Unsteady Flow in Vertical, Converging Tubes

INTRODUCTION The discharge ofliquids from vertical tubes with various con­ traction geometries was studied via the unsteady Bernoulli equa­ tion. The temporal variations of the exit velocity and fluid level in the tube were found from the numerical integration of nonlin­ ear differential equations. Sudden, quadratic, and exponential contraction geometries were considered. For inlet to exit area ra­ tios greater than two, the flow initially accelerates to a maximum speed and then it decelerates for the geometries studied. The exponential contraction has the shortest discharge time. The solutions also reveal that the largest possible velocity and the shortest discharge time are achieved in a non-converging tube. NOMENCLATURE A area AR area contraction ratio g gravitational acceleration h fluid level in the tube H h/L k VAR-l L tube length P pressure Q volume flow rate t time V velocity z vertical coordinate a In(AR) p density Subscripts e exit fs free surface I inlet 1,2 points on a streamline Unsteady turbulent flows, the subject of much current re­ search, can be easily generated in the laboratory by employing gravity as the driving mechanism. Discharging of a vertical tube filled with water, produces a jet in which the flow accelerates in time. In order to tailor the variation of acceleration with time, one may utilize various tube geometries. The exit velocity vari­ ation with time for a converging tube can be obtained from the integration of the unsteady Bernoulli equation. Since the flow typically starts from rest and lasts only a short time, the boundary layer is confined to the tube walls and has only modest growth. Furthermore, Lefebvre & White (1989) observed that the transition in accelerating pipe flows is delayed to very high Reynolds numbers (~ 5 X 105). Previous work on transient flow through pipes has typically concentrated on either the waterhammer problem or the veloc­ ity profile development in startup conditions. Szymanski (1932) found an exact solution of the Navier-Stokes equations for the laminar, incompressible, startup flow in a circular pipe. Durian (1986) studied the filling and emptying of small diameter tubes filled with viscous fluids. This study was mainly concerned with the time required to establish a fully-developed profile. The present work seeks to determine the velocity variation with time in vertical tubes having various contraction geometries. We are concerned with tubes of considerable diameter and short dis­ charge times where the velocity profile is nearly uniform except near the tube walls.