Dam-break Problem Research Articles

On Convergence of Finite-Difference Shock-Capturing Schemes in Regions of Shock Waves Influence

We perform a comparative accuracy study of the Rusanov, CABARETM, and WENO5 difference schemes used to compute the dam break problem for shallow water theory equations. We demonstrate that all three schemes have the first order of convergence inside the region occupied by a centered rarefaction wave, and the Rusanov scheme has the second order of convergence in the area of constant flow between the shock and the rarefaction wave, while in the CABARETM and WENO5 schemes there is no local convergence in this area. This is due to the fact that the numerical solutions obtained by the CABARETM and WENO5 schemes have undamped oscillations in the region of influence of the shock, the amplitude of which does not decrease with decreasing of the difference grid steps. As a result, taking into account the Lax-Wendroff theorem, the numerical solutions obtained by the conservative schemes CABARETM and WENO5 converge only weakly to the exact constant solution in the region of influence of the shock wave, in contrast to the Rusanov scheme, which locally converges with the second order to the exact solution in this region.

Numerical simulations of air-water flow and rigid-body motion based on two-liquid CLSVOF/IB method and overset mesh

For simulating air-water flows with rigid-body motion, the rigid boundary has to be body-fitted and discretized in the traditional dynamic mesh method, which will lead to a severe computational instability due to the mesh deformations when large displacements are involved. A two-liquid CLSVOF/IB (coupled Level Set and Volume of Fluid/Immersed Boundary method) method for directional numerical simulation of air-water flow and water-water mixing with a moving rigid-body is presented in this study, which is based on overset mesh method. In order to make the CLSVOF method applicable to two different cell sizes in overset mesh and background mesh respectively, the formula of the artificial time step size in the re-initialization equation of Level Set (LS) method is improved. A simple and effective IB method suitable for the overset mesh method and the dynamic mesh technique is also implemented. The IB method marks the cells around the rigid-body as forcing cells in the overset mesh, and then an artificial momentum forcing term is introduced into the momentum equation of the fluid to satisfy no-slip boundary condition in fluid-solid interface. The method has been validated with laboratory data, including two-liquid dam-break flow with wet bed and single-liquid water entry problem. The effects of various parameters such as ambient water depth and gate motion on mixing pattern and front propagation of water-water interface are investigated in the dam-break flow problems. The results have good agreement with the published experimental results and emphasize the necessity of considering the gate in numerical simulation of dam-break flow. In the numerical simulation of a free-falling rigid-body impacting the free surface of water, the simulation results, including the computed shape of the air cavity and displacement of free-falling rigid-body, agree well with experimental results.

Whitham modulation theory and exotic wave patterns of the good Jaulent–Miodek equation with step-like initial data

The complete classification of solutions to the Riemann problem of the good Jaulent–Miodek equation is investigated by Whitham modulation theory. The one-phase and two-phase periodic wave solutions and the corresponding Whitham equations are derived based on the Lax pair of the good Jaulent–Miodek equation and the finite-gap integration method. In particular, the N-phase periodic wave solutions are proposed by algebro-geometric approach. Then the basic wave structures of rarefaction waves and DSWs are proposed analytically and graphically, which makes it possible to establish the classification of all the possible wave patterns evolving from initial discontinuities. The asymptotic results given by Whitham modulation theory are in excellent agreement with direct numerical simulations. Finally, a detailed description of the shallow-water dam break problem is demonstrated to find the possible physical significance of the wave patterns found in this work.

