This paper was concerned to simulate both wet and dry bed dam break problems. A high-resolution finite volume method (FVM) was employed to solve the one-dimensional (1D) and two-dimensional (2D) shallow water equations (SWEs) using an unstructured Voronoi mesh grid. In this attempt, the robust local Lax-Friedrichs (LLxF) scheme was used for the calculating of the numerical flux at cells interfaces. The model named V-Break was run under the asymmetry partial and circular dam break conditions and then verified by comparing the model outputs with the documented results. Due to a precise agreement between those output and documented results, the V-Break could be considered as a reliable method for dealing with shallow water (SW) and shock problems, especially those having discontinuities. In addition, statistical observations indicated a good conformity between the V-Break and analytical results clearly.
Floods induced by dam failures can cause significant loss of human life and property damages, especially when located in highly populated regions. These entail numerical and laboratory investigations of dam break flows and their potential damage. The shallow water equations (SWEs) are conventionally used to describe the unsteady open channel flow such as dam break. These equations are named as Saint Venant equations for one-dimensional (1D) problem and also include the continuity and momentum equations for two-dimensional (2D) studies.
Many researchers studied the dam break problem, such as Toro [
Recently, Shamsai and Mousavi evaluated the dam break parameters such as breach width, side slop, time of failure and peak outflow, using 142 case studies of previous researches. Then, using those evaluations and also the BREACH and FLDWAV application software, the studies of Aidoghmosh earth dam breach were examined [
This paper attempts to present a novel development for 1D and 2D dam break problems in both wet and dry beds. A high-resolution FVM is employed to solve the SWEs on unstructured Voronoi mesh. The local Lax-Friedrichs (LLxF) scheme is used for the estimation of fluxes at cells and the numerical approximation of hyperbolic conservation laws.
The continuity and momentum equations of the SW can be written in different forms depending upon the requirements of the numerical solution of governing equations. The 2D SWEs with source terms are given in the vector form considering a rigid bed channel [
In this study, the friction slopes were estimated using the Manning’s formulas. In the case of the dam break flow, the influence of bottom roughness prevailed over the turbulent shear stress between cells. Therefore the effective stress terms were neglected in the computation.
The main advantage of the FVM is that volume integrals in a partial differential Equation (PDE) containing a divergence term are converted to surface integrals using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each FV. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the FVM is that it is easily formulated to allow for unstructured meshes. Unstructured grid methods utilize an arbitrary collection of elements to fill the domain. These types of grids typically utilize triangles in 2D and tetrahedrals in 3D, although quadrilateral, hexahedral, Voronoi, and Delaunay meshes can also be unstructured. In the Voronoi mesh, the chosen point has lower distance in the devoted domain rather than other points. If one point has the same distance from several domains, it will be divided between domains. Indeed, these points create Voronoi cell boundaries. Consequently, internal sections of the Voronoi mesh consist of nodes belonging to one domain and boundaries include nodes that belong to several domains [
In this paper, the studied domain was discretized using unstructured Voronoi meshes. Delaunay triangulation was created and then the Voronoi mesh was established using the Qhull program in MATLAB software. The governing equation was discretized applying the FVM. In this approach, the studied domain was divided into several separated control volumes without any overlapping. By integrating the governing differential equation over every control volume, the system of algebraic equations was created so that each of its formulations belonged to one control volume and each equation linked a parameter in the control volume node to different numbers of the parameter in adjacent nodes. This consequently led to the computation of the parameter in each node [
In order to solve discrete equations, the parameter in each node was computed considering its discrete equation and newest adjacent nodes’ values. Solution procedure can be expressed as follows. Assuming an initial value in each node as an initial condition. Calculating the value in a node considering its discrete equation. Performing pervious step for all nodes over the studied domain, one cycle is performed by repetition this step. Verifying the convergence clause. If this clause is satisfied, the computing will end otherwise the computations will be repeated from the second step.
As exact values of boundary conditions were not distinct, the Riemann boundary condition was utilized for computing the investigating parameter. Therefore, by assuming a layer which is close to the boundary layer,
Various methods can be used to discretize the governing equations, among which the FVM due to its ability to satisfy mass and momentum conservation is frequently adopted. In this research, the discretization of (
The 2D schematic Voronoi mesh cell used for describing the discretization of the governing equations.
By applying the Voronoi mesh, (
In shock capturing schemes, the location of discontinuity is captured automatically by the scheme as a part of the solution procedure. These slope-limiter or flux-limiter methods can be extended to systems of equations. In this paper, the algorithm is based upon central differences with comparable performance to Riemann type solvers when used to obtain a solution for PDE’s describing systems. FV and Finite Difference (FD) methods are closely related to central schemes like the most shock capturing schemes [
Many researchers (e.g., Lin et al. [
In this research, LLxF is used as a flux calculator. By expanding (
The CFD code named V-Break was prepared on the unstructured Voronoi mesh grid using MATLAB programming. Figure
The V-Break flowchart.
V-Break was then validated using Stoker’s analytical solution in 1D [
General summarization of the equipment used by some other researchers and present study.
