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This work relies on a compressible biglobal stability approach to describe the wave dynamics in a planar rocket chamber modeled as a porous channel. At first, the effectiveness of the compressible formulation is demonstrated by reproducing, in the absence of a mean flow, the Helmholtz frequencies and mode shapes. Next, the unsteady vorticity fluctuations, which intensify near the walls, are shown to be consistent with those associated with parietal vortex shedding. In this context, the penetration depth of vorticoacoustic waves is found to be strongly dependent on the penetration number. The latter gauges the cubic power of the injection speed to the product of kinematic viscosity, chamber half-height, and frequency squared. As for the strictly hydrodynamic modes, they seem to develop at the porous walls and grow in the core region, where the mean flow velocity is most appreciable. The ensuing modal analysis enables us to predict both longitudinal and transverse modes for several test cases, thus illustrating the tendency of hydrodynamic modes to intensify at higher injection speeds and longer chambers. Furthermore, by repeating the analysis with an active mean flow, one finds that successive increases in the injection speed gradually reduce the predicted frequencies relative to the eigenmodes obtained in a quiescent medium. Finally, recognizing that the spectral analysis is capable of recovering both longitudinal and transverse modes induced by acoustic and hydrodynamic disturbances, their coupled interactions, which often lead to specifically amplified frequencies in static tests, are robustly captured, namely, without resorting to any particular wave decomposition.
This work relies on a compressible biglobal stability approach to describe the wave dynamics in a planar rocket chamber modeled as a porous channel. At first, the effectiveness of the compressible formulation is demonstrated by reproducing, in the absence of a mean flow, the Helmholtz frequencies and mode shapes. Next, the unsteady vorticity fluctuations, which intensify near the walls, are shown to be consistent with those associated with parietal vortex shedding. In this context, the penetration depth of vorticoacoustic waves is found to be strongly dependent on the penetration number. The latter gauges the cubic power of the injection speed to the product of kinematic viscosity, chamber half-height, and frequency squared. As for the strictly hydrodynamic modes, they seem to develop at the porous walls and grow in the core region, where the mean flow velocity is most appreciable. The ensuing modal analysis enables us to predict both longitudinal and transverse modes for several test cases, thus illustrating the tendency of hydrodynamic modes to intensify at higher injection speeds and longer chambers. Furthermore, by repeating the analysis with an active mean flow, one finds that successive increases in the injection speed gradually reduce the predicted frequencies relative to the eigenmodes obtained in a quiescent medium. Finally, recognizing that the spectral analysis is capable of recovering both longitudinal and transverse modes induced by acoustic and hydrodynamic disturbances, their coupled interactions, which often lead to specifically amplified frequencies in static tests, are robustly captured, namely, without resorting to any particular wave decomposition.
This work considers a uniquely configured swirling motion that develops inside a porous tube due to sidewall injection. The bulk fluid motion is modeled as a steady inviscid Trkalian flow field with a swirl-velocity component that increases linearly along the axis of the chamber. The underlying procedure consists of solving the compressible Bragg–Hawthorne equation using a Rayleigh–Janzen expansion that produces a closed-form approximation for the stream function. Based on the latter, most remaining flow attributes may be readily inferred. Results are then compared to their counterparts obtained using a strictly incompressible Trkalian motion. They are also benchmarked against available compressible solutions in an effort to characterize the dilatational effects caused by flow acceleration in long chambers or chambers with sufficiently large sidewall injection. In addition to the stream function, the velocity, pressure, temperature, and density are evaluated over a range of physical parameters. Finally, the distortions affecting the velocity profiles are characterized and shown to result in a blunter motion near the center and a steeper curvature near the sidewall as a consequence of high-speed flow. In comparison with a non-swirling complex-lamellar solution, we find the Trkalian motion to be generally faster and therefore capable of reaching sonic conditions in a shorter distance from the headwall.
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