Lagrangian differencing dynamics for incompressible flows

A Lagrangian meshless method is introduced for numerical simulation of flows of incompressible fluids with free surface. Poisson Pressure Equation (PPE) formulation of Navier-Stokes equations is discretised in Lagrangian context using recently introduced spatial operators based on finite differences. The advection adjusts to moving or deforming boundaries, and volume-conservation is enforced by solving a set of geometrical constraints. A technique for detecting the edge or boundary of a point cloud is introduced, which is based on the Laplacian properties for detecting jumps. Since accurate and fast discrete versions of the operator are not straightforward to implement in meshless context, the technique exploits favourable properties of a discrete Laplacian that handles highly irregular arrangements of points. A geometrical scan-sphere test is optionally performed to verify that points are on the point–cloud boundary. The boundary–detection technique is validated on significantly deformed point clouds by artificial numerical experiments, and by simulating unsteady incompressible flows. The method was validated by simulating lid-driven cavity flow, dam-break and water-entry problems, and comparing the kinematics of flow and hydrodynamic loading during the impact. The method reproduced complex free surface shapes, violent fluid-structure interaction, and pressure fields with second-order accuracy.

Dam-break computations by a finite volume on dry regions

Abstract This paper presents numerical solvers, based on the finite volume method. This scheme solves dam break problems on the dry bottom in 2D configuration. The difficulty of the simulation of this type of problem lies in the propagation of shocks on the dry bottom. The equation model used is the shallow water equations written in conservative form. The scheme used is second order in space and time. The method is modified to treat dry bottoms. The validity of the method is demonstrated over the dam break example. A comparison with finite elements shows the weakness and robustness of each method.

Numerical modeling of two dimensional non-capacity model for sediment transport by an unstructured finite volume method with a new discretization of the source term

The main goal of this work is the resolution of the two-dimensional shallow water equations of water–sediment mixture coupled to the transport diffusion equation for the total sediment load, and bed change rate equation by Roe scheme with a new discretization of the source term. The proposed discretization is well-balanced with the flux gradient and uses data right and left on the interfaces between two control volumes and satisfies the C-property. The numerical method uses unstructured meshes and incorporates minmod limiter and Runge–Kutta method to reach second order spatial and temporal accuracy. We also use an adaptive mesh based on gradient concentration of sediments to refine the study domain with a lower computational cost. We present some numerical results in order to verify and validate the performance of the numerical scheme, particular attention is given to the treatment of the dam-break problem over mobile beds. The numerical scheme demonstrates the intended accuracy and robustness to modelize dam-break flows over erodible bed.

A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

We present a novel staggered semi-implicit hybrid finite volume / finite element method for the numerical solution of the shallow water equations at all Froude numbers on unstructured meshes. A semi-discretization in time of the conservative Saint-Venant equations with bottom friction terms leads to its decomposition into a first order hyperbolic subsystem containing the nonlinear convective term and a second order wave equation for the pressure. For the spatial discretization of the free surface elevation an unstructured mesh composed of triangular simplex elements is considered, whereas a dual grid of the edge-type is employed for the computation of the depth-averaged momentum vector. The first stage of the proposed algorithm consists in the solution of the nonlinear convective subsystem using an explicit Godunov-type finite volume method on the staggered dual grid. Next, a classical continuous finite element scheme provides the free surface elevation at the vertices of the primal simplex mesh. The semi-implicit strategy followed in this paper circumvents the contribution of the surface wave celerity to the CFL-type time step restriction, hence making the proposed algorithm well-suited for the solution of low Froude number flows. It can be shown that in the low Froude number limit the proposed algorithm reduces to a semi-implicit hybrid FV/FE projection method for the incompressible Navier-Stokes equations. At the same time, the conservative formulation of the governing equations also allows the discretization of high Froude number flows with shock waves. As such, the new hybrid FV/FE scheme can be considered an all-Froude number solver, able to deal simultaneously with both, subcritical as well as supercritical flows. Besides, the algorithm is well balanced by construction. The accuracy of the overall methodology is studied numerically and the C-property is proven theoretically and is also validated via numerical experiments. The numerical solution of several Riemann problems, including flat bottom and a bottom with jumps, attests the robustness of the new method to deal also with flows containing bores and discontinuities. Finally, a 3D dam break problem over a dry bottom is studied and our numerical results are successfully compared with numerical reference solutions obtained for the 3D free surface Navier-Stokes equations and with available experimental reference data.