Reference | Wang and Liu [ |
Liang et al. [ |
Loukili and Soulaïmani [ |
Baghlani [ |
V-Break |
---|---|---|---|---|---|
Numerical method | FVM | FVM | FVM | FVM | FVM |
Riemann solver | Roe-MUSCL, Roe-upwind, HLL-MUSCL, composite methods (CFLF8) | HLLC | Lax-Fredrichs, HLL, HLLC, WAF | FDS-FVS | Local Lax-Fredrichs |
Mesh grid | Unstructured triangular | Rectangular (quadtree) | Unstructured triangular, unstructured quadrilateral | Rectangular | Unstructured Voronoi |
Dimensional approach | 2D | 1D, 2D | 1D, 2D | 1D, 2D | 1D, 2D |
At the instant of the dam break, water is released through the breach, forming a positive wave that propagates downstream and a negative wave that moves upstream. Here, Stoker’s analytical solution of dam break problem can be used to illustrate the accuracy of the numerical schemes. Stoker derived this theory just for the wet bed dam break problem, but it is possible to develop it for dry bed considering a downstream water depth very close to zero [
1D Studied domain for verification.
Figures
Mann-Whitney test parameters for comparison between V-Break and stoker’s analytical solution results.
Output result type | Parameter | |||
---|---|---|---|---|
Mann-Whitney |
Wilcoxon |
|
|
|
Water depth at |
4314.500 | 8685.500 | −0.030 | 0.976 |
Water depth at |
5200.500 | 10453.500 | −0.004 | 0.997 |
Velocity at |
4269.500 | 8734.500 | −0.643 | 0.520 |
Velocity at |
5904.500 | 11899.500 | −0.082 | 0.934 |
1D Water depth values obtained using V-Break and Stoker’s analytical solution. (a) 1D water depth at
1D Velocity values obtained using V-Break and stoker’s analytical solution. (a) Velocities at
In terms of comparing two groups of data statistically, the null hypothesis (
In this research, all the above conditions were satisfied in Table
For this case study, a 2D partial dam break problem with asymmetrical breach was considered. The computational domain was defined in a channel with 200 m in length and 200 m in width. The breach is 75 m in length and the dam is 15 m in height. The initial upstream water depth is 10 m. The downstream water depth is 5 m in a wet bed and 0.1 m in a dry bed. The roughness coefficient was assumed zero implying a frictionless surface. In this example, a domain with
2D studied domain for verification.
2D asymmetrical partial dam break test in a frictionless, horizontal, and wet/dry bed domain at
Different conditions of the domain were simulated using V-Break. Figure
2D asymmetrical partial dam break test and in a frictionless horizontal domain at
The output results were then compared with the previous studies. A section was considered at
The comparison among Voronoi, rectangular [
As previously mentioned, Wang and Liu implemented four typical FVMs, including the Reo-MUSCL, Reo-upwind, HLL-MUSCL, and CFLF8 composite methods on unstructured triangular meshes to simulate a 2D dam break problem [
The comparison among various Riemann solvers used in Wang and Liu’s code (Reo-MUSCL, Reo-Upwind, HLL-MUSCL and CLFL8) and V-Break (LLxF).
In this test, 200 computational cells were used where initial conditions consist of two states separated by a circular discontinuity. The radius of the circle is 50 m and it is centered at
2D Circular dam break problem at
2D Circular dam break water depth contours obtained using V-Break.
2D Circular dam break problem at
Figures
In the current research, a novel and friendly user code named V-Break was written showing that the LLxF scheme along with the FVM on the unstructured Voronoi grid is a suitable combination in order to simulate 1D/2D dam break problems. The advantages of this method are very promising, especially in reconstructing the conducted tests. For 1D dam break, Stoker’s analytical solution was considered for validation. Illustrations and computed
Adjacent surface vector of the investigated Voronoi cell
Area of the adjacent surface vector of the investigated Voronoi cell (
The component of
The component of
Flux vector functions
The Voronoi cell normal flux vectors
The vector of conserved variables
Input and output fluxes to a Voronoi cell
The vector of source terms
Bed slope in the
Bed slope in the
Nodes of the both sides of the investigated Voronoi cell
Unit outward normal vector in each Voronoi cell
The unit tangent vector in each Voronoi cell
Joint surface element between investigated cell and other adjacent cells
Gravity acceleration
The water depth
The mean water depth
The upstream water depth at
The downstream water depth at
The boundary of the
The Manning’s roughness coefficient
The outward unit vector normal to the boundary
The central node of adjacent cells
Time
Velocity vector component in
The mean velocity vector component in
Velocity vector component in
The mean velocity vector component in
Horizontal coordinate component
Vertical coordinate component
The sum over the all Voronoi cells
Central node of investigated Voronoi cell
Time interval
The distance between the central node of investigated Voronoi cell and the adjacent cells
The null hypothesis
The value of
The critical value of
Sum of ranks for the first comparing group
Sum of ranks for the second comparing group
The significance level
The number of data in the first comparing group
The number of data in the second comparing group.
Denotes the parameter at the Voronoi cells side’s area
Counts all central control volumes
Counts all nodes of the central control volumes
Denotes parameters at the central node of adjacent Voronoi cells
Denotes parameters at central node of investigated Voronoi cell
Denotes the parameter outside the Voronoi cells side’s area
Denotes the parameter outside the Voronoi cells side’s area
The parameter component in
The parameter component in
Denotes parameters belonging to time of
Denotes parameters belonging to time of