Transverse wave dynamics in short tubes with axisymmetric headwall injection

Abstract

This work describes both traveling and standing vorticoacoustic waves in circular tubes that are driven by axisymmetric headwall injection. In this process, perturbation tools, field decomposition, and boundary-layer theory are jointly used. First, perturbation expansions are initiated to linearize the Navier–Stokes equations. Second, a Helmholtz decomposition of the first-order disturbances is pursued to identify a suitable set of acoustic wave equations. The last step consists of solving for the vortical mode using boundary-layer theory and a viscous expansion of the unsteady rotational set. At the outset, an explicit formulation for arbitrary headwall injection is obtained and confirmed both numerically and through limiting process verifications; the latter take into account special cases involving uniform and bell-shaped injection profiles. The resulting formulation is then described using both laminar and turbulent injection patterns. Using four canonical cases, the characteristics of the evolving vorticoacoustic wave, including its penetration depth, spatial wavelength, and overshoot factor, are systematically explored and discussed. Several fundamental flow features are also unraveled including the radial, tangential, and axial velocities of the time-dependent vortical field. Most rotational flow features are found to depend on the penetration number, the Strouhal number, and the distance from the centerline. The corresponding standing modes are expressed in closed form and shown to be appreciable in view of their amplitudes that twice exceed those associated with strictly traveling waves. Finally, by extending the boundary-layer analysis from the headwall to the sidewall, a uniformly valid wave approximation is achieved, which remains observant of the no-slip requirement everywhere.