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The time development of the symmetrical standing zones of recirculation, which is formed in the early stages of the impulsively started laminar flow over the square cylinder, have been studied numerically. The Reynolds number considered ranges from 25 to 1,000. Main flow characteristics of the developing recirculation region aft of the square cylinder and its interaction with the separating shear layer from the leading edges are studied through the developing streamlines. Other flow characteristics are analysed in terms of pressure contours, surface pressure coefficient, wake length and drag coefficient. Four main‐flow types and three subflow types of regimes are identified through a detailed analysis of the evolution of the flow characteristics. Typically, for a given Reynolds number, it is noted that flow starts with no separation (type I main‐flow). As time advances, symmetrical standing zone of recirculation develops aft of the square cylinder (type II main‐flow). The rate of growth in width, length and structure of the aft end eddies (sub‐flow (a)) depends on the Reynolds number. In time, separated flow from the leading edges of the square cylinder also develops (type III main‐flow) and forms growing separation bubbles (sub‐flow (b)) on the upper and lower surfaces of the square cylinder. As time advances, the separation bubbles on the upper and lower surfaces of the cylinder grow towards downstream regions and eventually merge with the swelling symmetrical eddies aft of the cylinder. This merging of the type II and type III flows created a complex type IV main‐flow regime with a disturbed tertiary flow zone (sub‐flow (c)) near the merging junction. Eventually, depending on the Reynolds number, the flow develops into a particular category of symmetrical standing recirculatory flow of specific characteristics.

A 3D numerical simulation of a circular cylinder wake is presented in this paper. The cylinder is harmonically forced in the stream-wise direction. The objective of the present work is to investigate the effect of the oscillation amplitude on the secondary transition of the wake. The frequency of the lift force is then linked to the form of the vortex shedding mode. The relation between these vortex shedding modes using POD analysis of the transverse velocity and the unsteady lift coefficient of 3D simulation is in good agreement with the 2D model. Results show that the 3D spanwise effect, which can change the wake structure, is suppressed at Re = 200 by streamwise oscillation of the cylinder. Thus the 2D analysis can effectively model the temporal instability of the wake flow.

The interaction between the cylinder motion and the wake is a complex feedback phenomenon in which the symmetry relationship between the wake and the cylinder motion plays a key role. Depending on the frequency of oscillation the symmetry relationship between the unforced von Karman wake and the imposed forced oscillations can induce a series of bifurcations. This detailed bifurcation behavior is the subject of study in the present work. 2D and 3D simulations are carried out for a Reynolds number Re=1000. As the inline cylinder forcing amplitude is increased, the wake undergoes a series of bifurcations and associated changes in the flow structure. Although the 2D analysis is clearly non-physical, it leads to a ‘simpler’ and more tractable model. Detailed comparison of the 2D and 3D POD modes provides insight into the forced wake dynamics. The 3D spatial mode shapes are significantly similar to those from 2D simulations. The relative modal energy distribution captures well the wake flow complexity. The first mode contains over 90% of the flow energy in the 2D simulations. This ratio drops significantly in the 3D case to around 45%. Clearly 3D effects are very important when it comes to energy distribution between the modes. However, the predominance of the first mode seems high enough to maintain the 2D-like dynamics. The wake flow is found to undergo two main transitions with increased forcing. The first is periodic shedding to chaotic shedding. The second is chaotic shedding to half-frequency shedding caused by a period-doubling bifurcation. The 2D simulations correctly predict these bifurcations — including the type and the number of bifurcations. The results suggest that the forced wake dynamics are primarily dominated by two-dimensional rather than 3D dynamics. However, 3D effects are important in determining the exact parameter values where bifurcations occur. A previously developed low order analytical model, based on 2D simulations, is also used to predict the wake bifurcation behavior. The relevance of the low order model has interesting implications for VIV control.