Citation: Pham Van Bang, D.; Uh Zapata, M.; Gauthier, G.; Gondret,P.; Zhang, W.; Nguyen, K.D. Two-Phase Flow Modeling for Bed Erosion by a Plane Jet Impingement. Water 2022, 14, 3290. https:// doi.org/10.3390/w14203290 Academic Editor: James Yang Received: 23 September 2022 Accepted: 17 October 2022 Published: 18 October 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). water Article Two-Phase Flow Modeling for Bed Erosion by a Plane Jet Impingement Damien Pham Van Bang 1,† , Miguel Uh Zapata 1,2, * ,† , Georges Gauthier 3,† , Philippe Gondret 3,† , Wei Zhang 4,† and Kim Dan Nguyen 5,† 1 Laboratory for Hydraulics and Environment, INRS-ETE, Quebec, QC G1K 9A9, Canada 2 CONACYT–Centro de Investigación en Matemáticas A. C., CIMAT Unidad Mérida, PCTY, Sierra Papacal, Merida 97302, Mexico 3 Laboratoire FAST, CNRS, Université Paris Sud, Bât. 502, Campus Universitaire, 91405 Orsay, France 4 Department of Civil Engineering, Xi’an Jiaotong-Liverpool University, Suzhou 215123, China 5 Laboratory for Hydraulics Saint Venant, École des Ponts, EDF-CEREMA, 78400 Chatou, France * Correspondence: angeluh@cimat.mx † These authors contributed equally to this work. Abstract: This paper presents experimental and numerical studies on the erosion of a horizontal granular bed by a two-dimensional plane vertical impinging jet to predict the eroded craters’ size scaling (depth and width). The simulations help understand the microscopic processes that govern erosion in this complex flow. A modified jet-bed distance, accounting for the plane jet virtual origin, is successfully used to obtain a unique relationship between the crater size and a local Shields parameter. This work develops a two-phase flow numerical model to reproduce the experimental results. The numerical techniques are based on a finite volume formulation to approximate spatial derivatives, a projection technique to calculate the pressure and velocity for each phase, and a staggered grid to avoid spurious oscillations. Different options for the sediment’s solid-to-liquid transition during erosion are proposed, tested, and discussed. One model is based on unified equations of continuum mechanics, others on modified closure equations for viscosity or momentum transfer. A good agreement between the numerical solutions and the experimental measurements is obtained. Keywords: jet erosion test; sediment transport; numerical modeling; two-phase flow model; experiments; water injection dredging 1. Introduction Vertical jet-induced scour erosion of soil has been studied for many industrial ap- plications, such as aerospace or hydraulic engineering [1–6]. As reported by Metzger et al. [1,2], in aerospace engineering, soil erosion and crater formation could generate problems for launching and landing spacecraft. In hydraulic engineering, we have sev- eral examples, such as the work of Rouse [3] for testing criteria on erosion, the study of Hanson and Cook [4] assessing in situ the erodibility of soil material, and the research developed by Hanson and Hunt [5] in the so-called jet erosion test (JET). We also have the work of Perng and Capart [6] for water injection dredging (WID) and trenching in harbors by jet translation and possible jet inclination. Experimental investigations of the jet normal impingement on plane surfaces have been conducted in various conditions. Although the jet nozzle section is circular in most cases [4,7–14], it can also be rectangular [15–18]. The circular and rectangular cases are called quasi-three-dimensional (3D) axisymmetric and two-dimensional (2D) plane jets, respectively. The jet can be either gas [1,2] or liquid [4,6] within unsubmerged [1,2] or submerged [8,9,11–18] conditions. Finally, the soil can be either cohesive [11] or non- cohesive [13,16]. Water 2022, 14, 3290. https://doi.org/10.3390/w14203290 https://www.mdpi.com/journal/water