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A numerical investigation has been conducted to explore the complex nonlinear nature of flow in a backward-facing step channel by the simulated results, which include the bifurcation diagram, limit cycle oscillations, power spectrums, and phase portraits. For small values of Reynolds number, the flow was steady and laminar. When the Reynolds number was amplified, the flow becomes unsteady with the initiation of a supercritical Hopf bifurcation. The flow path is trapped by the stable limit cycles, and this system is made to proceed with a sustained oscillation. As the Reynolds number was amplified further, the stability of the investigated system keeps decreasing through a sequence of frequency-doubling bifurcations. The fundamental frequency of high amplitude was identical to the most amplified mode of Kelvin-Helmholtz instability oscillations in the shear layer. Frequencies with a small amplitude result in a slower development of the Kelvin-Helmholtz instability and are responsible for the roll-up of the shear layer. The phase portrait shows the evolution of a chaotic attractor from a simple periodic attractor. Prior to the onset of the chaotic motion, pitchfork bifurcation showed its existence.

A quasi-implicit fractional-step method is presented for computing unsteady incompressible flow on unstructured grids. A non-staggered grid system is employed rather than a staggered grid system because of the simplicity and ease of extension to three-dimensional analysis. In this study, the momentum , is applied to problems on unstructured grids to resolve the pressure oscillation problem occurring in a non-staggered grid system. An implicit time advancing scheme is used in order to remove the time step restriction and to reduce the required CPU time for problems with complex geometries. The nonlinear equations resulting from this fully implicit scheme are linearized without deteriorating the overall time accuracy. The system matrices are solved using the CG family method, known with P-BiCGSTAB, for momentum equation and P-CG for pressure Poisson equation. The present numerical method is applied to solve four benchmark problems and the results show good agreement with previous experimental and numerical results.

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