Time-dependent flow dynamics within a cylinder with sidewall mass injection are investigated. A time-dependent injection velocity, prescribed along the sidewall boundary of a long, narrow, half-open cylinder, induces a low Mach number, high Reynolds number flow. The injection is the source of planar acoustic disturbances which interact with the injected fluid to produce vorticity on the sidewall in an inviscid manner. The analysis of these flow processes is based on the Navier–Stokes equations, which are reduced to simpler forms using a multiple-scale asymptotic analysis. The equations that arise from the analysis describe the leading-order vorticity dynamics. These nonlinear equations possess both wave and diffusion properties and are solved in an initial value sense. The results show that the vorticity produced at the sidewall convects toward the center of the cylinder, diffuses radially, and convects downstream.

A mathematical model is formulated to describe the initiation and evolution of intense unsteady vorticity in a low Mach number (M), weakly viscous internal flow sustained by mass addition through the sidewall of a long, narrow cylinder. An O(M) axial acoustic velocity disturbance, generated by a prescribed harmonic transient endwall velocity, interacts with the basically inviscid rotational steady injected flow to generate time-dependent vorticity at the sidewall. The steady radial velocity component convects the vorticity into the flow. The axial velocity associated with the vorticity field varies across the cylinder radius and in particular has an instantaneous oscillatory spatial distribution with a characteristic wavelength O(M) smaller than the radius. Weak viscous effects cause the vorticity to diffuse on the small radial length scale as it is convected from the wall toward the axis. The magnitude of the transient vorticity field is larger by O(M−1) than that in the steady flow.An initial-boundary-value formulation is employed to find nonlinear unsteady solutions when a pressure node exists at the downstream exit of the cylinder. The complete velocity consists of a superposition of the steady flow, an acoustic (irrotational) field and the rotational component, all of the same magnitude.

Three-dimensional internal flow dynamics are studied in a cylinder with mass injection from the sidewall. A time-dependent, harmonic, non-axisymmetric axial velocity disturbance is imposed on the endwall of the cylinder to create a non-axisymmetric velocity field. An asymptotic analysis is used to reduce the Navier-Stokes equations to more elementary forms in two regions adjacent to the endwall with distinct physical characteristics: an incompressible, inviscid and irrotational core near the endwall, and an incompressible viscous boundary layer containing all three components of vorticity adjacent to the sidewall. Solutions to these equations for disturbance frequencies associated with the lowest order axial acoustic modes show that the non-axisymmetric nature of the flow is confined to the core region adjacent to the endwall with a characteristic axial dimension on the order of the cylinder radius. Within the region, axial, radial, and azimuthal velocities exist. These non-axisymmetric effects decay exponentially fast so that only axisymmetric acoustic modes exist further downstream. These results are valid for driving frequencies below the “cut-off” value that one would find in a duct.