LINEAR-STABILITY OF LID-DRIVEN CAVITY FLOW

Previous experimental studies indicate that the steady two-dimensional flow in a lid-dri becomes unstable and goes through a sequence of tra nsitions before becoming turbulent. In this study, an analysis of this instability is undertaken. The two-dimensional base flow is computed numerically over a range of Reynolds numbers and is perturbed with thr ee-dimensional disturbances. The partial differential equations govern ing the evolution of these perturbations are then obtained using linea r stability analysis and normal mode analysis. Using a finite differen ce discretization, a generalized eigenvalue problem is formulated from these equations whose solution gives the dispersion relation between complex growth rate and wave number. An eigenvalue solver using simult aneous iteration is employed to identify the dominant eigenvalue which is indicative of the growth rate of these perturbations and the assoc iated eigenfunction which characterizes the secondary state. This pape r presents stability curves to identify the critical Reynolds number a nd the critical wavelength of the neutral mode and discusses the mecha nism of instability through energy calculations. This paper finds that the loss of stability of the base flow is due to a long wavelength mo de at a critical Reynolds number (Re) of 594. The mechanism is analyze d through a novel application of the Reynolds-Orr equations and shown to be due to a Goertler type instability. The stability curves are rel atively flat indicating that this state will be challenged by many sho rter wavelength modes at a slightly higher Reynolds number. In fact, a second competing mode with a wavelength close to the cavity width was found to be unstable at Re = 730. The present results of the reconstr ucted flow based on these eigenfunctions at the neutral state, show st riking similarities to the experimental observations.