Three-dimensional flow in a cylinder with sidewall mass addition

Abstract

Three-dimensional, time-dependent, nonlinear flow dynamics within a cylinder with sidewall mass injection are investigated. A nonaxisymmetric transient 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 drives nearly planar axial acoustic disturbances, which interact with the injected fluid to produce the azimuthal component of vorticity on the sidewall in an inviscid manner. A smaller, but important azimuthally dependent transient disturbance, driven by the nonaxisymmetric injection disturbance leads to the axial component of vorticity on the cylinder sidewall. Both components of vorticity are shown to convect toward the center of the cylinder, diffuse radially, and convect downstream. Other results show that the axial component of vorticity produced along the sidewall is largest near the maximum of a mass source distribution in the azimuthal dimension. The amplitude of the axial vorticity component decreases significantly away from the injection surface and at other azimuthal locations. The analysis of these flow processes is based on the full three-dimensional Navier–Stokes equations. A formal multiple-scale asymptotic analysis, based on the behavior of the axial Mach number M→0, is used to derive reduced equations. It is shown that the primary rotational flow response is described by a nonlinear, convection-diffusion equation.

Acoustic-Rotational Internal Flow Caused by Transient Sidewall Mass Addition

“…The lack of axial variation near the centerline is similar to a result of Staab and Kassoy [14], who find that azimuthal variations along the sidewall driven by nonaxisymmetric wall injection are damped out near the centerline. Both the model in Staab and Kassoy [14] and that used here are without azimuthal and axial diffusion terms, respectively.…”

“…Both the model in Staab and Kassoy [14] and that used here are without azimuthal and axial diffusion terms, respectively. These results show that axial convective terms are sufficient to produce these phenomena.…”

“…Equation (14) shows that the steady flow is incompressible, and the lack of viscous terms in (16) is noted. As shown by Staab et al [12], evaluation of (16) along the sidewall, r = 1, and use of the no-slip condition in (22) shows that a nonzero steady vorticity, Ω 0s , is generated on the sidewall,…”