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Hydrodynamic Instability of the Bidirectional Vortex

This work examines the hydrodynamic instability of the bidirectional vortex that is used to describe the flowfield engendered in NASA's cold-wall liquid-liquid vortex chamber. The mean flow profile to be analyzed is based on the analytical solution obtained recently by Vyas, Majdalani and Chiaverini (Vyas, A. B., Majdalani, J., and Chiaverini, M. J., "The Bidirectional Vortex. Part 2: Viscous Core Corrections," AIAA Paper 2003-5053, July 2003). To control the numerical accuracy, the Local Non-Parallel (LNP) approach is implemented such that axial, radial and, especially, tangential disturbances are accounted for. Both temporal and spatial stability theories are investigated and compared. At the outset, both time-dependent and spatial amplifications are independently characterized. The known singularity along the chamber axis is removed using Taylor series expansions. The corresponding boundary conditions are connected to the tangential wave number q to the extent that three classes of solutions are identified depending on whether q is 0, 1, or higher. For each frequency and spatial location, more than one eigenmode is captured. Also, the zeroth tangential mode is found to be the most unstable. Based on the most amplified mode, the range of stable frequencies and aspect ratios are determined for each Reynolds and swirl number. Based on the e n method for predicting transition to turbulence, the iso-n factor is calculated and found to be less than unity, specifically, too small to suggest that breakdown could be instigated. By comparison to the Taylor-Culick profile used in solid rocket motors, the characteristic stability curves associated with the bidirectional vortex are nearly reversed. Here, the most unstable region shifts from the headend to the downstream injection plane as the tangential mode number switches between 0 and 1. Increasing each of the Reynolds number, swirl number, or aspect ratio has a destabilizing effect although no region could be associated with sufficiently large amplification rates.