Concerning secondary flow in straight pipes

Abstract

In this paper the condition for the existence of a secondary flow in a straight non-circular pipe is determined according to the modified vorticity transfer theory, with Goldstein's assumed form for the tensorIt is shown that a secondary motion arises if the mixture length is not constant on the curves along which | gradu| is constant,ubeing the velocity parallel to the pipe axis.In problems of turbulent flow treated by means of the modified vorticity transfer theory, the quantitywhereis the mean value of the square of the velocity fluctuation andpthe mean pressure, plays a part analogous to the pressure in laminar flow. In two-dimensional flow through a channel the theory shows the existence of a gradient ofacross the channel from the central plane to each wall. Qualitative arguments such as are used to explain the existence of a secondary flow for laminar motion in a curved pipe are applied here to show that a secondary flow is to be expected near the short sides in the turbulent flow through a straight rectangular pipe of large length/breadth ratio.The equations to determine the secondary flow through an almost circular elliptic pipe are discussed, the mixture length being assumed constant on ellipses similar to and concentric with the pipe section. For a first approximation the problem is reduced to the numerical solution of three simultaneous ordinary linear differential equations.