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Global stability analysis of the axisymmetric wake past a spinning bullet-shaped body

We analyze the global linear stability of the axisymmetric flow around a spinning bullet-shaped body as a function of the Reynolds number, $Re=w_{\infty}D/ν$, and of the rotation parameter $Ω=ωD/(2 w_{\infty})$, in the ranges $Re<450$ and $0\leqΩ\leq 1$. Here, $w_{\infty}$ and $ω$ are the free-stream and the body rotation velocities respectively, and $ν$ is the fluid kinematic viscosity. The spectrum and the eigenfunctions obtained allow us to explain the different bifurcations from the axisymmetric state observed in previous numerical studies. Our results reveal that three global eigenmodes, denoted Low-Frequency (LF), Medium-Frequency (MF) and High-Frequency (HF) modes, become unstable in different regions of the $Re-Ω$ parameter plane. We provide precise computations of the corresponding neutral curves, that divide the $Re-Ω$ plane into four different regions: the stable axisymmetric flow prevails for small enough values of $Re$ and $Ω$, while three different frozen states, where the wake structures co-rotate with the body at different angular velocities, take place as a consequence of the destabilization of the LF, MF and HF modes. Several direct numerical simulations of the nonlinear state associated to the MF mode, identified here for the first time, are also reported to complement the linear stability results. Finally, we point out the important fact that, since the axisymmetric base flow is $SO(2)$-symmetric, the theory of equivariant bifurcations implies that the weakly non-linear regimes that emerge close to criticality must necessarily take the form of rotating-wave states. These states, previously referred to as frozen wakes in the literature, are thus shown to result from the base-flow symmetry.