Linear stability analysis of swirling flow in a finite domain

Abstract

The linear stability of swirling flows in a finite domain with non-periodic, realistic boundary conditions imposed on the inlet and outlet of the domain is systematically studied. This study is focused on a well-selected representative model of the circular Couette flow inside two coaxial rotating, straight, finite-length and circular pipes, aiming to fill gaps in the existing theory, which has been developed over the past two decades since the work of Wang and Rusak (1996a). Accurate and reliable numerical codes and novel analytical methods are developed to solve the stability equations in a combining way. Thus this study establishes the fundamental physical relationship between the swirling flow stability theory in a finite domain (governed by partial differential equations) and the swirling flow stability theory based on normal mode analysis (governed by ordinary differential equations). It is found for the first time that the circular Couette flow can be classified by the Rayleigh discriminant into three types of base flow based on an inviscid framework. Type 1 base flow exhibits the same stability nature as that found with the solid-body rotation flow (Wang & Rusak, 1996a), which results from the imbalance of spatial perturbation kinetic energy productions at various flow components, essentially beyond the reach of the classical stability mechanism. Type 2 base flow largely retains the instability mechanism of the normal mode (governed by ordinary differential equations). Its stability mechanism results from the competition of Rayleigh’s instability mechanism with the perturbation’s convective loss at the pipe outlet. Type 3 base flow exhibits the combined physical stability nature of Type 1 and Type 2 base flows. A numerical-analytical combined method is used to study the viscous stability for Type 1 base flow with a no-slip boundary condition imposed, and a general viscous growth rate correction formula from the relevant inviscid growth rate is thereby established. The stability mechanism of viscous flow is therefore essentially determined by the inviscid theory. This study clarifies the intrinsic connection between classical and current theories.