3-D finite-volume model of dam-break flow over uneven beds based on VOF method
Introduction
Dams are one of the common structures to collect and store waters streaming in rivers and to provide essential benefits including drinking water, power generation, flood protection, irrigation, and recreation. Despite the efforts dedicated to promote dam safety, dam break can occur as a result of overtopping, foundation defects, piping and seepage, human error, earthquake, etc. Dam-break incident is a catastrophic event that may cause significant loss of life and property, and environmental damages. Basically, dam-break flow studies can be categorized as theoretical analysis, experimental/field measurement, and numerical modeling. With increasing computer processing capacity, numerical studies of dam-break flows have become attractive and cost-effective.
Developing numerical models of dam-break flows has received considerable attention in the past few decades. Numerous efforts have been dedicated to the development of 1-D and depth-averaged 2-D models to simulate dam-break flows (e.g., [1], [11], [43]). These models are sufficient tools to simulate inundation depth and arrival time of dam-break flow, but their assumptions used to derive the governing equations, such as hydrostatic pressure distribution and insignificant vertical acceleration and free surface curvature, are inappropriate in the initial stages of dam-break flow, at dam-break wave front, and near in-stream structures. Simulating these complex flows must rely on vertical 2-D or 3-D models which solve the Reynolds-Averaged Navier–Stokes (RANS) equations.
In contrast to the depth-averaged 2-D models which consider the effects of uneven beds through the bed elevation term in the momentum equation, the bed elevation is not included in the governing equations solved by the vertical 2-D and 3-D models. Therefore special care must be taken in order to mesh the computational domain in the vertical direction. Most vertical 2-D and 3-D dam-break flow models are based on rectangular Cartesian grids [22], [39], which are not convenient to apply in cases with irregular bed topography and in-stream structures downstream of the dam. One approach to improve representation of the bed geometry in a Cartesian coordinate system is the cut cell [13] or immersed boundary method. These approaches are relatively tedious to set up the computational mesh in a large physical domain with irregular bed topography. Another approach to better represent the uneven beds is using an irregular mesh with fitting capability on solid boundaries. However, in this approach the non-orthogonality of computational cells introduced by the bed slope needs to be taken into account in the discretized governing equations.
In vertical 2-D and 3-D models, the water depth is not explicitly included either in the momentum equations or continuity equation and efforts must be devoted to trace the water surface. Smoothed Particle Hydrodynamics (SPH) introduced by Lucy [19] and Gingold and Monaghan [5] solves the governing equations without using a computational mesh by replacing the fluid with a set of particles that move with the flow motion, and thus can simulate the evolution of free surface. On the other hand, several techniques can be used to track the water surface movements on a computational mesh. Among these techniques the Volume-of-Fluid (VOF) is one of the most well-known interface-capturing methods. The SOLA-VOF scheme is the first VOF method which was introduced by Hirt and Nichols [7] and modified by Torey et al. [34], and has been adopted by most models of dam-break flows [13], [21], [39]. Since the SOLA-VOF scheme was originally developed for 1-D flows, the VOF advection equation for multi-dimensional flows is usually solved by operator splitting method which may not be convenient to apply on irregular and unstructured meshes [28]. To avoid applying the operator splitting method on multidimensional flows, Ubbink and Issa [35] developed the Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM), which solves the VOF advection equation using a multi-dimensional algorithm rather than operator splitting.
This paper presents a 3-D numerical model of dam-break flow over uneven bed, which solves the 3-D RANS equations explicitly using a finite-volume method based on the collocated irregular mesh with boundary-fitting capability. The VOF scheme used in the model is based on the explicit CICSAM scheme developed by Ubbink and Issa [35] for capturing fluid interfaces on arbitrary-shaped meshes. The model is tested using several multi-dimensional laboratory dam-break flows. The 3-D features of flow at initial stages of dam break and near in-stream structures are investigated by the model. In several test cases the accuracy of the present model to simulate dam-break flows over uneven beds is assessed by comparing the calculated results with those calculated by other 2-D and 3-D models in the literature. The governing equations, numerical methods, and test results of the developed model are presented in the following sections.
