Unsteady Effects in Flow Rate Measurement at the Entrance of a Pipe

Abstract

Unsteady flow in pipes and nozzles occur frequently in engineering applications and they pose special problems of measurement and calibration. When the Reynolds number is high the entrance region of a pipe (following a smooth contraction) is characterized by a thin boundary layer and the unsteady effects are then bound up in the unsteady behavior of the boundary layer. Woblesse and Farrell [1]2 have recently considered unsteady effects in laminar pipe entrance flows that start from rest by an integral method. Periodic disturbances also arise which require a different treatment. The primary interest of the present work is for thin entrance boundary layers subject to peridodic disturbances. In either case the ratio of the average velocity to the velocity in the potential core is V[sub]avg/V[sub]core = 1- 2[delta]*/R [equation 1] where [delta]* is the usual displacement thickness and R is the pipe radius. In steady flow this ratio is just the "discharge coefficient", c[sub]d. In unsteady flow it is very desirable to know how this ratio changes with time because many of the presently available experimental methods enable one to measure V[sub]core but not V[sub]avg readily. In this brief note we will estimate the unsteady effects of a periodic, fluctuating main flow on the displacement thickness of a laminar, flat plate boundary layer. It is assumed that the boundary layer is sufficiently thin compared to the radius of a pipe so that the pressure gradient caused by this effect in a pipe can be neglected; the results should then be directly applicable to equation (I).