An indirect technique for determining instantaneous flow rate from centerline velocity in unsteady duct flows

A new technique is presented for measuring instantaneous flow rates in unsteady laminar duct flows. It is based on a relationship between flow rate and centerline-velocity history that is an exact solution to the Navier–Stokes equations for parallel, fully-developed flow of constant-property Newtonian fluids undergoing arbitrary unsteadiness from an initially steady state. The method applies instantaneously in both forward and reversing flow and invokes the sole additional assumption that the flow is axisymmetric. The capabilities of the technique are demonstrated by using experimental measurements of the centerline velocity in a tube upstream of a fuel injector, during a series of sequential injections, to: (i) quantify the differences in the fuel masses injected in each event; and (ii) measure the cumulative masses injected over multiple events. They were found to be in good agreement with direct measurements made with a mass balance.