Laminar supersonic sphere wake unstable bifurcations
Résumé
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.
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