Extension of the two‐component pressure approach for modeling mixed free‐surface‐pressurized flows with the two‐dimensional shallow water equations

• Published in: International journal for numerical methods in fluids 2021-03, Vol.93 (3), p.628-652
• Main Authors: Cea, Luis, López‐Núñez, Alejandro
• Format: Article
• Language: English
• Subjects: 2D shallow water equations; Bridges; Confined flow; Discretization; Environmental risk; finite volume method; Flood management; Flooding; free surface flow; Free surfaces; Hydraulic structures; Mathematical models; Numerical models; Numerical stability; Pressurized flow; Risk management; Shallow water; Shallow water equations; Stability analysis; Transcritical flow; Two dimensional models; two‐component pressure approach; Underground structures
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.4902

Numerical models based on the two‐dimensional shallow water equations (2D‐SWE) are routinely used in flood risk management and inundation studies. However, most of these models do not adequately account for vertically confined flow conditions that can appear during inundations, due to the presence of hydraulic structures such as bridges, culverts, or underground river reaches. In this article we propose a new mathematical modification of the standard 2D‐SWE, inspired by the two‐component pressure approach for 1D flows, to address the issue of transient vertically confined flows including transitions between free surface and pressurized conditions. A finite volume discretization to solve the proposed system of equations is proposed and analyzed. Various test cases are used to show the numerical stability and accuracy of the discretization, and to validate the proposed formulation. Results show that the proposed method is numerically stable, accurate, mass conservative, and preserves the C‐property. It can also handle subcritical, supercritical, and transcritical flows under free surface or vertically confined conditions. Most shallow water models based on the two‐dimensional (2D) Saint Venant equations do not consider the vertical confinement of the flow that commonly appears during inundations caused by the presence of bridges, culverts, or other hydraulic structures. In this article we present a modification of the standard 2D shallow water equations to cope with transient vertically confined flows. We present and analyze a finite volume discretization of the proposed equations that is stable, mass‐conservative, and well balanced.