Abstract
The numerical modeling of one-dimensional (1D) domains joined by symmetric or asymmetric bifurcations or arbitrary junctions is still a challenge in the context of hyperbolic balance laws with application to flow in pipes, open channels or blood vessels, among others. The formulation of the Junction Riemann Problem (JRP) under subsonic conditions in 1D flow is clearly defined and solved by current methods, but they fail when sonic or supersonic conditions appear. Formulations coupling the 1D model for the vessels or pipes with other container-like formulations for junctions have been presented, requiring extra information such as assumed bulk mechanical properties and geometrical properties or the extension to more dimensions. To the best of our knowledge, in this work, the JRP is solved for the first time allowing solutions for all types of transitions and for any number of vessels, without requiring the definition of any extra information. The resulting JRP solver is theoretically well-founded, robust and simple, and returns the evolving state for the conserved variables in all vessels, allowing the use of any numerical method in the resolution of the inner cells used for the space-discretization of the vessels. The methodology of the proposed solver is presented in detail. The JRP solver is directly applicable if energy losses at the junctions are defined. Straightforward extension to other 1D hyperbolic flows can be performed.
Highlights
Numerical modeling of one-dimensional (1D) flow in networks of spatial domains joined by junctions offers a satisfactory compromise between the quality of the numerical predictions and the computational cost
Many numerical tests are presented to show the capabilities of the Junction Riemann Problem (JRP) solver, comparing analytical and numerical solutions under extreme flow conditions on collapsible tubes, subsonic–supersonic transitions, sonic blockage conditions, and the transmission of backward waves from downstream vessels through the junction to upstream supersonic vessels with supersonic conditions
The analytical solutions provided by the JRP solver are compared with numerical results computed using the JRP at the junction in combination with the HLLS scheme
Summary
Numerical modeling of one-dimensional (1D) flow in networks of spatial domains joined by junctions offers a satisfactory compromise between the quality of the numerical predictions and the computational cost. One-dimensional modeling has proven to be an accurate tool in complex applications, such as computational hemodynamics in deformable arterial models under pulsatile flow, leading to competitive results if compared with three-dimensional (3D) simulations [2,3]. Existing methods are based on coupling approaches for energy or momentum conservation to the continuity equation and the characteristic equations [12,14,15] considering only subcritical or subsonic flow conditions. Those methods assume that the Riemann problem solutions involve only rarefaction waves
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