The Mean Velocity of Discrete Particles in Turbulent Flow in a Pipe

The axial component of velocity, averaged over a long time, of a material element of fluid in a straight circular pipe is shown theoretically to be equal to the discharge velocity defined as the discharge, averaged over a long time, divided by the cross section of the pipe. This prediction was confirmed experimentally by injecting in turn into a water pipe a large number of solid spheres of different sizes having the same density as water. A satisfactory theory is developed to allow for the finite size of the spheres. The fluctuation in the time of travel of a particle between two fixed stations is examined, and it is shown how a few particles may be used to determine the discharge velocity within calculable limits of accuracy.