Section snippets
Governing equations
The governing equations are the 3-D continuity and RANS equations for incompressible flows, which can be written in the vector form aswhere u is the flow velocity, t is the time, p is the pressure, ρ is the water density, f denotes the external body force, and μ is the viscosity including the molecular and turbulent viscosities. The external body force considered is the gravity, so that f = ρg, where g is the acceleration of gravity. The above equations are in
Computational grid
The computational domain is represented by a mesh of hexahedral cells that fits the solid boundaries, such as bed and walls. Each cell is embraced by flat faces. Fig. 1 shows the 3-D sketch of a hexahedral cell where point P is the center of the control volume under consideration and point N is the center of a neighbor cell. Vector A is the outward-pointing face area vector normal to the face located between cell P and N. Vector shown in Fig. 1 as the dashed line connects points P and N.
Discretization of governing equations
Integrating the momentum equation (2) over the control volume centered at node P shown in Fig. 1 and using the Gauss’s theorem leads towhere Vp is the volume of the cell at node P, subscript f denotes the index counter of faces surrounding the cell at node P, and nb is the number of cell faces or neighboring nodes of node P.
The temporal term in Eq. (6) is discretized using the first-order Euler scheme. The second term on the left-hand side is
Pressure equation
Velocity values at cell faces are required to evaluate the cell fluxes used in Eq. (10). Because the non-staggered grid approach is used, linear interpolation between the velocity values at cell centers may result in checkerboard splitting and spurious oscillations [24]. In order to avoid this, the momentum interpolation method proposed by Rhie and Chow [27] is used to approximate the face velocity values using the discretized momentum equation (9) as follows:where
Surface-capturing method
In the present model the water surface movement is traced using the VOF method. A step function F(x, y, z, t) is defined to be unity at any cell occupied by fluid (fluid cells) and zero at cells occupied by air (empty cells). Cells with F-function between zero and one contain water surface (surface cells). For incompressible flows, the VOF method solves the following advection equation, which is in the conservative form, to compute the time evolution of the F-function:
Since F is a
Boundary conditions
The pressure of cells at inlet, outlet, or closed boundaries is defined using the Neumann condition with constant vertical gradient −g. The pressure at a water surface cell is calculated by interpolation between the pressure at the water surface (the atmospheric pressure) and the pressure of a neighbor fluid cell. The neighbor cell is chosen using the water surface orientation. The Youngs [42] reconstruction method is used to determine the unit normal vector of water surface in each surface
Model stability
The time step is determined by satisfying two stability conditions. The first condition is the Courant–Friedrichs–Lewy (CFL) condition, which assures the explicit solver of the RANS equations to have stable solutions. The second condition is a Courant-type restriction proposed by Weymouth and Yue [37] to ensure that the explicit CICSAM solver of multi-dimensional flows does not overfill or overempty the computational cells in each time step. Therefore, the stability condition can be written in
Model testing
The developed model has been tested using seven small- and large-scale laboratory experiments. The first test aims to investigate the initial stages of dam-break flow over a wet bed. The sensitivity of the model to the mesh size and the constant coefficient of the Smagorinsky turbulence model are also investigated. The ability of the model to simulate dam-break flows over uneven beds is investigated using dam-break flows over trapezoidal and triangular steps in the next two test cases and a
Conclusions
In this paper, a 3-D model of dam-break flows over uneven beds has been developed. The model solves the RANS equations using an explicit finite-volume method based on collocated mesh with hexahedral cells that fit solid boundaries. The turbulent viscosity is calculated by the Smagorinsky model. The velocity and pressure coupling is achieved by using the PISO method and the Rhie and Chow’s [27] momentum interpolation method. The CICSAM-VOF surface-capturing method for explicit calculation is
Acknowledgments
This research was supported by the US Department of Homeland Security-sponsored Southeast Region Research Initiative (SERRI) at the US Department of Energy’s Oak Ridge National Laboratory, and by the USDA-ARS Specific Research Agreement No. 58-6408-7-236 (monitored by the USDA-ARS National Sedimentation Laboratory). The first author also acknowledges the Summer Graduate Research Assistantship and Dissertation Fellowship provided by The University of Mississippi.
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Formerly at NCCHE, The University of Mississippi, MS 38677, USA.