One-way pressure-independent valves with hysteresis and chattering

Abstract

We propose a model for the one-dimensional isothermal gas flow through a valve in a pipe. The valve aims at keeping the outgoing flow in an assigned interval; moreover, its opening and closing depend on a further parameter. The model is realized by a system of two conservation laws with suitable conditions coupling the ingoing and the outgoing flow. We study the corresponding Riemann problem, where non-classical shocks show up. Although simple, the model allows to study two key oscillatory behaviors of the flow, which are due, respectively, to the chattering of the valve, especially seen in the numerical simulations, and to the stabilizing hysteresis induced by the valve operation. We characterize the states leading to chattering and show how they depend on the parameters of the model. Through several numerical simulations, we study the interplay between the two above phenomena.