Two-Dimensional Pipe Model for Laminar Flow

Abstract

The one-dimensional Zielke model of the energy loss in laminar pipe flow is exact but gives no information about the velocity profile. Here a two-dimensional pipe model is presented which gives the two-dimensional velocity profile in the time domain for an unstationary pipe flow of a compressible fluid that follows an equation of state. The continuity and the motion equations are projected over two sets of functions accounting for the radial and the axial dependence. A set of ordinary differential equations for the time-dependent coefficients is obtained, which is numerically integrated according to the boundary conditions at the pipe ends needed in practical applications. The model reproduces the experimental results of a water hammer and the analytical transfer functions over a wide range of frequencies.