Hopf bifurcations in the wake of a square cylinder

Abstract

In the present numerical study we report growths of four different unsteady modes of Hopf bifurcations in the wake of a square cylinder. Extensive three-dimensional simulations of the flow are conducted with six different Reynolds numbers (60, 100, 115, 125, 200, and 300) and three different (nondimensional) cylinder lengths (7.5, 10, and 12). At a near-critical Reynolds number (Re), 60, the initiation of near-wake spanwise pressure oscillation along the coreline of the shedded vortices, and formation of a local pressure maximum over its midsection led to the growth of a single mode Hopf bifurcation. At Re=100, the enhanced spanwise oscillation of pressure, and the resultant occurrence of two distinct local pressure maxima along the vortex corelines facilitated the growth of a second mode. For 100≤Re≤125 the spanwise wake was noted evolve predominantly through this nearly symmetrically distributed second mode. As the Reynolds number gradually increased, the spanwise asymmetric pressure and velocity oscillations along the vortex corelines gained momentum, and owing to enhanced flow instability, higher order bifurcation modes started to grow. For supercritical transitional flows with Re≥200 there occurred frequent switching between the transient bifurcation modes, and the instantaneous wake thereby evolved through single/multiple modes of Hopf bifurcations with variable length scales. The growth of a number of local pressure maxima over the spanwise extended vortex corelines and gradual decrease in pressure along their two sides are found to be responsible for initiating these bifurcations. Our study further reveals that the formation of a Karman vortex street in the supercritical transitional regime is essentially governed by the spontaneous generations of sequences of Hopf bifurcations in the near wake.