Two-dimensional modal and non-modal instabilities in straight-diverging-straight channel flow

Abstract

A systematic study of a two-dimensional viscous flow through the straight-diverging-straight (SDS) channel defined by two straight-walled sections of different widths and a divergent section in-between is presented here. It has the plane Poiseuille flow (PPF) and the symmetric sudden expansion flow as the limiting cases. The topology of steady laminar flows and its bifurcations are characterized in the multi-parametric space formed by the divergence angle, the expansion ratio, and the Reynolds number. Three different steady flow regimes with two symmetric zones of recirculation, two asymmetric zones of recirculation, and the one with an additional third recirculation zone are observed with increasing Reynolds number. Modal stability analysis shows that the asymmetric flows remain stable at least up to Re = 300, regardless of the divergence angle and expansion ratio. Non-modal stability analyses are applied to SDS flows in the three topology regimes. A remarkable potential for transient amplification due to the Orr mechanism is found even for relatively low Reynolds numbers, which is related to the flow topology. The optimal energy amplification grows exponentially with the Reynolds number, as opposed to the substantially weaker Re2 scaling known for the lift-up mechanism dominant for PPF. This scaling holds for all divergence angles and is further increased by the expansion ratio, resulting in energy amplifications Gmax ∼ 104 for Reynolds numbers as low as Re ∼ 300. Present results suggest that the sub-critical transition due to transient growth is the most likely scenario for SDS flows at low Reynolds numbers.