Abstract
This work seeks to provide a closed-form analytical solution for the transverse vortical wave generated at the sidewall of a circular cylinder with headwall injection. This particular configuration mimics the conditions leading to the onset of traveling radial and tangential waves in an idealized liquid rocket engine chamber. Assuming a short cylindrical enclosure with axisymmetric injection, regular perturbations are used to linearize the problem’s conservation equations. Flow decomposition is subsequently applied to the first-order disturbance equations, thus giving rise to a compressible, inviscid, acoustic set responsible for driving the unsteady motion and to an incompressible, viscous, vortical set driven by virtue of coupling with the acoustic mode along both the sidewall and headwall. While the acoustic mode is readily recovered from the wave equation, the induced vortical mode is resolved using boundary-layer theory and an expansion of the rotational equations with respect to a small viscous parameter, . At the outset, an explicit formulation for the leading-order vortical field is derived and verified numerically. A radial penetration number akin to the Stokes or Womersley numbers is identified and found to control the penetration depth of the viscous boundary layer forming above the inert sidewall. This parameter is based on the transverse oscillation mode frequency and scales with the squared ratio of the Stokes layer and the chamber’s characteristic radius.
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