Abstract
A lot of fruitful applications of model-based methods to measurement of unsteady fluid flow in a pipe have been reported recently. Those applications, however, are limited to the measurement of the fully-developed flow although the application to the entry flow is strongly needed. The limitation comes from the difficulty in making a mathematical model of the entry flow: a nonlinear problem has to be solved to make the model.
This paper presents a unified method of making a linear model for the entry flow as well as for the fully-developed flow to derive a linear estimator of velocity profiles of the flows in both the outside and the inside of the entrance region of a circular pipe. The linear model is derived by, first, modifying the Navier-Stokes' equation to a linear partial differential equation having an unknown function as an input and, then, lumping its parameters by the finite element method. The Kalman filter theory is applied to this linear lumped parameter model with unknown inputs to yield a linear estimator of an inverse Kalman filter form.
The effectiveness of the estimator in flow measurement is shown by simulation and experiment done for many cases. The experiment is done by using a beam scanning laser Doppler velocimeter which can measure velocity profile in a pipe by scanning the laser beam. This velocimeter was developed originally in the authors' laboratory.
