Axisymmetric stability of the flow between two exactly counter-rotating disks with large aspect ratio
Résumé
We study the first bifurcation in the axisymmetric flow between two exactly counterrotating
disks with very large aspect ratio Γ ≡R/H, where R is the disk radius and
2H is the inter-disk spacing. The scaling law for the critical Reynolds number is found
to be Rec ∝ Γ(−1/2), with Re=ΩH2/ν, Ω being the magnitude of the angular velocity
and ν the kinematic viscosity. An asymptotic analysis for large Γ is developed, in
which curvature is neglected, but the centrifugal acceleration term is retained. The
NavierStokes equations then reduce to leading order to those in a Cartesian frame,
and the axisymmetric base flow to a parallel flow. This allows us locally to use a
Fourier decomposition along the radial direction. In this framework, we explain the
physical mechanism of the instability invoking the linear azimuthal velocity profile
and the effect of centrifugal acceleration.