Rational Approximation of Unsteady Friction Weighting Functions in the Laplace Domain

Abstract

This paper aims at improving the weighting function based-method (WFB) for modeling the transient behavior of a laminar flow in cylindrical pipes in a one-dimensional approach. Two improvements for the numerical computation of the unsteady friction term are presented. First, a rational approximation of the weighting function in the Laplace domain is preferred instead of an exponential series fit in the time domain. It allows the WFB method to be improved in terms of validity for small time steps, accuracy, and computational efficiency. Second, the use of auxiliary differential equations to compute convolution makes the high order time-integration of the frequency-dependent friction term straightforward, without the assumption of a constant acceleration during the time step. The simulation results for a well-known experimental test case show a good agreement of the derived methods with the experiment. Finally, the time stability of the discretized problem is fully analyzed, and a stability condition for the WFB method is brought out.