Three-dimensional instability of a ow past a sphere: Mach evolution of the regular and Hopf bifurcations
Résumé
A fully three-dimensional linear stability analysis is carried out to investigate the
unstable bifurcations of a compressible viscous fluid past a sphere. A time-stepper
technique is used to compute both equilibrium states and leading eigenmodes. In
agreement with previous studies, the numerical results reveal a regular bifurcation
under the action of a steady mode and a supercritical Hopf bifurcation that
causes the onset of unsteadiness but also illustrate the limitations of previous
linear approaches, based on parallel and axisymmetric base flow assumptions, or
weakly nonlinear theories. The evolution of the unstable bifurcations is investigated
up to low-supersonic speeds. For increasing Mach numbers, the thresholds move
towards higher Reynolds numbers. The unsteady fluctuations are weakened and
an axisymmetrization of the base flow occurs. For a sufficiently high Reynolds
number, the regular bifurcation disappears and the flow directly passes from an
unsteady planar-symmetric solution to a stationary axisymmetric stable one when
the Mach number is increased. A stability map is drawn by tracking the bifurcation
boundaries for different Reynolds and Mach numbers. When supersonic conditions
are reached, the flow becomes globally stable and switches to a noise-amplifier
system. A continuous Gaussian white noise forcing is applied in front of the shock
to examine the convective nature of the flow. A Fourier analysis and a dynamic
mode decomposition show a modal response that recalls that of the incompressible
unsteady cases. Although transition in the wake does not occur for the chosen
Reynolds number and forcing amplitude, this suggests a link between subsonic and
supersonic dynamics.
Domaines
Sciences de l'ingénieur [physics]
Origine : Fichiers produits par l'(les) auteur(s)
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