Laminar supersonic sphere wake unstable bifurcations

Abstract

The laminar sphere unstable bifurcations are sought at a Mach number of M∞ = 1.2. Global stability performed on steady axisymmetric base flows determines the regular bifurcation critical Reynolds number at Rereg cr = 650, identifying a steady planar-symmetric mode to cause the loss of the wake axisymmetry. When global stability is performed on steady planar-symmetric base flows, a Hopf bifurcation is found at ReHopf cr = 875 and an oscillatory planar-symmetric mode is temporally amplified. Despite some differences due to highly compressible effects, the supersonic unstable bifurcations present remarkably similar characteristics to their incompressible counterparts, indicating a robust laminar wake behavior over a large range of flow speeds. A new bifurcation for steady planar-symmetric base flow solutions is found above Re > 1000, caused by an anti-symmetric mode consisting of a 90○ rotation of the dominant mode. To investigate this reflectional symmetry breaking bifurcation in the nonlinear framework, unsteady nonlinear calculations are carried out up to Re = 1300 and dynamic mode decomposition (DMD) based on the combination of input data low-dimensionalization and compressive sensing is used. While the DMD analysis confirms dominance and correspondence in terms of modal spatial distribution with respect to the global stability mode responsible for the Hopf bifurcation, no reflectional symmetry breaking DMD modes were found, asserting that the reflectional symmetry breaking instability is not observable in the nonlinear dynamics. The increased complexity of the wake dynamics at Re = 1300 can be instead explained by nonlinear interactions that suggest the low-frequency unsteadiness to be linked to the destabilization of the hairpin vortex shedding limit cycle.