level. With the decrease of Re, the stable state of Mode A structure gradually becomes less well-defined and eventually disappears. With the decrease of the disturbance level, the stable state of Mode A structure emerges at a higher Re.

Two-dimensional direct numerical simulation and Floquet stability analysis have been performed at moderate Keulegan–Carpenter number ( $KC$ ) and low Reynolds number ( $Re$ ) for a square cross-section cylinder with its face normal to the oscillatory flow. Based on the numerical simulations a map of flow regimes is formed and compared to the map of flow around an oscillating circular cylinder by Tatsuno & Bearman (J. Fluid Mech., vol. 211, 1990, pp. 157–182). Two new flow regimes have been observed, namely A $^{\prime }$ and F $^{\prime }$ . The regime A $^{\prime }$ found at low $KC$ is characterised by the transverse convection of fluid particles perpendicular to the motion; and the regime F $^{\prime }$ found at high $KC$ shows a quasi-periodic feature with a well-defined secondary period, which is larger than the oscillation period. The Floquet analysis demonstrates that when the two-dimensional flow breaks the reflection symmetry about the axis of oscillation, the quasi-periodic instability and the synchronous instability with the imposed oscillation occur alternately for the square cylinder along the curve of marginal stability. This alternate pattern in instabilities leads to four distinct flow regimes. When compared to the vortex shedding in otherwise unidirectional flow, the two quasi-periodic flow regimes are observed when the oscillation frequency is close to the Strouhal frequency (or to half of it). Both the flow regimes and marginal stability curve shift in the $(Re,KC)$ -space compared to the oscillatory flow around a circular cylinder and this shift appears to be consistent with the change in vortex formation time associated with the lower Strouhal frequency of the square cylinder.