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.

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The effects of yaw angle on wake characteristics of a stationary square cylinder were investigated in terms of the hydrodynamic forces, the vortex shedding frequency, and the vortical structures using direct numerical simulations (DNS) at a Reynolds number of 1000. In total, four yaw angles, namely, α = 0°, 15°, 30°, and 45°, were considered. The three-dimensional (3D) Navier–Stokes equations were solved directly using the finite volume method in OpenFOAM. It was found that the first-order statistics of the drag coefficient and the Strouhal number satisfied the independence principle (IP) closely. However, the second-order statistics of the drag and lift coefficients deviated apparently from the IP for α ≥ 25°. The iso-surfaces of the spanwise vorticity gradually disorganized and the magnitudes of the spanwise vorticity contour decreased as the yaw angle α was increased from 0° to 45°. By contrast, the streamwise vorticity iso-surfaces were found to become more organized and the magnitudes of the spanwise velocity contour became larger as a result of the increase in yaw angle, indicating the impairment of the quasi-two-dimensionality and the enhancement of the three-dimensionality of the wake flow. Extensive comparisons were also made with previous DNS results for a yawed circular cylinder, and both similarities and differences between these two kinds of cylinder wakes are discussed.