Biglobal Instability of the Bidirectional Vortex. Part 1: Formulation

Abstract

Hydrodynamic instability research has begun to shift from the one-dimensional analysis of the Orr-Sommerfeld, local nonparallel, and parabolic type to the two-dimensional, biglobal analysis. Classic methods, such as the local nonparallel approach, specify a one-dimensional amplitude function a priori so that streamwise and azimuthal variations become characterized by a complex wave and integer mode number, respectively. This formulation is heavily dependent on the so-called parallel flow assumption. In brief, this property requires the streamlines of the baseflow to be parallel or nearly parallel, a condition that may be quite restrictive at times. It can therefore be seen that the use of biglobal analysis becomes necessary to encompass a broader range of baseflows. The relatively novel, two-dimensional characterization of the hydrodynamic waveform gives rise to a more complete analysis that intrinsically captures spatial instability behavior while remaining unrestricted to special geometries, such as elongated rocket chambers. The biglobal approach only requires periodicity in the tangential direction and seems to be tailor-made for cylindrical axisymmetric flows, such as those arising in the bidirectional vortex engine. Moreover, for short rocket chambers such as those associated with bidirectional vortex engines, the parallel flow assumption cannot be justified, and one finds the biglobal analysis as a more suitable alternative to the local nonparallel (LNP) approach used extensively in solid rocket motor stability investigations. However, being multi-dimensional, the biglobal technique requires more effort than its predecessor, specifically in the analytical derivation, numerical discretization, and spectral collocation tools necessitated by the treatment of the resulting partial differential equations. The present work serves to offer a detailed description of the theoretical framework and numerical tools needed to tackle this problem in the context of a confined vortex chamber such as that investigated by NASA/ORBITEC.

Characterization of the Bidirectional Vortex Using Particle Image Velocimetry

“…The same may be said of acoustic and/or combustion instability of vortex-fired engines. Preliminary studies using the biglobal stability approach by Batterson & Majdalani (2011a;b) show that stability is promoted with successive increases in swirl. Nonetheless, their study is limited to a narrow range of chamber aspect ratios and the hydrodynamic instability effects only represent one of the factors that must be accounted for in studying engine stability.…”

“…In this vortex-driven engine, the swirling motion of the oxidizer insulates the sidewalls against thermal loading, thus leading to a substantial reduction in engine weight. The swirling motion also has a mitigating impact on pressure oscillations in the chamber as shown by Batterson & Majdalani (2011a;b). The mathematical character of this application is described in the following section.…”

Biglobal Stability of Compressible Flowfields. Part 1: Planar Formulation

“…For a more detailed description of the numerical methods that are typically used in stability analyses, the reader is directed to Batterson [29]. His work serves as a comprehensive and concise overview of the different methods currently in use.…”

“…These wall disturbances may include defects in the propellant or intersegments, which are widely used in solid rocket motors. More recently, Batterson and Majdalani [23,24] successfully capture hydrodynamic structures in the bidirectional vortex rocket engine using the biglobal approach. Elliott et al [25] also manage to study the biglobal stability of both solid and hybrid rockets with headwall injection.…”

Two-Phase Flow Stability of Cylindrically-Shaped Hybrid and Solid Rockets with Particle Entrainment

“…Following previous works, 26,27,35 Chebyshev collocation is selected over other discretization methods due to its simplicity and straightforward implementation. We define a discrete Chebyshev polynomial of order N as …”