Pipe Networks: Coupling Constants in a Junction for the Isentropic Euler Equations

Abstract

The modelling of junctions in pipe networks with subsonic flows is discussed, where pipes are described by one-dimensional, single-phase isentropic flow models. We first study the Riemann problem in a pipe to understand what information is needed to couple two pipes in a flat junction. Using this insight, we generalise the Riemann problem to an arbitrary number of pipes meeting together at a junction. Three coupling strategies found in the literature are presented, where only one leads to physically sound solutions for all the selected test cases. The theoretical derivation is performed in previously published literature.

The junction is considered to be a point with no volume. The three coupling strategies are, first, to impose all the pipe sections to be at the same pressure at the junction. The second is to impose equal momentum fluxes at the inlet of all the pipes coupled to the junction. The third is to impose all the pipe sections to reach the junction at a unique stagnation enthalpy, that is, equal for all of them. Only the latter satisfies the second law of thermodynamics, expressed through an entropy condition, in all the test cases run in the study. For the two former coupling strategies, test cases where the entropy condition is violated could be found and are presented.

The different coupling strategies are implemented in a numerical model. The one-dimensional models for the pipe sections are solved using a Roe scheme. We illustrate with numerical cases that we can find initial conditions for which the entropy condition is violated for the two first coupling strategies, while the third verifies it in all the cases.