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

publication date

  • April 2014

start page

  • 302

end page

  • 327

volume

  • 748

international standard serial number (ISSN)

  • 0022-1120

electronic international standard serial number (EISSN)

  • 1469-7645

abstract

  • We analyze the global linear stability of the axisymmetric flow around a spinning bullet-shaped body of length-to-diameter ratio L/D=2, as a function of the Reynolds number, Re=rhow∞ D/mu, and of the rotation parameter Omega=omega D/(2 w∞ ), in the ranges Re<450 and 0 ≤ Omega ≤ 1. Here, w∞ and omega are the free-stream and the body rotation velocities respectively, and rho and mu are the fluid density and viscosity. The two-dimensional eigenvalue problem (EVP) is solved numerically to find the spectrum of complex eigenvalues and their associated eigenfunctions, allowing us to explain the different bifurcations from the axisymmetric state observed in previous numerical studies. Our results reveal that, for the parameter ranges investigated herein, three global eigenmodes, denoted low-frequency (LF), medium-frequency (MF) and high-frequency (HF) modes, become unstable in different regions of the (Re,Omega)-parameter plane. We provide precise computations of the corresponding neutral curves, that divide the (Re,Omega)-plane into four different regions: the stable axisymmetric flow prevails for small enough values of Re and Omega, 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 (DNS) of the nonlinear state associated with 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 nonlinear 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.

keywords

  • instability; separated flows; wakes; aerodynamics; bifurcation (mathematics); plasma stability; reynolds number; stability; wakes; axisymmetric base flow; axisymmetric state; complex eigenvalues; equivariant bifurcation; global stability analysis; length to diameter ratio; rotation parameters; separated flows; eigenvalues and eigenfunctions; bifurcation; eigenvalue; fluid dynamics; instability; rotating flow; separation; wake