A Fourier-Galerkin-Newton Method for Periodic Nonlinear Transmission Line Problems

The problem of pressure and flow-rate oscillations in periodically excited transmission line systems arises in number of fluid power applications. The fluid-borne noise problem in the suction and delivery lines of pumps has been studied by a large number of authors in the past. More recent applications can be found in the simulation of common-rail diesel injection and hydraulic valve actuation systems in the automotive industry. In the case of laminar flow with small pressure oscillations around a stationary operating point, a linear model can be used and the system answer to periodic excitations can be computed efficiently in the frequency domain. Even in the case of nonlinear boundary conditions. e.g. a valve with a quadratic pressure drop, combined time and frequency domain methods have been used by a number of authors in order to maintain the benefits of frequency domain modelling of transmission lines. A much harder problem arises if the nonlinearity is distributed along the transmission line. In this paper, a Fourier-Galerkin-Newton method is applied to laminar transmission line flow with a nonlinear compressibility law. The viscous effects are modelled by the linear, frequency dependent friction model in the form due to Kagawa et al. For the spatial discretisation a Galerkin approach with a staggered grid is taken from literature. In order to treat periodic problems efficiently, the solution is parameterised by Fourier series. The approach results in a large scale, nonlinear system of equations to be solved for the periodic system response to periodic excitations.