On the compressible biglobal stability of the mean flow motion in porous tubes

Abstract

In this work, a compressible biglobal stability approach is used to investigate the growth characteristics of hydrodynamic and vorticoacoustic waves in porous tubes with uniform wall injection. The retention of compressibility effects enables us to construct a physics-based formulation that is capable of predicting both hydrodynamic and vorticoacoustic wave motions simultaneously with no need for mode decomposition. At first, we show that, in the absence of a mean flow, the stability framework reproduces traditional Helmholtz frequencies and modal shapes. This confirms the embedment of the wave equation within the compressible Navier–Stokes framework. We then proceed to simulate the idealized motion in solid rocket motors, often modeled as porous tubes, where a mean flow expression is available. Specifically, using the compressible Taylor–Culick profile as a base flow, our solver produces a comprehensive frequency spectrum that returns both hydrodynamic and vorticoacoustic modes in one swoop with the added benefit of pinpointing the flow-induced longitudinal, radial, and mixed frequencies at user-prescribed tangential modes. Moreover, we find that increasing the flow Mach number leads to a slight reduction in the vorticoacoustic frequencies relative to their strictly acoustic counterparts. Similar results are obtained while increasing the Reynolds number and aspect ratio, thus affirming the origin of frequency shifts often observed in motor firings. Finally, the vorticoacoustic velocity fluctuations are shown to resemble those obtained asymptotically. Particularly, their depths of penetration appear to be controlled by the penetration number, a dimensionless parameter that combines the effects of sidewall injection, oscillatory frequency, viscosity, and chamber radius.