Secondary Finite Amplitude Flow in a Rotating Pipe and its Instability

Abstract

We consider here the solution of two problems concerning the flow of a rotating fluid. In the first problem, we investigate the nonlinear stability of a viscous incompressible flow in a circular pipe rotating about its own axis. We solve the initial boundary value problem for the unsteady three-dimensional Navier-Stokes equations by the Bubnov-Galerkin method.^1–5 A series of methodological investigations is made. The nonlinear evolution of the periodic self-oscillating regimes is studied, and their characteristic stabilization times, amplitudes, and other integral and fluctuation characteristics found. The secondary instability of these fmiteamplitude wave motions is examined It is established that the secondary instability is initially weak and linear in character; the corresponding growth times are approximately an order greater than for the primary perturbations. There is the possibility of a sharp, explosive restructuring of the motion when the secondary perturbations reach a certain critical amplitude. A “survival curve”⁵ is constructed, which makes it possible to determine the preferred perturbation, distinguishable from the rest if the initial perturbation amplitudes are equal, and the critical amplitude values starting from which the other perturbations may prevail even over the preferred one. The range of these surviving perturbations is obtained. It is shown that as a result of the nonlinear interaction of several perturbations at low levels of supercriticality, a periodic motion in the form of a single traveling wave is generated.

The second problem we investigate is the shear instability of a rotating vertical stratified column of fluid.