Comparing convolution-integral models with analytical pipe- flow solutions

This paper presents and discusses known analytical solutions for accelerated laminar pipe flow. On the basis of these solutions, formulas are given that enable the determination of shear stresses on the pipe walls together with a formula for coefficients of non-stationary friction losses. A small extension is introduced, which will make it possible to analyse flow in pipes inclined at any angle. Besides solutions related to acceleration, this paper also examines the analytical solution for decelerated laminar flow and the solution for flow changing its direction in both a stepwise and a linear manner. Calculations based on the analytical formulas are compared with numerical results obtained by Zielke's convolution-integral method. The convolution integral is treated both in the classical way and in a novel more efficient way. The classical approach requires the knowledge of the complete history of the flow, which makes it computationally inefficient and memory-wise burdensome. In the efficient calculation, only the velocity changes taking place in the last three time steps are needed.