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Global stability of the two-dimensional flow over a backward-facing step

Abstract The two-dimensional, incompressible flow over a backward-facing step is considered for a systematic variation of the geometry covering expansion ratios (step to outlet height) from 0.25 to 0.975. A global temporal linear stability analysis shows that the basic flow becomes unstable to different three-dimensional modes depending on the expansion ratio. All critical modes are essentially confined to the region behind the step extending downstream up to the reattachment point of the separated eddy. An energy-transfer analysis is applied to understand the physical nature of the instabilities. If scaled appropriately, the critical Reynolds number approaches a finite asymptotic value for very large step heights. In that case centrifugal forces destabilize the flow with respect to an oscillatory critical mode. For moderately large expansion ratios an elliptical instability mechanism is identified. If the step height is further decreased the critical mode changes from oscillatory to stationary. In addition to the elliptical mechanism, the strong shear in the layer emanating from the sharp corner of the step supports the amplification process of the critical mode. For very small step heights the basic state becomes unstable due to the lift-up mechanism, which feeds back on itself via the recirculating eddy behind the step, resulting in a steady critical mode comprising pronounced slow and fast streaks.