Transition in a 2-D lid-driven cavity flow

Direct numerical simulations about the transition process from laminar to chaotic flow in square lid-driven cavity flows are considered in this paper. The chaotic flow regime is reached after a sequence of successive supercritical Hopf bifurcations to periodic, quasi-periodic, inverse period-doubling, period-doubling, and chaotic self-sustained flow regimes. The numerical experiments are conducted by solving the 2-D incompressible Navier-Stokes equations with increasing Reynolds numbers (Re). The spatial discretization consists of a seventh-order upwind-biased method for the convection term and a sixth-order central method for the diffusive term. The numerical experiments reveal that the first Hopf bifurcation takes place at Re equal to 7402±4%, and a consequent periodic flow with the frequency equal to 0.59 is obtained. As Re is increased to 10,300, a new fundamental frequency (FF) is added to the velocity spectrum and a quasi-periodic flow regime is reached. For slightly higher Re (10,325), the new FF disappears and the flow returns to a periodic regime. Furthermore, the flow experiences an inverse period doubling at 10,325 <Re< 10,700 and a period-doubling regime at 10,600 <Re< 10,900. Eventually, for flows with Re greater than 11,000, a scenario for the onset of chaotic flow is obtained. The transition processes are illustrated by increasing Re using time-velocity histories, Fourier power spectra, and the phase-space trajectories. In view of the conducted grid independent study, the values of the critical Re presented above are estimated to be accurate within ±4%.