Dynamics of Pipes Conveying Fluid With a Non-Uniform Velocity Profile

Abstract

Previous work on stability of fluid-conveying cantilever pipes assumed a uniform velocity profile for the conveyed fluid. In real fluid flows, the presence of viscosity leads to a sheared region near the wall. Earlier studies correctly note that viscous forces drop out of the system’s dynamics since the force of fluid shear on the wall is precisely balanced by pressure drop in the conveyed fluid. The effect of shear has therefore not been ignored in these studies. However, a uniform velocity profile assumes that the sheared region is infinitely thin. Prior analysis was extended to account for a fully developed non-uniform profile such as would be encountered in real fluid flows. A modified equation of motion was derived to account for the reduced momentum carried by the sheared fluid. Numerical analysis was carried out to determine a number of velocity profiles over the Reynolds number range of interest and a simple set of curve fits was used when finer discretization was required. Stability analysis of a pipe conveying fluid with these profiles was performed, and the results were compared to a uniform profile. The mass ratio, β, is the ratio of the fluid mass to the total system mass. At β = 0.2, the non-uniform case becomes unstable at a critical velocity, ucr, that is 5.4% lower than the uniform case. The critical frequency, fcr, is 0.36% higher than the uniform case. A more sensitive region exists near β = 0.32. There, the nonuniform velocity ucr is 23% lower than the uniform case and the non-uniform critical frequency fcr is 49% of the uniform case.