Data Assimilation of Water Elevation in Shallow Water Flow Based on the Extended Kalman Filter FEM Using Measurement Data from Image Analysis

In this paper, we present a data assimilation analysis in a shallow water flow field considering shoreline movement, based on the extended Kalman filter finite element method (extended Kalman filter FEM). It is known that if the combined method of the Kalman filter and the finite element method(FEM) is employed, a solution can be obtained that is closer to the practical observed value than that based on normal FEM. A dam-break problem is targeted in numerical experiments. In this study, the observed values used in the extended Kalman filter FEM are obtained by image analysis, and we investigate the estimation accuracy by changing the governing equation for shallow water flow.

Accelerated derivative-free nonlinear least-squares applied to the estimation of Manning coefficients

A general framework for solving nonlinear least squares problems without the employment of derivatives is proposed in the present paper together with a new general global convergence theory. With the aim to cope with the case in which the number of variables is big (for the standards of derivative-free optimization), two dimension-reduction procedures are introduced. One of them is based on iterative subspace minimization and the other one is based on spline interpolation with variable nodes. Each iteration based on those procedures is followed by an acceleration step inspired in the Sequential Secant Method. The practical motivation for this work is the estimation of parameters in Hydraulic models applied to dam breaking problems. Numerical examples of the application of the new method to those problems are given.

A Nonintrusive Reduced-Order Model for Uncertainty Quantification in Numerical Solution of One-Dimensional Free-Surface Water Flows Over Stochastic Beds

Free-surface water flows over stochastic beds are complex due to the uncertainties in topography profiles being highly heterogeneous and imprecisely measured. In this study, the propagation and influence of several uncertainty parameters are quantified in a class of numerical methods for one-dimensional free-surface flows. The governing equations consist of both single-layer and two-layer shallow water equations on either flat or nonflat topography. For this purpose, the free-surface profiles are computed for different realizations of the random variables when the bed is excited with sources whose statistics are well defined. Many research studies have been dedicated to the development of numerical methods to achieve some order of accuracy in free-surface flows. However, little concern was given to examine the performance of these numerical methods in the presence of uncertainty. This work addresses this specific area in computational hydraulics with regards to the uncertainty generated from bathymetric forces. As numerical solvers for the one-dimensional shallow water equations, we implement four finite volume methods. To reduce the required number of samples for uncertainty quantification, we combine the proper orthogonal decomposition method with the polynomial chaos expansions for efficient uncertainty quantifications of complex hydraulic problems with large number of random variables. Numerical results are shown for several test examples including dam-break problems for single-layer and two-layer shallow water flows. The problem of flow exchange through the Strait of Gibraltar is also solved in this study. The obtained results demonstrate that in some hydraulic applications, a highly accurate numerical method yields an increase in its uncertainty and makes it very demanding to use in an operational manner with measured data from the field. On the other hand, when the complexity of physics increases, these highly accurate numerical methods display less uncertainty compared to the low accurate methods.

Study on incompressible fluid analysis by three-dimensional particle method with finite volume techniques

Simulating violent free-surface flows by particle methods is effective. In recent years, the SPH (Smoothed Particle Hydrodynamics) method and the MPS(Moving Particle Simulation method) method are considered representative especially in the fields of ship engineering and civil engineering but these methods need various artificial parameters in calculation and may have unnatural large numerical oscillation in the calculated pressure values. In order to avoid these problems, the authors have proposed another particle method based on the finite volume techniques. The usefulness of the proposed method in two-dimensional problems has been verified by comparing the pressure fluctuation when a water tank of the two-dimensional model is forcibly oscillated with the corresponding experiment [1]. Therefore, in this research, the authors extended the algorithm so that the calculation formula can be applied to three-dimensional problems. The effectiveness of the calculation is demonstrated by applying to the dam break problem.

