Numerical schemes for quasi-1D steady nozzle flows

Abstract

In this work we construct high-resolution numerical schemes for the calculation of the quasi-1D unsteady flow in pipes with variable cross-sectional area. This is an example of non-homogeneous hyperbolic systems of conservation laws that admit stationary solutions. We use the strategy developed by the authors in [1] which is to transform the non-homogeneous system into homogeneous writing the source term in divergence form, so that it can be incorporated into the flux vector of the homogeneous system and discretized in the same way. As a result, the source terms are automatically discretized to achieve perfect balance with flux terms, obtaining well-balanced schemes that produce very robust and accurate solutions. Concretely, the mentioned strategy will be used to extend the flux limiter technique [2] and the Harten, Lax and van Leer (HLL) Riemann solver [3] to the quasi-1D flow in ducts of variable cross-section. The numerical results confirm the capacity of these methods to construct well-balanced schemes.