The Linear Stability Analysis of the Lamb–Oseen Vortex in a Finite-Length Pipe

This article extends the perturbation method introduced by Wang (2008, “A Novel Method for Analyzing the Global Stability of Inviscid Columnar Swirling Flow in a Finite Pipe,” Phys. Fluids, 20(7), p. 074101) to determine the global stability of a swirling flow in a straight circular pipe with specified inlet and outlet boundary conditions. To accurately compute the flow stability characteristics, a general procedure to treat the complexity arising from high-order terms is developed. It extends the previous fourth-order method to an eighth-order method. The technique is first applied to the benchmark case of a solid-body rotation flow with a uniform axial speed. It is demonstrated that the eighth-order method is sufficient to construct the growth rate curve between the first and second critical swirls of this flow. Note that this range of swirl is crucial for the study of the vortex breakdown phenomenon since the base flow is unstable and starts the initial stage of transition to a breakdown state. The method is then applied to the Lamb–Oseen vortex to construct the growth rate curve between the first and second critical swirls of this flow. Calculated results are compared with the growth rate curve computed from direct numerical simulations and an overall agreement between the two computations is found. This demonstrates that the Wang and Rusak (1996, “On the Stability of an Axisymmetric Rotating Flow in a Pipe,” Phys. Fluids, 8(4), pp. 1007–1076) instability mechanism captures quantitatively the initial growth of disturbance, which eventually evolves into a breakdown state.