On steady rotational high speed flows: The swirling compressible Trkalian profile with headwall injection

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.