Numerical Analysis on Effects of Wall Structures on Bubble Groups

With the volume of fluid (VOF) method for a dam-break problem, the effects of wall structures on compressible bubble groups were studied through measurement of the spatial average pressure on the wall. An obstacle was set up at the bottom of the tank, which helps create air bubbles in the collapsing water impacting on it. Three kinds of structures were set up on the left wall, namely, a cuboidal structure, an ellipsoidal structure and a conical structure. It is found that when water hits the left wall, the topology of the bubble wrapped in the water will be changed by the wall structure, which leads to the change of pressure on the wall. The example analysis shows that, the cuboidal structure has the maximum effect in reducing the average pressure amplitude on the wall among those three kinds of wall structures. Especially, a proper adjustment of the position and the size of the cuboidal structure can eliminate the oscillation of the wall pressure.

A Role of Interfacial Drag Force to Simulate Dam-Break Problem with Plant System Analysis Code, Trace

A Role of Interfacial Drag Force to Simulate Dam-Break Problem with Plant System Analysis Code, Trace

A Role of Interfacial Drag Force to Simulate Dam-Break Problem with Plant System Analysis Code, Trace

A Role of Interfacial Drag Force to Simulate Dam-Break Problem with Plant System Analysis Code, Trace

Evolution of initial discontinuities in the Riemann problem for the Jaulent–Miodek equation with positive dispersion

On the basis of Whitham modulation theory, we consider the complete classification of solutions of the initial discontinuities for the Jaulent–Miodek (JM) equation with positive dispersion. The Whitham equations of the JM system are obtained with the finite-gap integration theory, in which dispersive shock waves (DSW) can be described with one-phase modulated traveling wave solution. Since the JM system with positive dispersion is modulationally unstable, the corresponding behaviors of Riemann problem are more complicated than the stable JM system. All the possible wave configurations evolving from initial discontinuities are combinations of the plateau, vacuum, rarefaction wave, DSW, as well as overlap of rarefaction wave and DSW. The results of JM-Whitham system are applied to describe the wave structure in physical model cases, the dam breaking problem and piston problem.

Solid Particle Simulation with a New Particle Interaction

A particle method is a state-of-the-art method in computational fluid dynamics. The specialty of the particle method is tracking each particle that carries its own physical quantities, such as velocity, pressure, and position. Therefore, the large deformation of a free surface can be simulated. In this paper, the concept of the fully Lagrangian approach is extended in its applicability to the solid particle with new particle interaction models, which are the friction and drag models. The newly developed particle method with the new particle interaction models simulated mobile sea-bed behavior under very violent waves due to the broken dam problem. The numerical results were compared to corresponding experimental results for validation. The validated particle method demonstrated the mobile sea-bed behavior with various densities of soil particles. Furthermore, solid-particle interaction model results were also compared with those of fluid models to show the difference and to prove the implementation.

A parallel cell-centered adaptive level set framework for efficient simulation of two-phase flows with subcycling and non-subcycling

We develop a unified adaptive level set (LS) framework using the multi-level collocated grid for incompressible two-phase flows. This framework allows us to advance all variables level by level using either the subcycling or the non-subcycling method such that the data advancement on each level is fully decoupled. A series of synchronization operations are designed to keep the momentum and mass conserved across all levels. A multi-level re-initialization method for the LS function is also proposed, which greatly improves the mass conservation of the two-phase flows. The collocated grid allows the use of a single set of differential schemes and interpolation operations for all variables, which greatly simplifies the numerical implementation. The capability and robustness of the computational framework are validated by a variety of canonical problems, including the inviscid shear layer, gravity wave, rising bubble, and Rayleigh-Taylor instability. It is shown that the present multi-level scheme can accurately resolve the interfaces of the two-phase flows with gravitational and surface tension effects while having good momentum and energy conservation. At last, a three-dimensional dam breaking problem is simulated to show the efficiency and significant speedup of the proposed framework.

