Closure to “Three-Dimensional Numerical Modeling of Dam-Break Flows with Sediment Transport over Movable Beds” by Reza Marsooli and Weiming Wu

The authors of the discussed paper appreciate the discussers’ interest in this paper and their sharing their opinions about 3D nonhydrostatic pressure modeling of dam-break flows and sediment transport. For the benefit of readers, the authors of the discussed paper would like to clarify some of the issues addressed by the discussers. The discussers are acknowledged for agreeing with the necessity of considering the effects of nonhydrostatic pressure and 3D flow features on dam-break fluvial processes. The importance of these factors has been proven in the original paper by comparing the results of the present 3D model with those of the past 1D and depth-averaged 2D models in the test cases. As stated in the original paper’s conclusions, a 3D nonhydrostatic pressure model is preferred for more accurately calculating the sediment transport and morphological changes in the initial stage of dam-break flow, near the wavefront, and around instream structures. The vertical dynamic pressure gradient in Eq. (25) of the original paper, ∂pd=∂z, is near, not at the bed. It is determined using the vertical gradient at the center of the first cell near the bed, as shown in Fig. 1. The bottom face of the cell, denoted as b, represents the bed surface, and a ghost cell, B, outside of the computational domain is used for the convenience of numerical implementation. The present 3D model solves the governing equations on a collocated mesh with all variables, including the pressure, stored at cell centers. It is true that the Neumann condition, Eqs. (3) or (4) in the discussers’ paper, gives a zero gradient of dynamic pressure at the river bed, i.e., on the b face. However, the dynamic pressure gradient at Cell Center P is calculated by ðpd;t − pd;bÞ=Δz. The Neumann condition indicates that the dynamic pressure at the bottom face of the cell, pd;b, is equal to the dynamic pressure at Center P, whereas the dynamic pressure at the top face of the cell, pd;t, is calculated by linear interpolation between the values at Cell Centers P and T. Since pT can be different from pP, the dynamic pressure gradient at the Cell Center P is not necessarily zero. The discussers proposed an approach expressed in Eq. (5) in their paper to determine the vertical dynamic pressure gradient ∂pd=∂z. However, their formulation considers the sediment layer below the bed surface, which is not susceptible for motion or entrainment. The approach in the original paper for ∂pd=∂z applies to the sediment particles on the bed surface, which are subject to transport. These two approaches are totally different. As demonstrated on page 316 of Wu (2007), in the case of jet impinging to the bed or flow around a bridge pier, localized dynamic pressure is built up at the impinging point and at the bed front of the bridge pier. The vertical dynamic pressure gradient near the bed has a negative value, which leads to a reduction of the net weight of the sediment particles and in turn the critical shear stress for incipient motion. However, the formulation proposed by the discussers leads to a positive gradient of dynamic pressure in the bed sediment layer, which increases the critical shear stress for incipient motion. One can see these two approaches can have opposite effects. Readers should be aware of this difference. The authors of the original paper believe the discussers’ approach may not be applicable for the sediment transport in the water column near the bed surface. By assuming that the thickness of the first vertical layer of the computational mesh is equal to the bed-load layer thickness, it could be possible to avoid solving the hydrodynamic and suspended-load transport equations on two distinct computational