Distortion of pipe-flow development by boundary layer growth and unconstrained inlet conditions

Abstract

This study extends the analysis of the canonical developing pipe-flow problem to realistic inlet conditions affecting emerging jets. A comparison of simulations to existing theory reveals adverse phenomena caused by the inlet: the velocity profile inversion and flow separation (vena contracta) at a sharp inlet. Beginning with the simple uniform inflow, the inversion is shown to persists at significantly higher Re (Re = 2000) than previously reported. It is found to be caused by the theory's neglected radial velocity, resulting from the boundary layer's displacement effect. Rescaling of the inlet axial coordinate is shown to collapse all centerline velocity curves above Re = 100, thus elucidating the known weak dependence of entrance-length on Re. The sharp-inlet separation bubble is found not to occur below Re ≅ 320 although this inlet profile overrides the boundary layer's effect. Furthermore, the bubble's downstream length increases rapidly with Re, whereas its upstream length grows gradually and proportionally to its thickness - here identified as its characteristic-scale. Beyond the bubble, the profile relaxes to a monotonic form - captured beyond x/(Re·R) = 0.005, if theory is modified using the bubble's characteristic-scale. This scale also sets the threshold which differentiates between a sharp-inlet regime, accompanied by a separation bubble, and a rounded-inlet one without it. The latter regime relaxes more rapidly to the monotonic profile - captured already beyond x = 2R. Finally, the modified idealized theory is demonstrated as a useful design tool - explicitly relating nozzle length to characteristics of emerging free-surface and submerged jets.