Womersley's solution for the measurement of volume flow rates in transient laminar flow tubes

Abstract

The characterization of transient flows within the Reynolds number range of Re = 10–100 and the Womersley number range α = 0.8–10 is required for the ongoing development of green monopropellant thrusters. However, at the ml/min scale flow rates of interest, these measurements are outside the capabilities of current commercial flow meters. It is proposed here that transient flows under the required dynamic conditions can be characterized via Womersley's solution for transient flow in a rigid tube. This solution method requires only the measured transient pressure gradient within a controlled laminar flow section and can be accomplished using existing commercial pressure measurement hardware. Experiments were performed where flow similarity was maintained with the ultimate thruster characterization application, but the radius of the flow passage was increased so that flows could be simultaneously characterized by both the proposed solution method and a commercial ultrasonic flow meter. It was shown that across the range of interest, applying Womersley's solution to a measured pressure gradient was an effective method of transient flow characterization. Additionally, it was shown that non-periodic flows can be characterized except for the initial flow startup transient with solution convergence times approximating the analytical solution to starting flow in a pipe. While these results were expected due to an experimental design matching the assumptions required for Womersley's analytic solution, this work demonstrates that this method is practically feasible as novel instrumentation enabling previously unobtainable measurements.