Vortex breakdown, linear global instability and sensitivity of pipe bifurcation flows

Abstract

Pipe bifurcations are common flow configurations in both natural and man-made systems. This study follows our previous report (Chen et al., Phys. Fluids, vol. 27, 2015, 034107) by describing three aspects of flows through junction angles of $70^{\circ }$, $90^{\circ }$ and $110^{\circ }$, with a square cross-section. First, the inflow creates tightly spiralling vortices in four quadrants of the junction. For sufficiently large Reynolds number $Re$, these vortices undergo behaviour resembling steady near-axisymmetric breakdown. With increasing $Re$, the flow through the $90^{\circ }$ junction remains steady and stable until the first Hopf bifurcation. Beyond the Hopf bifurcation, the vortices undergo a helical instability. The $70^{\circ }$ and $110^{\circ }$ junctions, however, first exhibit pitchfork bifurcations leading to asymmetric solutions. Second, the direct eigenmodes of the linearised flow are large in vortices in the outlet pipes, whereas the adjoint eigenmodes primarily reside in a small region in the inlet and the junction, near the front and back walls. Third, the sensitivities of the eigenvalues to spatially localised feedback and base flow modifications are greatest in and near the junction vortices. We highlight the regions of high growth rate and frequency sensitivity, as well as regions where the production and transport of perturbations by modifications of the base flow contribute most to the base flow sensitivity. The flow separation at the corners of the junction does not coincide with the eigenmodes or sensitivity regions.