An improved meshless artificial viscosity technology combined with local radial point interpolation method for 2D shallow water equations

The two-dimensional shallow water equations (SWEs) are a hyperbolic system of first-order nonlinear partial differential equations which have a characteristic of strong gradient. In this study, a newly-developed numerical model, based on local radial point interpolation method (LRPIM), is adopted to simulate discontinuity in shallow water flows. In order to accurately capture the information of wave propagation, the LRPIM is combined with the split-coefficient matrix (SCM) method to transform the SWEs into a characteristic form and the selection of the direction of local support domain is introduced into the LRPIM. An improved meshless artificial viscosity (MAV) technique is developed to minimize the non-physical oscillations near the discontinuities. Then, the LRPIM and the second-order Runge–Kutta method are adopted for spatial and temporal discretization of the SWEs, respectively. The feasibility and validity of the proposed numerical model are verified by the classical dam-break problem and the mixed flow pattern problem. The comparison of the obtained results with the analytical solution and other numerical results showed that the MAV method combined with LRPIM can accurately capture the shocks and has high accuracy in dealing with discontinuous flow by adding appropriate viscosity to the equations in the discontinuous region.

A well-balanced positivity-preserving numerical scheme for shallow water models with variable density

We develop an unstructured numerical scheme for a coupled system modeling shallow water flows and solute transport over variable topography. A novel algorithm is introduced for the reconstructions of the variables of the system with variable density. These reconstructions are used in combination with the expression of the relative density of the mixture to guarantee the positivity required for some physical parameters of the coupled model. New discretization techniques are developed to guarantee the well-balanced property of the scheme and the consistency between the transport equation and the continuity equation at the discrete level. We prove the well-balanced property of the proposed method as well as the positivity preserving property of the scheme for both the water depth and concentrations. The performance of the proposed scheme is tested on a number of numerical examples, among which we consider nontrivial analytical solutions for the equilibrium with mixture constituents, parabolic wave with pollutant transport, and dam-break problem for modeling solute transport in rapidly varying flow. The numerical results confirm stability and well-balanced property of the scheme, consistency between the discretizations of the continuity and transport equations, positivity preserving property required for some physical parameters, and accuracy of the proposed method in modeling the dynamics of water flow and scalar transport.

A Massively Parallel Hybrid Finite Volume/Finite Element Scheme for Computational Fluid Dynamics

In this paper, we propose a novel family of semi-implicit hybrid finite volume/finite element schemes for computational fluid dynamics (CFD), in particular for the approximate solution of the incompressible and compressible Navier-Stokes equations, as well as for the shallow water equations on staggered unstructured meshes in two and three space dimensions. The key features of the method are the use of an edge-based/face-based staggered dual mesh for the discretization of the nonlinear convective terms at the aid of explicit high resolution Godunov-type finite volume schemes, while pressure terms are discretized implicitly using classical continuous Lagrange finite elements on the primal simplex mesh. The resulting pressure system is symmetric positive definite and can thus be very efficiently solved at the aid of classical Krylov subspace methods, such as a matrix-free conjugate gradient method. For the compressible Navier-Stokes equations, the schemes are by construction asymptotic preserving in the low Mach number limit of the equations, hence a consistent hybrid FV/FE method for the incompressible equations is retrieved. All parts of the algorithm can be efficiently parallelized, i.e., the explicit finite volume step as well as the matrix-vector product in the implicit pressure solver. Concerning parallel implementation, we employ the Message-Passing Interface (MPI) standard in combination with spatial domain decomposition based on the free software package METIS. To show the versatility of the proposed schemes, we present a wide range of applications, starting from environmental and geophysical flows, such as dambreak problems and natural convection, over direct numerical simulations of turbulent incompressible flows to high Mach number compressible flows with shock waves. An excellent agreement with exact analytical, numerical or experimental reference solutions is achieved in all cases. Most of the simulations are run with millions of degrees of freedom on thousands of CPU cores. We show strong scaling results for the hybrid FV/FE scheme applied to the 3D incompressible Navier-Stokes equations, using millions of degrees of freedom and up to 4096 CPU cores. The largest simulation shown in this paper is the well-known 3D Taylor-Green vortex benchmark run on 671 million tetrahedral elements on 32,768 CPU cores, showing clearly the suitability of the presented algorithm for the solution of large CFD problems on modern massively parallel distributed memory supercomputers.