Abstract:
The classical criterion for the inviscid linear stability of swirling flows in a pipe was established by Rayleigh (1916) [32] and more recently, it has been proved by Wang (2009) [39] through the Arnol’d’s energy Casimir stability method [2]-[4] that this criterion is also a necessary condition for the inviscid nonlinear stability. The aim of this current research is to extend the nonlinear stability criterion for viscous steady swirling flows, which essentially combines the work by Wang (2009) [39] and Koolsterziel (2010) [21]. We follow the same mathematical model as in Wang (2009) [39] and Szeri & Holmes (1988) [38]. This requires the axisymmetric perturbations to be incompressible, satisfy the non-penetration boundary condition on the boundary of the pipe and must also be periodic along the axial direction, in addition to the non slip boundary condition. In particular, the base flow that we are interested in is the Taylor Couette flow. To obtain sufficient conditions regarding its nonlinear stability, we first subject this base flow with viscous perturbations and then investigate their corresponding time evolutions under the dynamics of the full nonlinear perturbation equations. Lyapunov’s second method is used, which is a direct analytical approach for the verification of stability. The major difficulty in applying this method is that a Lyapunov functional is required, and it is often uncertain as to how to construct or find such a functional. It turns out that the change of the reduced Arnol’d functional Ard in [39] for the inviscid nonlinear stability can be rewritten as a weighted kinetic energy of the perturbation, hence this serves as a suitable candidate for a Lyapunov functional in the viscous case. As a result, the nonlinear stability of Taylor Couette flow is established and the corresponding sufficient conditions for nonlinear stability with respect to viscous axisymmetric perturbations are also deduced.