Dynamics of pipes conveying fluid with non-uniform turbulent and laminar velocity profiles

Previous analytical work on stability of fluid-conveying 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 do not affect the dynamics of the system since these forces are balanced by pressure drop in the conveyed fluid. Although viscous shear has not been ignored in these studies, 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, highly tractable equation of motion was derived, which includes a single additional parameter to account for the true momentum of the fluid. This empirical parameter was determined by numerical analysis over the Reynolds number range of interest. The stability of cantilever pipes conveying fluid with two types of non-uniform velocity profile was assessed. In the first case, the profile was a function of Reynolds number and transition to turbulence occurred before the onset of flutter instability. This case had stability properties similar to the uniform velocity case except in specific narrow regions of the parameter space. The second case required that the Reynolds number be such that the flow was always laminar. For this case, lower fluid velocity was required to achieve instability, and the oscillation frequency at instability was considerably lower over much of the parameter space, compared to the uniform case.